Make the Switch to Blackjack Switch

            Over the years, numerous inventors have attempted to tinker with the game of Blackjack.  I warn them to tread very carefully when doing this.  Of all the games in the casino, blackjack strategy has probably become the best learnt strategy.  With the proliferation of computer generated strategies, you see far less splitting of 10's/faces and far less awful choices by the average Player.  You'll still occasionally find the novice who isn't happy until their own hand is 17 or better, even if that means busting it, but you'll now get a collective groan out of the remaining Players instead of several following suit.

            This is where the trouble started for creating a blackjack variant.  Players knew that original Blackjack had a payback of 99.5% (give or take) and they had learned the strategy fairly well.  When someone created some form of blackjack with a twist, they guessed it meant a lower payback (otherwise, why would the casinos offer it?) and it meant a new strategy.  Just like in video poker, if you don't adapt your strategy for the rules of the game, you can't earn the top payback. 

            So, once in a while a new game would hit the floor, Players would give it a try, but, without the right strategy, the theory on payback turned into a self-fulfilling prophecy - and the Player invariably lost more playing the new version than the original.  The new game might have been a bit more exciting than Blackjack, but not enough to overcome the extra losses the Player had to endure.

            As well all know, over the years a few blackjack variants have stuck.  Spanish 21 is likely the most successful of these variants.  It removes the 10's (not the face cards) from the deck.  As this hurts the Players, it returns this missing payback to the Player by offering more liberal rules and some bonus payouts for some novel hands.  This added more excitement to the game and offered the Player some opportunities for something other than mostly even money payouts.   While Spanish 21 is past its prime, it continues to boast a significant presence in the casinos.   It's payback is actually quite comparable to blackjack, but the need to learn a new strategy has kept the casinos happy by having Player error contribute to the hold of the game.

            More recently, Blackjack Switch has also entered the market.  It has roughly 100 tables in the marketplace.  Blackjack Switch uses a unique method to alter the game.  If the Dealer busts with a 22, all Player non-busted hands (except a natural Blackjack) are a push.  This costs the Player several percentage points.  But, to make up for this, Blackjack Switch allows the Player to 'switch' the 2nd card dealt in each of his two hands.  So, if dealt a 5-10 and a 10-6, the 10 and 6 can be swapped to turn the hands into an 11 and a 20.  From two stiffs to two strong hands.  The payback again is comparable to regular blackjack, albeit you must play two hands at a time.

            Blackjack Switch requires not only learning the strategy for the 'Push 22' rule, but you must also learn when to switch cards.  Much of the time it will be fairly obvious as in my earlier example.  In others, less so.  Imagine being dealt a 10-7 and an 8-10 vs a Dealer face card.  What is the right play?  You have two pat hands or you can 'switch' and have a total bust (15) and one strong hand (20).  When we look at the expected values of each of these hands, there is not much of a choice.  17's and 18's against a Dealer 10 are sitting ducks in any blackjack game.  We do the swap and the combined expected value of our hands goes from 1.3 to 1.97.  If you never switched cards, you'd take a 7-8% hit in payback.  No one would ever (hopefully) play this bad, but if you go by the seat of your pants, you're likely to take a 2-3% hit.  Throw in not knowing how to alter your strategy for the Push 22 rule and you could easily take Switch down to a 97% payback from its 99.5+% payback.
            Just like in video poker, there is a simple solution for this.  LEARN THE STRATEGY.  To help you with this, my booklet Expert Strategy for Blackjack Switch comes with a full-color pocket-sized strategy card that you can bring with you into the casino.  One side has the expected values for every hand to help you decide when to switch.  The other contains the hit/stick strategy for Push 22.  The retail price is $6.95 for the booklet and the card, but for a limited time, I'll offer them to GT readers for only $5.95.  You can also order ADDITIONAL strategy cards for $1.00 each.  If you would like to order, please send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.

Four Play

            Four Card Poker has a special place in my heart because it sort of launched my career as a gaming analyst.  Ironically, I didn't analyze it as it was being developed.  Rather, I wrote about it right here in Gaming Today way back in February 2004.  The column got noticed by the then President of Shuffle Master, who put me in touch with Roger Snow, the inventor of the game and at the time, the Manager of Table Games for Shuffle Master.  That introduction was the beginning of what has been a very successful collaboration which has included blockbuster games such as Ultimate Texas Hold'em and Mississippi Stud, along with countless sidebets for virtually every game in the casino.

            Four Card Poker was also an important game for the evolution of proprietary table games.  By the time Four Card Poker hit stride, there had been a bit of a lull in table game creation.  The casino floor had already changed a good deal with Let It Ride, Caribbean Stud Poker, Three Card Poker and Spanish 21, but those games were all already several years old.  Perhaps there were some other games in between that I am unaware of.  Admittedly, this lull I speak of, occurred after my father passed away and before I entered the field.

            The game itself didn't really break any new ground in terms of betting structure or rules.   The new ground was broken by Four Card Poker's 'crazy' cousin - Crazy 4 Poker, which introduced the Super Bonus wager - which is more commonly known as the Blind wager on more recent Shuffle Master games.  This wager will push if the Player wins with a poor or so-so hand and will win odds if the Player wins with a strong hand.  I'll cover more about Crazy 4 Poker in a few weeks.  Crazy 4 Poker has about 100 tables in the marketplace as compared to Four Card Poker which has about 250.

            Four Card Poker utilizes the same betting structure as Three Card Poker.  There are two separate wagers - Aces Up and Ante/Play.   The Aces Up pays on a pair of Aces or better and is not concerned with the Dealer's hand at all.   The Ante/Play is the wager where you are playing head to head against the Dealer's hand.  You make an Ante wager to begin play and you are dealt your hand which you can review.  Now you can either make a Play wager of 1x - 3x your Ante or Fold, forfeiting your Ante wager.  If you beat the Dealer's hand, you are paid even money.  If you don't you lose both wagers.   Also, similar to Three Card Poker are the Ante Bonuses.  These pay the Player whether he wins or loses against the Dealer - if the Player can achieve a Four of a Kind, Straight Flush or Three of a Kind.  They pay 25, 20 and 2, respectively.

            So, by this point, if you are not familiar with Four Card Poker already, you're probably guessing that the Player and Dealer each get 4 cards and you might be wondering what hand the Dealer needs to qualify.  WRONG!  The name comes from the size of the hand the Player makes.  He is dealt FIVE cards to make a FOUR card hand.  The Dealer is dealt SIX Cards to make a FOUR card hand.  Thanks to this little benefit, the Dealer does NOT need to qualify in Four Card Poker.  Every hand plays. 

            In Three Card Poker, many people follow a strategy to just do what the Dealer does - and play any hand that is Queen or better.  This is a little below perfect, but will not hurt your bankroll significantly.  If you want to play like an Expert, you go with Queen-6-4 as the lowest hand you Play.  So, with the Dealer qualifying on every hand in Four Card Poker, you have nothing to guide you at all.  Adding to the dilemma is when to Play 1x vs. Play 3x.  As is normally, the case, we NEVER bet 2x.  We either cut our losses (FOLD), hedge (Play 1x) or slam on the gas (Play 3x).

            When Four Card Poker was introduced, Shuffle Master supplied information cards that included a basic strategy on them.  This strategy produced a 98.41% payback and includes only 3 rules.  In my analysis of the game, I took that strategy a bit further and produced one with about 7 rules (admittedly, more complex rules too) that takes the payback up to 98.60%.  Even this strategy is not absolutely perfect as it does not take into account specific suit make up of the Player's hand nor go any further than the first 'kicker' in the Player's hand.  It is my expert opinion that to do so would only get the Player an additional 0.01 - 0.02% in payback, but it would also greatly increase the probability of errors by making the strategy that much more complex.

            Without further ado, I present the basic strategy which Shuffle Master initially developed and I have verified.

·         Fold with a Pair of 2's or Less
·         Bet 1X with a Pair of 3's thru 9's
·         Bet 3X with a Pair of 10's or Better
            It's that simple if you want to earn the 98.41% payback which is respectable.   Expect to Fold a good amount of the time - just under half.  Four Card Poker was designed to be quite a bit more volatile than Three Card Poker.   As Roger told me way back in 2004, "one of three things typically happens.  One, you double up.  Two, you get crushed.  Three, both one and two, and not necessarily in that order." 

            If you'd like to learn more about Four Card Poker, including the Expert Strategy, I highly recommend my Expert Strategy for Four Card Poker.  You can order it by sending $5.95 to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.  This price includes free shipping and handling.

Deuces Gone Wild

            I love getting fan mail and/or e-mails from readers.  There are two reasons for this.  The first is that it is always nice to know that someone is actually reading my column.  It is especially gratifying when someone tells me that they ALWAYS read my column.  The second reason is that a question from a reader can frequently become the basis for a particular week's article.  There are times I sit down to write my column and I simply don't know what I want to write.  I think of a topic and realized I covered it at some point.  Of course, being that I have now been writing for Gaming Today for more than 8 years, it is possible that I last wrote about the topic in 2005 and by now there may be some new readers.

            This week, I received an e-mail from someone who was questioning some of the strategy for Full Pay Deuces Wild.   Full Pay Deuces can be found in several casinos in Las Vegas.  As is the case for most 100+% machines, you won't find them on the strip.  You're going to have to head to some of the local casinos (such as Station Casinos) if you want to find them.  My source ( shows that there should be 100+ machines scattered about at a variety of denominations up to a quarter. 

            While I've never been a big fan of Deuces Wild, this is just a personal choice.  Any game that pays 100.6+% is hard to criticize and is a good game for the regular Player to learn and master.  The strategy table is rather long, but when you break it down by the number of wild cards, you realize that it is not really a hard strategy to learn.  With a paytable that begins paying at Three of a Kind, you don't have to worry about counting High Cards.  The one thing, I strongly advise the beginner to learn is how to recognize hands with lots of wild cards in them.  It can become very easy to not realize that 2 6D 9D 10D KH is a 4-Card Inside Straight Flush with 1 Wild Card. 

            The question I received this week was specifically about holding a 4-Card Inside Straight (presumably with no Wild Cards) versus throwing all 5 cards as a Razgu.  The strategy table tells us that we hold the 4-Card Inside Straight.  If you look at the strategy table in Winning Strategies for Video Poker, however, it lists both hands as having an expected value 0.3+ - although it does list the 4-Card Inside Straight higher meaning its "+" is greater than the Razgu "+".

            While I write extensively on Video Poker in Gaming Today, I spend most of my time analyzing table games.  Many years ago I did write my own video poker engine that allows me to analyze most video poker paytables.  One of the limitations is that it does NOT do wild card games.  Fortunately, I have both other resources available to me and the ability to quickly create a program to help determine exactly how much those "+" are worth.

            Calculating the expected value of the 4-card Inside Straight was very easy.  There are 8 ways to draw the Straight (4 Wild Cards plus 4 of the 'natural' way to complete the Straight).  Each pays 2 units so we have a total return of 16 units.  Divide this by 47 ways to draw and we have an expected value of 0.3404.

            The Razgu is a bit more complicated.  As I've written about in the past, the overall expected value as shown in a strategy table for a hand like a Razgu is the actually the AVERAGE of all the possible hands of that type.  Often, no single hand will actually have EXACTLY the expected value shown. 
            About 20% of all hands in Deuces Wild are classified as a Razgu, each with their own subtleties.  The exact make up of suits and ranks will have some impact on the exact expected value.  For each 10 through Aces that is in the hand, there will be less chances to make a Natural Royal.   The exact suit composition of the initial deal will impact the number of possible Flushes that can be made if we discard all five cards. 

            In this particular case, however, the reader was talking about a 4-Card Inside Straight, which does limit the possibilities.  In order to get a more exact expected value, I quickly set up a program that had the initial deal set to 3D 4C 5H 7S 8S.  I figured that by leaving in all of the High Cards I would leave the expected value about as High as it could go and we could see just how close of a decision this really is.

            The expected value of this specific Razgu came back at 0.3267.   So, it would be more accurate to say that a Razgu is about 0.33- and a 4-Card Inside Straight is 0.34+.  It is not exactly a canyon between the two expected value, but there is a clearly superior choice.

            To help me prove my work, I realized that we also sell a Deuces Wild tipsheet that my father created a long time ago.  It has more detailed numbers on it.  It actually lists the expected value to two decimal places.  It lists the 4-Card Inside Straight 0.34 and the Razgu at 0.32.  (When all the possible Razgus are considered, the average must wind up at below 0.325).   It was good to know that my quick and dirty program was able to produce accurate results!

            If you're interested in learning the strategy on Deuces Wild, we have the tipsheet for $2.95.  It includes the strategy tables for Deuces Wild, Double Pay Deuces Wild and Triple Pay Deuces Wild.  Or you can order Winning Strategies for Video Poker which includes these 3 paytables plus dozens more for only $5.  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89128.

There is Such a Thing as a Free Breakfast!

             Last Friday morning, my wife and I went to see one of the first showings of The Hunger Games at the Red Rock Station.  She had read the book a couple of months ago and has been very excited about the movie coming out.  I can't remember the last time we went to see a movie on opening day.  Maybe on the Sunday of opening weekend, but at 9:30 a.m. show on Friday?  We got there by about 8:30 a.m. to make sure we got tickets.  Fortunately, the 6 people ahead of us didn't buy them all and try to scalp them!  So, by 8:31 a.m. we had our tickets and now had tome to go get some breakfast.

            We walked over to the buffet and found out that if you have one of their Boarding Pass frequent Player cards, they'll take $3 off the cost.  I didn't have mine on me, but I knew I had one, so we walked over to the Customer Service area, got a new card and saved our $6.  While we were getting our new cards, I noticed a promotion that they were running today.  If you earn 300 points, you get a free box of Girl Scout Cookies.  Limit 2 per person and they DON'T take any of your points for the box of cookies.  If we were looking for a place to play, this might have swayed us to stay - all other things being equal.  We usually buy Girl Scout Cookies every year.  They cost about $4 per box, so there is some real value to us.  If we were diabetic, or I found they only have the one variety of cookies we don't like, then the value of this promotion has to be discounted from the $4 price.

            I'm a firm believer in taking advantage of any/all promotions that the casinos want to throw at the Player.  But, you have to be smart about deciding the value that each has to you.  If your local supermarket runs a half-price sale on cold breakfast cereal, but you like oatmeal in the morning, the sale has little value to you.  If you're the type of person who is going to stock up anyhow - just in case - well, you're going to be doing the casino a big favor by chasing everything. 

            After the movie was over, we stopped at the Rampart Casino on our way home.  It is completely on the way to our place.  They are running a promotion that gives me $10 in free slot play - which of course, includes video poker as the casinos continue to erroneously equate the two.  The rule is you have to play the full $10, but you get to keep any of the return.  I sat down at a nickel Bonus Poker machine and played max-coin.  40 hands later, the machine said $12 and I left.  It took about 5-10 minutes.  Am I that hard up for $12?  Certainly not.  But for 5 minutes of my time, the Rampart picked up the tab for breakfast for me and my wife!  If I had sat down for an hour and played their short-pay machines and lost the $10 AND another $20, I would've played right into the casinos hands.

            These are the little benefits the casinos give to Players to try to bring them into the casino.  They'll give you a discount on meals.  They'll give you some free play.  It is no different than what the supermarkets do.  They put milk on sale and hope you'll pay full price for bread, pasta and cheese.  No one has ever complained about the supermarkets doing it, so no one should be surprised that the casinos do it as well.

            Just like the supermarkets, the casinos now give you points when you play.  I've heard some Players voice concern that the casinos use your Rewards card in less savory ways.   Some are worried that the casino will have you win less often if you use your card because they are in essence paying you to play.  From a computer perspective, this is certainly possible.   But, do people really believe that the casino will cheat (aka. break the law) to not give the 1/4% back to the Player that they are telling the Player they can have.  Caesar's Entertainment (formerly known as Bally's and Harrah's) has a market cap of over $1.5 BILLION.  I don't think they are going to risk that type of money so that they can cheat some Player out of his $5 cashback!

            The lesson for today is get the Rewards card.  Get everyone you can.  Keep a little box of them with you when you head out to the casinos.  When you play in a casino, ALWAYS use your card.  That said, be smart when determining the value of the rewards when you decide where to play.  Are you better off playing at a casino that gives 50% more rewards, but has lower paybacks?  If there is a promotion running that allows you to get a free meal at a steakhouse, but you're a vegetarian, it has no value.  If it is a steakhouse you love to eat at, then it is worth its face value. 

            Especially for video poker players who are (hopefully) playing at 99% paybacks and higher, taking that extra 1/4% - 1/2% from the casino can put you very close to an even game.  It will certainly cut the house advantage down considerably and your bankroll will certainly notice.

Surrender and Insurance

            As I’ve described many times, the concepts of Expert Strategy apply to more than just video poker.  Essentially, they apply to every game in the casino (except slots of course).  You should always know which games to play, what strategy to play them with and what to expect.  Most games in the casino do not require learning very significant strategies to play them properly.  Two that do – video poker and blackjack (and its variants Blackjack Switch and Spanish 21 – require some serious effort to learn them correctly.  The reward for doing so is a payback that is above 99.5%.

            In order to achieve the theoretical payback, you have to learn ALL of the strategy including the less well-known parts and even the parts we might find ‘offensive.’  For blackjack this would be the concepts of insurance and surrender.  The idea of ‘surrender’ is the one that you may find to be ‘offensive’, but there are times it is the right play.

            First let’s begin with the definition of the Insurance bet.  When the Dealer has an Ace up, he will offer everyone at the table the option to make an Insurance wager (which must be ½ of your base blackjack wager).  In reality, it is nothing more than a proposition bet.  If the Dealer has a blackjack, then you win 2 to 1.  If he doesn’t you lose your Insurance wager.  Assuming you have not been counting cards, then the odds of the Dealer having Blackjack is roughly 4 out of 13 (I’m ignoring his upcard ‘Ace’ and any of the cards you can see).  Paying 2 to 1, gets us back 12 out of every 13 units wagered for a payback of about 92.31%.  Obviously, you can do some light card counting and only make this wager when it is more in your favor, but it will take a lot of non-10s/Faces to turn the deck in your favor.

            Sometimes you will hear a Player who has a Blackjack to ask for ‘even money’ when the Dealer has a Blackjack.  This is really the equivalent of the Player making the Insurance wager.  If he makes it and the Dealer does NOT have Blackjack he will win 3 for 2 on his base wager, but would have lost 1/2 unit on Insurance leaving him having won even money.  If the Dealer DOES have blackjack, he pushes his blackjack wager and wins his Insurance wager, which will pay 2 to 1 of the INSURANCE wager which is equal to his base wager – in other words, even money on the base wager.  To keep things moving along, most casinos will just allow the Player to call “even money” and get paid 1 to 1 on his blackjack wager. 

            In reality, this is no better a decision than making the Insurance wager under any other situation.  However, from an emotional standpoint, many Players hate the idea of a total push when getting a Blackjack.  This would be the outcome if you don’t take the Insurance Wager AND the Dealer has Blackjack.  The proper play is to stay unemotional and never take even money.  This situation should only occur about 1 in 275 hands (approximately) which would mean once every 9 hours of play.  For some strange reason, I seem to get it about 3 times an hour?!

            Next up is the Surrender rule.  Many of you may never have heard of it.  The casinos don’t really advertise it much.  You have the right to Surrender your hand before you take any other action by forfeiting half of your initial wager.  Once you hit, split, double down, etc… you can no longer Surrender your hand.  There are two different variations of Surrenders.  The first called Early Surrender is rarely offered.  It allows you to Surrender BEFORE the Dealer checks for a Blackjack when he has a 10/Face or an Ace up.  Thus, even if the Dealer has a Blackjack, you would have forfeited only half of your wager.  This is a big advantage to the Player which explains its rarity.  The other variation is called Late Surrender.  This version has the Dealer checking for Blackjack and only after it is confirmed that he does NOT have one can the Player opt to Surrender.

            Unlike the Insurance Wager, this is not a proposition wager better left ignored.  If that were the case, the casino would have it on the felt in big bold letters “PLEASE SURRENDER!”  Instead it is an option you need to take on occasion and you almost have to ask the casino permission to do so.  From a mathematical perspective, the decision is easy.  If you are going to win less than 25% of the time with your starting two cards, you Surrender.  At a 25%, you would win back exactly half of your initial wager which is what you’ll have left after Surrendering.  Hence, that is why this is the decision point.  There are slightly different strategies depending on whether the Dealer hits or sticks on  Soft 17. 

            You should always Surrender a Hard 16 to a Dealer 9, 10 or Ace.  You should also always Surrender a Hard 15 to a Dealer 10.  If the Dealer hits a Soft 17, you also Surrender a Hard 15 to a Dealer Ace and a Hard 17 (yes, I said 17) to a Dealer Ace.  If the Dealer has a 6 underneath, he gets to keep going and is that much more likely to wind up beating you.   These rules apply to larger shoes of 4-8 decks. 

            The impact of properly Surrendering is that the payback is increased by 0.07%.  This may not sound like a lot, but looked at differently, it can cut the house edge by about 15%.

Get Up to SPEED - Let It Ride and Mississippi Stud


            Comparing Let It Ride to Mississippi Stud gives us a great opportunity to understand how a subtle difference in betting structure can greatly alter the strategy of the game and thus radically change a game that is otherwise rather similar.  The subtle difference in this case is that in Let It Ride the '1' and '2' wagers are completely optional (essentially, they can be 'checked') and in Mississippi Stud the choice is to Play or Fold.  No checking allowed.

            To best compare these games we need to realize that the idea that the you can take your wager down in Let It Ride doesn't change the game.  The rules of the game could have simply made the '1' and '2' wagers simple optional wagers.  You can either 'check' or you can make the wager. 

            Mississippi Stud also differs in that your first decision is after seeing only 2 cards instead of the 3 as in Let It Ride.  Mississippi Stud's paytable also goes down to a Pair of 6's, whereas Let It Ride goes to a Pair of 10's.   After 2 cards, the Mississippi Stud Player must decide whether to make at least an additional 1-unit wager or to Fold.  Mathematically, this is vastly different than the decision to check or Play.  When we have the decision to check or Play the question becomes one of whether or not the Player will win more than he loses on THAT specific wager.  Prior and future wagers play no part in the equation.  When the choice is to Play or Fold, the question becomes one of whether we can win back at least the amount we are about to wager when we consider ALL other wagers - both those already made and those we might make during the hand.  This is because if we choose NOT to Play, we are forfeiting all past wagers and the right to make all future wagers.

            The impact to this becomes most evident when we compare the '1' wager in Let It Ride to the 4th street Wager in Misssissippi Stud.  In this case, both hands consists of 3 cards and we are deciding whether to/how much to wager on the 4th card.  In Let It Ride, we find ourselves making the wager very infrequently.  We are willing to leave the wager in place only on sure winners (Pair 10's or Better or Trips), 3-Card Royals and 3-Card Straight Flushes (Open or Inside, NOT Double Inside).   We make this wager only 7% of the time.

            In Mississippi Stud, by the time we get to 3 cards, we have already wagered our Ante and at least 1 unit on the 3rd Street Wager.  If we Fold, we are forfeiting both of these  wagers.  We will also end our hand right then and there.  So, we also forfeit the right to potentially benefit from our next wager (5th Street).  The decision to Play 3x is similar to our Let It Ride decision.  Once you are going to win more than you are going to lose on a specific wager, you wager as much as the house lets you.  So, we find that we wager 3x on all sure winners, 3-Card Royals and a variety of 3-Card Straight Flushes.  We still go ahead and wager 1x on a whole lot of hands that sound like they're going to need some help to become winners.  This includes all Low Pairs, 3-Card Flushes and hands with the right combination of High and Medium cards.

            The net result is that we very rarely fold at this decision point.  The overall fold rate for Mississippi Stud is 44%.   31% (or nearly 75% of the total folds) occur after you see the first 2 cards.  Of the remaining 69% of hands that go to 3 cards, you will fold only 12% of the time.

            In Let It Ride, you will let the '2' wager stay up 16% of the time.  In Mississippi Stud, you will make a wager at 5th Street more than 90% of the hands that go that far.  This happens for two major reasons.  The weakest hands were folded very early on.  A hand that started as two Low Cards was dropped early, which makes weaker hands that much less frequent later on.  In Let It Ride, even the weakest hands have a chance to make it to the end of the hand.  The second reason is that when you have 3 units already wagered and you are compelled to either Fold or make another 1-unit wager, it does NOT take a high win frequency to make it worthwhile to make that additional 1-unit wager.   With just 1 High card and 2 Medium Cards or 2 High Cards in hand, it still pays to make this wager.

            With 2 High Cards, the Player still has 6 chances to draw a High Pair which will return 8 units the Player (each).  With 48 cards remaining in the deck this amounts to an expected value of exactly 1.0, which is the cutoff for determining whether or not to make the wager.  Throw in a Medium card as well and he gets 3 more chances to pick up 4 units and the expected value is now 1.25.  If this were a check or Play decision as in Let It Ride, the decision would clearly be to pull it back with these types of hands.

            There is a reason why I've coined Mississippi Stud to be Let It Ride on SPEED.  The games are very similar in how they play but vastly different in strategy and size of bankroll needed to sit and play.  I can't quite cover all the differences or all the strategy here, but for a limited time, I'm offering up a buy one get one special on my two booklets for these games.  Buy Expert Strategy for Mississippi Stud for $5.95 and get Expert Strategy for Let It Ride for free.  Just send check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133 and I'll get them both out to you ASAP.

Clarity on a Little Advice

Last week’s column gave some simplistic advice to beginners who are not yet ready to sit down and really learn the strategy for video poker. It discussed the relative rankings of four of the most common hands – high pair, four-card flush, low pair and four-card straight.
While I gave the expected values for each of these hands, along with some explanations as to why the rankings are what they are, this week I want to stress that these explanations are not the critical part of the process.
The strategy for video poker is based on one thing – math.
We don’t keep a high pair over a 4-card flush because the high pair is a sure winner. If this were the case, we’d keep a high pair over a four-card straight flush, too (but we don’t!). The fact that the high pair is a sure winner explains why its expected value is as strong as it is, but it is the actual value of this expected value that puts the high pair where it does.
So what is this "expected value" I keep talking about?
It is the average amount of coins we expect to win over the long run from that hand.
How is it calculated?
It is calculated by looking at EVERY possible draw given the 5-cards already dealt.
Say what?
There are 2,598,960 ways to deal five cards from a 52-card deck. For each of these ways, there are 32 different ways to play each – ranging from discarding all the cards to discarding none of them. For each of these 32 ways to play a hand, there is a varying number of possible draws.
If we discard one card, then there are 47 possible draws (each of the 47 remaining cards). If we discard three cards, then there are 16,215 possible draws (choosing three cards from 47). A computer program goes through every possible draw and tallies up the winning hands for each of the 32 ways to play a hand.
It then computes the average number of coins returned for that way. This is the expected value for that particular way of drawing. It compares the expected values for each of the 32 ways and whichever has the highest one is the proper play for that deal and is deemed the expected value for that deal.
An example usually helps to shed some light on this process. Assume you are dealt: 4 of clubs, 5 of hearts, 5 of clubs, 5 of spades, 7 of diamonds.
We recognize the three-of-a-kind (5’s), the EV of which is calculated as follows:
Drawing two cards from the 47 remaining in the deck will create 46 four-of-a-kind winners (a five combined with each of 46 remaining cards). Sixty-six draws will end as full houses (six pairs in all ranks but 4, 5, and 7; 3 pairs of 4 and 7) while the remaining 969 draws do not improve the hand but instead leave it as a three-of-a-kind.
In summary we have:
46 4-of-a-Kind paying 25 each,
66 Full Houses paying 9 each,
969 3-of-a-Kind paying 3 each,
We calculate the total payout as 4,651, which is an average of 4.30 for each of the 1,081 possible draws. Therefore, the expected value of this deal/draw combination is 4.30.
As should be fairly obvious, if we try to play this hand in any of the other 31 ways, the expected value will NOT be any higher than 4.30 and thus this is also the expected value of this deal.
As all three-of-a-kinds have exactly the same expected value, this is ALSO the expected value of all. We will find this value on our strategy table.
Next week, I’ll walk through the four hands (high pair, low pair, four-card flush and four-card straight) I used in last week’s column. This will explain why the strategy I described last week doesn’t just make some sort of logical sense but is the right play mathematically.
I’d like to take this opportunity to wish everyone a happy holiday and a very happy and healthy 2012!

A Little Advice

            Last week's column was a gambling related philosophical debate about perfect vs. good enough.  This week, I'm going to the other end of the spectrum.  It is nearly impossible to define a 'bad' strategy as there really is no end to how bad a Player can play most games.  Playing every hand in Three Card Poker would probably meet the definition of a bad strategy, but is it worse than Folding every hand below a Pair?  Probably not, and I'm not going to waste my time to try to find out.

            This is not to say that every strategy that isn't perfect or as per last week's column 'good enough' would necessarily meet the definition of 'bad'.  I don't consider playing Three Card Poker with the strategy of Play any hand with a Queen to be good enough, but I can't really call it a bad strategy either.  With a game like Three Card Poker, there isn't really much to learn so you draw your line in the sand where you do and that's how you play it.

            A game like video poker is far different.  For anyone that doesn't use Expert Strategy, you might be hard pressed to find two people who used identical strategies.  In reality, they may be TRYING to use Expert Strategy (or some other particular strategy) but due to its complexity, they make a variety of errors along the way.  Then there are the multitudes of Players who just play by the seat of their pants, pretty much oblivious to the math that should be guiding them.  To these Players, getting them to even good enough will be quite a challenge.

            But, no matter what level they play at, if they just learn a few simple strategy points that might help them get a little closer to Expert Strategy then at least it is a step in the right direction.  So, today's column is for these Players.  I would like you all to consider learning just this small part of the strategy and trying to implement it.  You may still be a long ways away from playing Expertly, but hopefully, we can save you just a few bucks along the way and add to your enjoyment too.

            Here goes:
            1)  High Pair
            2)  4-Card Flush
            3)  Low Pair
            4)  4-Card Straight

            This strategy only means something on the hands that are either a 4-Card Straight or a 4-Card Flush and are also a Pair.  Approximately 25% of all 4-Card Straights and Flushes fall into this category, so these hands are fairly common.  This is why it is imperative that these hands be played correctly.  Let's take a closer look at why you should play the hands as described above and learn how these are NOT close calls.

            The High Pair is the only sure winner in the bunch, but this is NOT the reason it is at the top of the chart.  The determining factor is always the expected value of the hand, which is the average amount we expect to win with that hand over the long run.  Sometimes, the sure winner is not the right answer, but in this case it is.  The expected value of our High Pair is 1.54 which reflects the opportunities to turn this into Two Pair, Trips, Full House and Quads. 

            Next up is the 4-Card Flush which will win for us in the long run.  This is NOT to say that we will have more winning hands than losing hands.  With 9 opportunities to complete a Flush and perhaps a few more to complete a High Pair (depending on the exact makeup of the 4-Card Flush), we can expect to win with this hand only 20-30% of the time.  But since many of these will win with a Flush, the wins will be significant.  The expected value of a 4-Card Flush is 1.22.  It will be a smidge higher if you have 1 or 2 High cards and a bit lower if you have none.  If you have 3 High Cards, you have a 3-Card Royal and that takes precedence over the 4-Card Flush, but not the High Pair.

            While the Low Pair has the exact same probabilities as the High Pair of winding up as Two Pair, Trips, Full House or Quads, the fact that it starts as a losing hand is enough to bring its expected value all the way down to 0.82.  That means in the long run, this is a losing hand.  It is the second strongest losing hand (behind the relatively rare 10-J-Q-K Straight, which is also the ONLY exception to the rule I'm presenting here as you hold this 4-Card Straight over a Low Pair, which can only happen with a Pair of 10's).  The Low Pair is also BY FAR the most common hand in video poker, accounting for nearly 30% of all hands.

            Lastly, we have the 4-Card Straights.  While a 4-Card Straight with 2 High Cards ranks only slightly below the Low Pair with an expected value 0.81, it is still below it.  It only gets worse with 4-Card Straights with 1 High Card or 0 High Cards with expected value of 0.74 and 0.68, respectively.  These may not seem like big differences, but they will eat at your bankroll over time.

            It would still be far better for anyone reading this to become a truly Expert Player, but any improvements in your strategy are still better than none.  To help you on your way, we continue with our holiday special.  We are offering Winning Strategies for Video Poker, Video Poker: America's National Game of Chance and Expert Video Poker for Las Vegas for $5 each, which includes postage and handling.  Feel free to order as many as you'd like as they make great stocking stuffers!  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.  We'll do our best to get them to you before the holidays.

Perfection is the Enemy of Good Enough

            Recently, while my teenage son and I were debating something, he responded with "perfection is the enemy of good enough."  My initial response was to shoot back "good enough is the enemy of perfection."  Since this highly philosophical discussion, I've given both of these phrases a lot of thought.

            I'm very well aware that I am a perfectionist who was raised by a perfectionist.  If you brought home a 99 on a test, my father wanted to know why you didn't get a 100.  If there is such a thing, however, as a realistic perfectionist, I think both by dad and I would qualify.  We strive for perfection, but also realize that it is often not realistic to truly attain it all the time.  I think this is why I found the aforementioned quotes to be both interesting and a little befuddling.

            My initial reaction that good enough is the enemy of perfection goes to my basic notion that we should always strive to be perfect.  Over the years, I've been asked many times regarding the strategy for Three Card Poker and if it really matters if you go with Q-6-4 or just Q-High.  The impact to payback is barely noticeable.  You might play for hours before getting a hand that Plays under one strategy but not the other.  Yet, the notion of settling for the easier Q-High frustrates me so.  Clearly the strategy is 'good enough.'  But, is remembering Q-6-4 SO hard that you one needs to go with Q-High?  To me, this is a case where good enough became the enemy of perfection.

            There were times my father's work on video poker was criticized (mildly) by other analysts for being less than perfect.  On one hand, my father was not prone to doing things less than perfectly - especially math work.  On the other hand, he taught himself how to program a computer at age 60, so this was not totally his comfort zone.  In a nod to that realistic perfectionism I mentioned earlier, my father's strategies for video poker were not designed to be 100% perfect.  They were designed to be played by humans.  And, not a bunch of rocket scientists, but the masses.

            The process that my father used to analyze video poker was rather similar to the same one I use, which is most likely not all that different from the ones created by anyone else.  We all have different degrees of shortcuts we use to speed up the process but the basic idea is the same.  We look at each of the 2,598,960 possible initial 5-card deals from a 52-card deck.  We then analyze each of the 32 possible ways to discard and review each of the myriad ways to draw to each of these 32.  Whichever of these 32 ways results in the highest expected value is the proper way to play the hand. 

            The calculation to do the above is absolute and assuming no error in the process will be 100% accurate.  In other words, it will be PERFECT.  So, in a perfect world, a Player could sit down at a video poker machine, press the Deal button and then enter the five cards he was dealt into an APP on his phone, which would run the process I just mentioned and tell him exactly which cards to discard.

            Unfortunately, the casinos are not too keen on this idea.  In fact, I was recently sitting at a Blackjack table and pulled out my phone to check e-mails while the Dealer was shuffling and got reprimanded.  I knew you couldn't use such devices at the table, but I assumed this meant while the game was in progress, not while waiting for the shuffle!  So, sitting at a video poker machine with your tablet in your hand will probably not be allowed.

            Because of this, the next best thing is that the results of analyzing all of these hands need to be summarized a bit.  This is what we call a strategy table that lists the rankings of all the hands in order of their expected value.  Certain hands become essentially 'exceptions to rules' when we try to summarize the hands.  These exceptions could be listed as their own rows on the strategy table, but what would the impact be if the strategy table grew to be 50 or 60 rows instead of the usual 35 or so?   By ignoring these exceptions we cost ourselves MAYBE 0.01% or 0.02% of payback, but we greatly simplify the strategy table, thus reducing the probability of errors.

            In this case, my son was right as perfection could be the enemy of good enough.  My father could have put together a perfect strategy table, but if learning it became that much harder so that the likelihood of errors increased to the point where an average person would lose more in errors than he would gain in playing 'perfectly' - would this still really be 'perfect'?

            At the end of the debate, it would appear that my father had already resolved the issue for us - and we were both right!

            As we are approaching the holiday season, Gambatria would like to offer to all of our readers a deal that may not be perfect, but is certainly better than good enough.  We are offering Winning Strategies for Video Poker, Video Poker: America's National Game of Chance and Expert Video Poker for Las Vegas for $5 each, which includes postage and handling.  Feel free to order as many as you'd like as they make great stocking stuffers!  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.  They'll ship 1st class mail (or priority mail in some cases) so you can get them in time for the holidays.

Giving Thanks

            I apologize to those of you who have been looking for my column the past couple of weeks and couldn't find it.  As some of you may have heard by now, my mother (and wife of Lenny Frome), passed away two weeks ago.  After the funeral, my brother recounted a story to me that I had never heard before.

            When my father passed away in 1998, my brother was the first one who headed out to Las Vegas to be with our mom.  It took a day or two before all the arrangements were made for them to come back East for my dad's funeral.  Yet, of course, they still had to eat.  My brother asked my mom where she wanted to go to dinner and she responded with Hugo's Cellar at the Four Queens.  My family had already made that a regular dinner spot when anyone came to town - and it is a tradition that carries through until today. 

            As they walked through the casino from the parking garage to the restaurant, they passed by two women playing video poker.  They were each holding a copy of one of my dad's books.  My brother said he could not have staged it any better if he tried.  This was clearly a sign.  My father's impact to the industry would continue long after he was gone.

            My father was informally called "the Godfather of Video Poker" by many in the industry.  To be sure, he played NO part in the invention of the game.  At the same time, no one can deny the impact he had on popularizing it.  Even if you are not a video poker expert or even a regular, I can't help but imagine that your play isn't just a tiny bit better from having read his articles - or any of the numerous writers who came after him - including me!  Would video poker have had the staying power if there wasn't someone telling the early Players how to play it?  Would video poker have eaten up as large a percentage of the casino floor as it does today?

            Of course, my father could just have easily been called "the Godfather of Proprietary Table Games".  He had a hand in the development of Let It Ride, Three Card Poker, Spanish 21 and Caribbean Stud Poker.  At their respective peaks, there must have been a combined 2500-3000 of these tables.  As I consider myself an extension of my father's work, we can add on Ultimate Texas Hold'em, Mississippi Stud and a host of smaller games to the total.  This brings the total to perhaps as high as 4000 proprietary tables that my father directly or indirectly had a hand in.  Imagine the casino floor without any of these games.

            While my father was the public face of everything that went on, everyone that knew them (both personally and professionally) knew that my parents were always together.  My dad brought my mom to business meetings to size up the potential client.  My mother was the proofreader for all of my dad's books and booklets.  She was responsible for shipping orders and for the accounting.  In fact, it was my mother who was always listed as the "President" of their company. 

            With the help of Catherine Jaeger, the editor of Midwest Gaming and Travel, we have launched a campaign to get my father into the American Gaming Association's (AGA) Gaming Hall of Fame in 2012.  No disrespect to Blue Man Group (one of the inductees for this past year), but I truly believe Lenny Frome's impact on the industry has been far greater.  To this end, we are asking people to write to the AGA and urge them to induct my father into the Hall of Fame in 2012.

            There are a number of ways to make your voice heard.  You can copy the sentence below or use your own experience to explain why you believe the time has come for Lenny Frome to be inducted into the Gaming Hall of Fame. "Because of his many significant contributions to casino gaming, I respectfully request your consideration of Lenny Frome for induction into the Gaming Hall of Fame."

Mail it to:
American Gaming Association
Frank J. Fahrenkopf, Jr., President/CEO
1299 Pennsylvania Avenue, NW, Suite 1175
Washington, DC 20004

E-mail to:
Brian Lehman/Communications Manager-AGA

            Over this Thanksgiving weekend, my family and I one again dined at Hugo's Cellar.  This time, for the first time we toasted the memory of both my father and my mother.  My dad may have been the "Godfather of Video Poker", but most importantly, they were the "Father/Mother and Grandfather/Grandmother of the Frome family."  Once again, they are "always together."

Video Poker Progressives

            A couple of weeks ago, I described in detail how the math behind Progressives work.  In that column, I mentioned how video poker progressives works just a little different.  There are still two paybacks to be concerned with - the long term theoretical that the casino is concerned with and the specific payback at any point in time that should be the attention of the Player.  The majority of the calculation is still the same in that we multiply the payout of a winning hands by the frequency of the winning hands  and sum up these values.

            What is different about video poker is that the frequencies of the different winning hands can vary as the amount on the meter changes.  For those of you who are video poker Players, this should be no surprise.  For years, I've been telling you that a single unit change in the payout of a hand not only changes the payback but can change the strategy.  Each time you change the strategy you potentially increase the frequency of some hands at the expense of others.

            As a very simple example of this, imagine how the strategy changes as we go from a standard full-pay jacks or better machine to a Double Double Bonus machine.   Because the payout for Four Aces is so high, we actually find that the Player should discard Two Pair in favor of a single Pair of Aces.  This will obviously reduce greatly the frequency of Two Pairs and Full Houses and increase the frequency of Three of a Kinds and Four of a Kinds. 

            So, it should be no surprise that as the jackpot for a Royal increases above 800 that the strategy will begin to shift.  Hands with the potential to be a Royal will have their expected values increase.  This will lead to more Straights, Flushes and of course Royals and the expense of Pairs, Trips and Quads.  Of course, we will also throw away a variety of partial Straights or Flushes to go for the Royal, so this will work against the Straights and Flushes and might increase the number of High Pairs. 

            Thus, pinpointing the exact frequencies can be a bit tricky.  Fortunately, the far easier of the paybacks to determine is the payback at any point in time.  This is because at any point in time, we can know the exact amount of the Progressive jackpot and use this number to determine the exact strategy and in turn the exact frequency of each hand.

            The most common video poker Progressive is an 8-5 machine, meaning it pays about 97.3% when the jackpot is reset to 800 (per unit wagered).  At this level, the frequency of the Royal is about 1 in 40,200 hands.  If the jackpot were to climb to 1600 (per unit wagered) then the payback of the game will go up to about 99.5% and the frequency of the Royal goes up to 1 in 32,700 hands.

            Of course it is rather unlikely that you're going to see a Progressive for a Royal get this high.  With only 1% of the amount wagered (at most) going to the meter, the average amount that will be added to the Progressive Jackpot is somewhere between $327 and $402 (1% of the previously mentioned frequencies).  Of course, something that can occur 1 in 40,000 hands or so can easily occur every 10,000 hand or 80,000 hands.  So, it is not impossible to see the progressive meter go to 1600.  It would have to go to about 1800 for the game to become positive (payback over 100%).  This is not impossible, but not very likely.

            As the payback goes up, the strategy changes and the frequency of the Royal increases, making it harder and harder for the jackpot to keep increasing as the likelihood that it gets hit goes up.  Because of this, it is a bit harder to calculate easily the long term theoretical payback.  It is reasonable, however to approximate it using the same process used for regular progressives.

            In this case, I would take the frequency of each hand using the reset value of the jackpot and multiply each by the payout of the hand and sum these up.  Lastly we would add the percent of each wager going to the jackpot to the total.  This means that the long term theoretical payback of a Royal paying 8-5 with an 800 unit reset amount is about 98.3%.

            I have to admit, if I were designing a paytable for a video poker progressive, I would probably make the likelihood of the game going over 100% a bit more common.  I think it would be a lot of fun to the frenzy that would/should occur each time the payback at any point in time goes over 100%.


Don't Be Foolish!

            It was almost a year ago that I launched this blog (  I was very nervous about launching it.  If there is one thing I've learned about the internet over the years is that pretty much any idiot can have a blog - and quite frankly, I didn't want to be 'any idiot.'  I'd like to think that the name "Frome" is the gold standard in the industry where math analysis is concerned.  To our credit, we have Three Card Poker, Let It Ride, Caribbean Stud Poker, Spanish 21, Ultimate Texas Hold'em, Mississippi Stud, Imperial Pai Gow and countless sidebets.  That's a lot of the casino floor whose math was done by Leonard Frome or Elliot Frome. 

            So, I was quite surprised this past week when I came across a financial blog that was very unimpressed this year's G2E where table games were concerned.  Admittedly, I did write a column last year that called on more inventors to display their ideas at the G2E.  I recognize that the cost of even a small booth can be rather prohibitive for the individual inventor, but what a great opportunity to show your game to people in the industry.  I was pleasantly surprised to see at least two new inventors displaying their games and larger booths from some of the more established companies.

            What I found amazing about this financial blog, however, was not that the writer looked over every game and found none of them to his liking.  That would've been one thing.  Instead he essentially takes table game companies to task for "designing games that the gambler has no hope of beating, but they force the gambler to take the time to learn how to play them!"  This blew me away!  Does he truly expect the casino to introduce games that the Player can easily beat?  That's not going to happen.  The only game that has ever been put on the floor that can readily be beaten are certain variations of video poker. 

            Further, our blogger is annoyed that you have to take time to learn how to play them.  The only game which requires ZERO time to learn how to play them is perhaps slot machines.  As I've recounted in my column many times, I can't even figure out when I've won or lost anymore in today's video slots, but since all you need to do is press the 'spin' button and we can assume that the machine will properly tally your win or loss, I assume this meets the requirement of not needing to take time to learn how to play them.

            Thus, we can conclude from our blogger that what he is looking for is a slot machine with a 100%+ payback.  Perhaps he should've read my column from two weeks ago where I talked about a company that provides the payback information for their slot machines.  This WOULD necessitate learning how to use the smartphone 'app', so I don't know if this meets his strict criteria.

            A couple of days after this first column appeared, our blogger was back with more information for us.  First, he repeats some of his thoughts from the previous column, decrying the lack of innovation from table game companies and then stating, "how the gaming industry has not seen a blockbuster table game since blackjack, and how the industry may not see one until somebody steps up and creates a game that is theoretically beatable."

            That is quite a statement.  According to Wikipedia, blackjack's origins may be as much as 400 years old.  The game as it is played in most jurisdictions is hardly beatable - or at least not easily.  Yes, we're all aware of the MIT team that did it, but this took a rather significant effort on the part of a focused group of individuals. 

            In 1991, the table half of the casino floor consisted of nothing but blackjack, craps and roulette.  Twenty years later, it is estimated that as much as 15-20% of the tables in the U.S. market may be those that were invented AFTER blackjack.  Twenty years from now, I have little doubt that blackjack will make up an even smaller percent of that floor.   Let's not forget that a blackjack table is essentially FREE to the casino while they have to pay to put a proprietary table game on their floor.

            As a gaming analyst - and one that focuses mostly on table games - I am keenly aware of the math of the games.  Most of the newer games that are being introduced have paybacks in the higher 98% to low 99% range.  Yes, they do require that you 'learn' how to play them to achieve these paybacks.  No one, not myself, not the inventors nor the casinos will try and let you believe that the games are beatable in the long run.  That does NOT, however mean that you cannot have winning sessions in the short run and enjoy the entertainment value that they can provide.  Most table games are developed to have the Player win about 35-45% of the time over a 3-hour session - assuming you are willing to 'take the time to learn how to play them'

            Best of all, I won't "force" you to do this, but I'll give you the opportunity to!  There are now 7 books in the Expert Strategy series for table games (Let It Ride, Three Card Poker, Four Card Poker, Spanish 21, Caribbean Stud Poker, Mississippi Stud and Blackjack Switch) and for a limited time, you can order the entire set for $20 which includes postage and handling.  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.


            Last week, I alluded to the seemingly complex math associated with games that offer progressive payouts (i.e. "progressives").  Progressives are games where the top pays are not fixed dollar amounts or odds payouts, but rather have variable payouts that increase as more wagers are made since the last time the prize was won.

            Progressives have become very popular for table games sidebets.  They have long been used for some video poker machines for payouts on Royal Flushes.  Most commonly they are found on slot machines, which love to use a progressives ability to create a very large payout for a very rare occurrence.  As is always the case with a random event, the cycle between hits can frequently become far larger than 'average' and thus create an even larger than normal jackpot.

            As I described last week, Progressives essentially have two different paybacks.  The first is the long-term payback which is what concerns the casino.  The second is the payback of the wager at any point in time which is what should concern the Player.  Let's take a closer look at how these are calculated and why there are two different paybacks.

            Normally, to calculate payback, we take the frequency of a winning hand, multiply it by the payout of this hand which gives us the contribution rate for the hand.  We then sum up these contribution rates to arrive at the overall payback.  For most wagers, the frequency of a particular winning hand is fixed as it is unaffected by strategy.  So, if we are playing Caribbean Stud Poker, we don't have to worry about the strategy of Folding and Playing for the sidebet because you would never Fold a hand that is strong enough to earn a bonus.  Video Poker presents an additional challenge in that you can alter you strategy depending on the payouts and thus alter the frequency of winning hands.

            So, to calculate the payback of a Progressive at a particular point in time, we follow the calculation I just described.  For example, let's assume the following paytable at a particular point in time for a $1 wager:

Pays (For 1)
Royal Flush
Straight Flush
Four of a Kind
Full House
Three of a Kind
Two Pair


            If we perform the calculation described, we get the following:

Pays (For 1)
Contribution Rate
Royal Flush
Straight Flush
Four of a Kind
Full House
Three of a Kind
Two Pair


            So, if you were to walk up to a table and see these payouts, the payback of the game at that very point in time is 81.46%

            But, the payback to the casino could be vastly different.  Let's assume that the Royal Flush is seeded at $50,000.  This means that every time someone wins the jackpot, the prize for the Royal Flush will be reset to $50,000.  Further, let's assume that for every $1 wager that is made, the Progressive increases by 10 cents (i.e. 10% of the wager).

            There are two changes that we must now make to calculate the payback for the casino.  The first is that we always use the seed amount as the payout for that hand.  Thus, we repeat the calculation shown above but we use $50,000 as the payout for the Royal Flush.  This is the amount that the casino itself directly paying out each time the jackpot is won.  When we do this, we find that the payback of this wager is 79.08%.

            However, we must now ADD to this payback the amount of each wager this added to the Progressive meter - in this case 10%.  Eventually this 10% will go back to a Player.  It might happen while the jackpot is at $50,000.10 or it might happen when it is at $120,000 or more.  From the casino's standpoint, it doesn't matter.  That 10% belongs to the players.  Essentially all the Players that don't win the jackpot are handing those dimes to the person who finally does.  So, when we add that 10% to the 79.08% we find that to the casino this wager really has an 89.09% theoretical payback.  Over time, the casino will keep 10.91% of every dollar wagered.

            So, if you were to play this wager while the Jackpot is $65,473, you would actually be playing it on the 'low side' of the average jackpot.  How big is the average jackpot?  To calculate that, we take the average number of hands between jackpots (in this case 649,740) and multiply it by that 10%.  On average the jackpot will grow by $64,974 before it is hit.  We add this to the seed amount and find that the average jackpot will be $114, 974.  At that point, the payback of the wager is the same as theoretical payback of the wager. 

            If the jackpot grows to be above $185,930 (which is very likely at times), then the payback of the wager at that point will actually be OVER 100%.  The only problem with this is that it will only be over 100% for the ONE person who actually wins the jackpot.  Everyone else will just be feeding dimes to the one person who wins.

Coming Soon to a Casino Near You?

           Every inventor thinks their new game is going to be a sure-fire hit.  It's taken Three Card Poker about 15 years to get to 1500 or so tables.  Everyone else is sure they can do it in 2 to 3 years.  So what, if so far, no other game has even come close.  I believe the record for fastest game to 100 tables belongs to Ultimate Texas Hold'em and that took just over a year.

            So, in reality, I can't say with any certainty that any of the games I'm going to discuss today will make it to a casino any time soon.  They will, however, be on display at the Global Gaming Expo (G2E) next week, at the Shuffle Master booth.  I didn't work on these games, so I can't give much insight into the strategy or the math (yet).  If you're going to the show this year, make sure to check these games out.  Next week, for the G2E edition, I'll review a couple of games I did the math on and be able to go more in depth on each.

Cincinnati 7 Card Stud

The base game is a simple game going head-to-head against the Dealer.  You make an Ante and Blind wager.  You get to look at the first 6 cards of what will eventually be a 7-card hand.  After reviewing your 6-cards, you can Fold or Play 1x or 2x your Ante.  The Dealer reveals his 7-cards and you get to see your 7th card.  If the Dealer beats you (best 5 out of 7 cards), you lose all your wagers.  If you beat the Dealer, the Ante and Play pay even money and the Blind pays according to the paytable.  There is also a bonus sidebet that pays if your 7-card hand is Three of a Kind or better.

The twist in Cincinnati 7 is the second optional sidebet.  Here, you are playing against all the other Players as well.  Top hand takes the enitre pot - as long as it at least a Two Pair or Better.   The Dealer participates just like every other Player - including putting up a wager each hand.  If nobody has Two Pair or better, all the wagers carry over to the next round.  Obviously, you can't jump into the middle of a pot.  If you skip a round, you're out until someone wins the pot.  Get a mini hot streak and you can increase your bankroll quickly.  It should also be noted that this sidebet has NO house advantage.  You're playing true odds against everyone.

Six-Card Poker

Another relatively simple to understand game against the Dealer.  Player makes an Ante Wager and is dealt 6 cards.  The Dealer is dealt 6 cards as well and turns over three of them, face up.  Player can now Fold or Play, making another wager equal to the Ante.  The Dealer reveals the rest of his hand.  If the Dealer does not have an A-K or better, the Ante pushes and the Play wager is won or lost depending on who has the best hand (best 5 out of 6 cards).  If the Dealer does have an A-K or better, then both Ante and Play wagers are won or lost depending on who has the best hand.

There is also an Aces Up sidebet that pays if the Player's hand is a Pair of Aces or better.  This sidebet pays even if the Player folds.  Yes, you will Fold with a Pair of Aces if the Dealer's three upcards are Three of a Kind.

Money Market

This one is a bit more complex and a little reminiscent of Mississippi Stud.  To begin Play, you make an Ante Wager and get 4 cards.  You must now either Fold or discard 1 card AND make a wager of 1-4x your Ante.  The Dealer will now expose the 1st of 3 community cards.  You must now either Fold or make a wager of 1-3x your Ante.  Dealer will expose the 2nd of 3 community cards and you will either Fold or make a wager 1-2x your Ante.

The Dealer will expose the final community card and then expose his 3 cards.  Best 5 out of 6 cards wins.  The Ante will pay according to the paytable.  All other wagers pay even money.

The betting structure on this one means you'll be wagering at least 4 units if you stay in until the end and might wager as much 10 units.  Unlike Mississippi Stud, you might have a likely winner, but you will rarely have a guaranteed winner.

There is also a one-way bad beat sidebet.  If you lose with a Pair of Jacks or Better, you win this sidebet.  This wager stays in action even if you Fold your base wager.

If you make it to the show and get to check out these games, feel free to let me know your thoughts about any or all of them.  You can reach me at or on my blog at

Deja Vu All Over Again!

            How many times does a person move from northern New Jersey to Las Vegas in a lifetime?

            It was the summer of 1985 and the plans had been in the works for months.  My parents called me in my dorm one night to tell me that they’ve decided they were retiring to Las Vegas that summer.  At the time, they thought that I might transfer to UNLV or one of the UC schools.  But, having made many friends and in the middle of pursuing my degree, transferring just didn’t seem prudent.  I decided that I would stay at SUNY@Albany. 

            When I got home for summer break, I found much of our house already packed up.  Because they were moving across country, I convinced my parents to allow me to live off campus for my final two years, figuring I would need a place to stay at times when the dorms were closed.  In June, we took a trip up to Albany to set up my new room with much of my furniture from my room at home.  In early August, we began the 10-day drive across country.  We went thru Wilkes-Barre, Toledo, Chicago, St. Louis, Tulsa, Amarillo, Albuquerque, Flagstaff and Kingman before arriving in Las Vegas. 

            Once we arrived, I spent an additional two to three weeks in Las Vegas before flying back to New York to get ready for school.  For the next two years, Las Vegas was essentially my home.  For the following ten years or so, I would visit 2-3 times a year.  My parents had an incredible ‘retirement’ in Las Vegas.  Well, maybe retirement isn’t the right word.  My father would go on to become the ‘godfather’ of video poker and change the casino floor forever with his work on games like Three Card Poker, Let It Ride, Caribbean Stud and Spanish 21.

            It is now the summer of 2011, 26 years later.  To quote Yogi Berra – it’s déjà vu all over again.  My wife and I have spent the last 6 months staging our house and packing up our stuff in anticipation of our move to Las Vegas.  We promised that once my eldest son was in college that we would head out of the New York area.  After researching countless cities, we decided that Las Vegas had the most to offer us.  Most of our friends think we’re going because of my profession.  There are benefits there as well.  The ability to see games in person will certainly help me write about games and develop new games.  But, the primary reasons dealt with the quality of life that Las Vegas affords us.

            As I write this column, we are 2-3 days away from the ‘hurricane of the century’ hitting us almost directly.  Of course, it is expected to come in as a Category 1 hurricane, so what we will endure will be seem like a light rain compared to what those in New Orleans dealt with a few years ago.  I’m likely to see more rain this weekend than I will the next 2-3 years in the Las Vegas valley.  If this wasn’t bad enough, we actually had an earthquake here too this past week.  I personally didn’t feel a thing, but about an hour before it hit, we were at the top of the Empire State Building, where I am told it WAS felt.  I can’t really say that I won’t deal with the same in Las Vegas.  I was there in 1992 when a significant earthquake hit between Los Angeles and Las Vegas and felt my parent’s apartment get shaken up quite a bit.

            Several months ago, I announced that I was changing the name of Compu-Flyers to Gambatria.  I knew then, that that was the beginning of a good deal of change in our lives.  In about 2-3 weeks when we arrive in Las Vegas, the end phase of that change will begin.  Compu-Flyers, now known as Gambatria will return to Las Vegas after a 13-year hiatus.  From a base of operations in Las Vegas, I hope to be able to  write about more up and coming games and to write in more detail about what I see going on in the casinos.

            I hope that Las Vegas will be as good to my family as it was to my parents.  I hope that I can be as good for Las Vegas as my father was. 

The Payback Mirage

             The battle between the short-term and the long-term, where gambling is concerned, is an epic struggle and so totally misunderstood by most.  Why is this a problem?  Because it is critical to understand what to expect when you’re playing.  If you don’t, you may begin to believe that something is wrong about what you are doing and this can lead you to deviate from proper play.  When you do that, you may help yourself in the short-term, but this will eventually give way to damage done in the long run.

            To try and prove this point, I ran some video poker simulations.  I played 100,000 3-hour sessions of video poker.  I assumed each session consisted of 2100 hands of video poker (700 hands per hours).  I started with a full-pay jacks or better game.  What did I find?

            Well, in total, 210 million hands of video poker were played.  At the end of all these hands, the payback was essentially exactly where we would expect it to be – 99.52%.  If the simulation used max-coin quarters, the result would be a loss of about 1.26 million dollars.  Of course, based on 210 million hands, it would also take 34 years of 24 hour/day play to get to this point.  Quite frankly, this is MORE than a lifetime of play.

            When we look at some short term results, we find that the Player will lose about 68.5% of the session and win 31.5% of the time.  So, even when playing a full-pay jacks or better machines, the Player can expect to lose 2 out of 3 times when playing for 3 hours.  Even though the edge is less than 0.5% for the house, the Player will walk away a loser far more often than a winner.

            So, is it any wonder that I advocate playing games with a 100% payback or better.  To prove this point, I created a fictitious machine whereby the payouts are the same as full-pay, EXCEPT the Four of a Kind pays 30 instead of only 25.  The simulation showed that after 210 million hands, the overall payback was 100.70%, which is what we would expect.  So, this game is a bit more positive than full-pay is negative.  Thus, the results of our sessions should probably be flip-flopped from our full-pay version, right?

            Not exactly.   We find that even with the payback of 100.7%, the Player will STILL lose 58% of his sessions!  That’s right.  The Player will still lose nearly 6 out of 10 sessions while playing a game that is significantly in his favor.   Despite this 1.2% turnaround (from 0.5% negative to 0.7% positive), the Player will wind up winning only an additional 1 session out of 10 and still lose a significant majority of his sessions.  How can this be?

            These results occur because when playing video poker, our wins will, on average be larger than our losses.  Of course, even this is a bit deceiving.  What really happens is that every so often we have a HUGE victory, while our losses tend to be more moderate.   In sessions where we hit a Royal Flush, our winnings will be far larger than virtually ANY loss we would ever have.  As a result, we lose more sessions than we win, but those big winning nights tip the scale back in our favor.  When we play a 99.5% game, it only is enough to bring it back closer to even.  If we play a game with a payback of OVER 100%, those big wins are enough to turn the game positive in the long run, even if in the short run we are losing more than winning – in terms of sessions, not dollars.

            As I said earlier, it is critical to understand how this all works.  Otherwise, it is way too easy to simply give up on playing the right strategy if you feel you are losing too often.  While we all play in ‘sessions’, in the end, all that matters is how we are doing over the long run.  In the second example (the 100.7% game), would you really be upset to lose 58% of your session, but  wind up winning 1.85 million dollars over a lifetime?

            One last point for those of you who would try to use the information here as ‘proof’ that the long run is really too long.  I ran each machine for a mere 1000 sessions or 3000 hours of play.  This could be 3-5 years of play for a local in Las Vegas.  While there is a bit more deviation from the long term expectations, on the whole the numbers still prove my point.  The overall paybacks for the games were 99.26% and 100.49% respectively.  The win frequencies for a session were 70.8% and 58.7%, respectively.  So, even over a much shorter period than multiple lifetimes, we will begin to see a pattern develop whereby the Player loses more sessions than he wins, but can still end up a winner in the long run.

Royal Alterations

            Last week’s column discussed how by altering your strategy, you can make Royals appear more often.  Let’s face it, by altering your strategy, you can make any hand you want to appear more often.  Just because Royal Flushes are the highest paying hand does NOT mean that by getting more of them you will automatically win.  If your goal is the bragging rights as the King or Queen of Royal Flushes, it might be worth it to you.  In reality, however, you’ll also be ‘flushing’ your bankroll by doing this.

            That isn’t to say that there isn’t a right time to alter your strategy in order to make a Royal appear sooner.  The obvious case of this is when you are playing a Progressive, where the meter is considerably above the normal 800 for 1 payout.  The intriguing part about playing Progressives is that the strategy keeps changing as the meter increases.  Even under normal circumstances it would be unusual for the meter to get to double the normal payout, but nowadays with some professionals monitoring progressive payouts, the likelihood is even less.  As soon as the meter gets to the point where the game is positive, a team of Players can hit a bank of machines and just keep playing until the jackpot is hit. 

            The Expert Player realizes that as the jackpot goes up, the strategy changes and the frequency of a Royal Flush can increase, which can push the payback up even more.  Using Expert Strategy for a full-pay jacks or better machine will result in a Royal (on average) every 40,400 hands.   If the Royal is paying 1600 for 1, we alter our strategy to make a Royal appear (on average) every 32,700 hands.  This increase in frequency allows us to extract an additional 0.9% of payback out of the Royal Flush hand.  Of course, this change in strategy costs us about 0.7% of payback on all the other hands.  The net increase is 0.2%, however.  So, you can play the Progressive using the altered strategy at 99.5% or you can use standard (8-5) strategy and play it at 99.3%.  It doesn’t seem like much of a choice to me.

            So, what are some of the changes we use when playing a Progressive paying 1600 for 1 on a Royal?  One of the biggest is that the 3-Card Royal now outranks a High Pair.  Yep, this one is going to hurt.  You going to throw away a sure winner (High Pair) and go for the Royal Flush.  Your odds of hitting that Royal is a bit more than 1000 to 1.  But, it’s paying 1600 for 1!  Throw in the fact that you still have many chances to hit a Straight Flush, a Flush, a Straight, Trips, Two Pair and a High Pair and quite frankly, the math isn’t even close.  The 3-Card Royal has an expected value of more than 2, while the High Pair is down at 1.5.

            Another significant change in our strategy is that the A-10 Royal is now playable.  Normally, when playing jacks or better, we do NOT hold a 2-Card Royal consisting of A-10.  We only have 1 way to fill it for Straights and/or the Royal Flush (with the JQK), which greatly reduces its expected value.  However, with the Royal Flush’s payout pumped up to 1600, we’re still better off holding the 2-Card Royal vs. holding just the Ace.  Keep in mind, however, that this hand is just barely playable.  This means that many other combinations of cards might be held instead (such as a 3-Card Straight Flush), so don’t forget to look at your WHOLE hand before getting overly excited about a suited A-10.
            Besides learning some of the changes to the strategy for a Progressive, another key point is learned.  Every change to the paytable can impact the strategy.  Now, if you sit down and play a full-pay bonus poker game using jacks or better strategy, I’m not saying you’ll get wiped out in 10 minutes.  But, what is the point of learning strategy if you’re just going to wing it when you change which type of game you’re playing.  0.1% or 0.2% might not seem like a lot to give up – but in reality, this may increase your loss rate by 20-50%!

            One of the best ways to learn how to play all the different games out there is to learn the strategy tables from a book like Winning Strategies for Video Poker and then practice what you’ve learned on your PC using Masque’s Video Poker Strategy Pro.  For a limited time, we’re offering a package of both the book and the software for only $14.95.  For an additional $5 ($19.95 in total) we’ll also include Video Poker: America’s National Game of Chance which is 200 pages of Lenny Frome’s best articles, quizzes and stories.  If you’d like to order, please send a check or money order to Compu-Flyers, P.O. Box 132, Bogota, NJ 07603. 

Royal Appearance

            Last week’s column was prompted by a reader who raised some concerns that Players who use their frequent player cards are somehow cheated by casinos.  The ‘proof’ of this is that some locals (i.e. frequent Players) don’t seem to get as many Royals as the tourists.  Previously, I had cited at least two reason for this. 

            The first is selective memory.  We tend to remember things we want to remember.  When we go through an extended cold streak, every other scream of “Royal” is burned into our brains.  I’m guessing that in the week you hit your last Royal, someone else did too that week, but it didn’t bother you one bit.  If you’ve gone a year without one, everytime someone gets one, it hits you like a ton of bricks.

            Secondly, even if you’re trying to be relatively objective about it, you also have to remember that ‘you’ are greatly outnumbered by ‘them’.  Even if there are a couple of you playing together, there are dozens if not hundreds of other people playing around you.  It is no surprise that they WILL actually get more Royals than your group will.

            There is, of course, another possibility – other people ARE actually get more Royals than you are!  So, am I buying into the whole ‘rigged’ video poker machine theory?  ABSOLUTELY NOT! 

            But, the number of Royals you get over an extended period of time is greatly influenced by the strategy you use.  So, there are two possibilities.  YOU may be using the wrong strategy which is reducing the probability of a Royal OR the other guy is using the wrong strategy which might INCREASE the probability of a Royal.

            Let’s look at these two scenarios.  The proper strategy for any particular video poker machine is one that maximizes the overall payback, not one that maximizes the probability of hitting a Royal.  What do you do when you’re dealt the following?

A♥       Q♥       10♥      5♥        5♠

            Do you hold the Low-Pair?  The 4-Card Flush?  The 3-Card Royal?

            The correct answer is the 3-Card Royal.  If you’re playing one of the other two, not only are you hurting yourself from a payback perspective, you’re lowering your chances of hitting a Royal.  By the way, the decision is not even close.  The expected value (EV) of the 3-Card Royal is 1.41.  The 4-Card Flush has an EV of 1.22 and the Low Pair a meager 0.82.

            So, if you’re not playing this hand correctly, don’t be surprised if some others around you are hitting more Royals.  Of course, they may have their own issues.  What do you do when you’re dealt the following?

A♥       Q♥       10♥      5♥        Q♠

            The correct answer is hold the High Pair with an expected value of 1.54.  Now, you may be doing this, but that ‘tourist’ behind you may not.  What is the impact of holding the 3-Card Royal.  Well, he’ll lower his overall expected payback, BUT he will increase his probability of hitting a Royal.

            The examples I used here are not the most common occurrences, so these will not make a big difference to the frequency of a Royal happening.  Far more common are the hands that include a 2-Card Royal that also include 3-Card and 4-Card Flushes and Straights.  I have little doubt that there are many novice Players who get Royal fever and just play every 2-Card Royal instead of 4-Card Straights and Flushes or 3-Card Straight Flushes.  Doing so, will make them hit more Royals than you will, but they won’t be any richer for it. 

            Under normal circumstances, for a jacks or better machine, a Royal should appear about once every 40,000 hands or so.  By altering one’s strategy it is very easy to reduce this to once every 30,000 hands or so, which is considerably more frequent.  But, it will come at a cost of lowering the payback by a significant amount too.  So, the next time you’re upset that someone else got a Royal, start worrying about how you’re playing and not what’s happening around the corner.  They may pay dearly for their Royal appearance.


That's Why They Call It Gambling

            I’m in Las Vegas this week, penning this column from my hotel room.  The other night, I was playing video poker at Sam’s Town, next to a guy who was playing single-line Multi-Strike video poker.  I’m familiar with how the game works, but I have to admit, my knowledge of the strategy changes for this intriguing game is extremely limited.  I know that you have to alter your strategy to increase win frequency at the expense of payback when you are on the lower 3 lines without having received a ‘Free Ride’.

            For those unfamiliar with the game, allow me to try and explain the game.  There are four ‘levels’ in Multi-Strike.  To move up to the next level, you have to get a winning hand on the prior level.  Each of the levels pay progressively more than the previous one.  Thus hands on Level 1 pay 1 times the paytable.  Level 2 hands pay 2 times the paytable.  Level 3 hands pay 4 times the paytable and Level 4 hands pay 8 times the paytable.  To play the game you have to wager at least 1 unit on EACH level.  Thus to play ‘max-coin’ you have to wager 20 units – 5 coins times 4 levels.  This means you are paying for a level that you may never reach for each hand.  On each level, you play a brand new hand of video poker. 

            Roughly speaking, a Player playing proper ‘normal’ video poker strategy will win 45% of his hands.  This can be raised a bit if you tweak the strategy to focus a bit more on winning as opposed to how much you win.  However, at 46-47%, you would get slaughtered playing Multi-Strike because the odds of winning the 3 hands at Levels 1 through 3 would not be enough to be worth putting up the extra coin each time.  Thus, the game also incorporates what is called a ‘Free Ride’.  This is randomly generated by the machine to give the Player an automatic trip to the next level.  The Player continues to play the level that gives him the Free Ride, but even if he loses the hand, he still proceeds to the next highest level.  The impact of this feature is to bring the win frequency very close to 50%.

            I’ve never analyzed Multi-Strike, so I can’t provide you with a payback of the game.  Also, there are numerous versions of the game to correspond to regular games (i.e. Jacks or Better, Bonus, Double Double, etc…).  Additionally, the game does not clearly provide the frequency of the Free Ride feature at each level which is required to calculation an accurate payback.  I have seen published numbers from IGT (maker of the game), but there is no way to know for sure if there aren’t different variations and which games are programmed for what frequency.

            Then again, the point of this particular column was not necessarily an analysis of Multi-Strike.  The Player I mentioned earlier came across an interesting hand.  He was dealt an Ace High Straight that was also a 4-Card Royal on Level 3.  The Straight paid 4 units times 4 (for Level 3) for a total of 16 units (I didn’t notice what denomination the guy was playing).  He now faced the choice of sticking with that win and guaranteeing a shot at Level 4, OR going for the Royal Flush which would pay 1000 units (250 times 4).  By going for the Royal, he would also risk not winning at all and thus, not being given an opportunity to play the Level 4 hand.

            First, I’d like to look at this as if it didn’t happen in Multi-Strike.  So, the question is, when dealt a Straight that is also a 4-Card Royal, what is the right play?  Keep in mind, in this particular case, the Player was NOT playing max-coin, so the payout for the Royal was ‘only’ 250.   To fully analyze this situation, we need to look at every possible outcome of going for the Royal.  However, even at a quick glance, we get our answer.  The Player is essentially risking 16 units to win 1000, which is more than a 60-fold increase.  With 47 cards remaining in the deck, he has a 1 in 47 chance of hitting the Royal, which means his potential winnings are greater than the risk.  This tells us that he should go for the Royal.  When we realize that he will also have an additional chance to get a Straight Flush, 7 more ways to get a Flush, 5 more ways to get a Straight and 9 ways to get a High Pair, the decision to go for the Royal becomes an easy one.   The expected value of going for the Royal is about 8.19, while holding the Straight was only 4.

            Of course, in the specific case I’ve spelled out, the decision was a bit tougher.  By going for the Royal, he still has 23 out of 47 chances to wind up a winner and get to Play Level 4.  But, by holding the Straight, he has a 100% chance of playing Level 4.  We cannot dismiss this from the equation.  The expected value for Level 4 is about 7.84 (assuming a 98% payback multiplied by 8).  However, this assumes that we definitely get to play it.  In the case of going for the Royal, we need to multiply this by 23 and divide by 47 to account for the probability of getting to Level 4.  This is only 3.84. 

            So, we need to add these amounts to the respective EVs stated earlier.  While the decision gets quite a bit closer, going for the Royal still edges out the Straight by about 0.19.  I have to admit that I didn’t exactly do this calculation in my head when the guy looked my way (not knowing who I was) and I said “I’d go for it.”  Good thing for me and for the guy playing that he hit the Royal!  Yes, folks – that’s why they call it gambling!

As Plain as the Nose on my Face!

            I have frequently stated in my column that the biggest difference between slots and video poker is that in video poker ‘everything is known’.  What does this mean?  Well, it DOES NOT mean that anyone knows exactly which cards are about to be dealt or drawn.  What it DOES mean is that because the cards are random, we know what will happen over the long run and we know the probability of winning hands forming.  Thus, we are able to create a strategy that maximizes the amount of money we can win by using these probabilities and the payouts of these winning hands.

            When you walk up to a Roulette Wheel, everything is known also – and fairly simplistic.  If you bet a single number (and assuming a single zero wheel), you have a 1 in 37 chance of winning.  If you bet ‘Odd’ or ‘Black’, you have an 18 in 37 chance of winning.  If you sit down at a Blackjack table you know that the probability of you getting a blackjack is about 4.75%.  This information is all known because you’re dealing with real life objects that have a clear probability and are completely random.

            The same is true of video poker.  The fact that it is a digital deck does not change the randomness.  Everything about the game would be the same if you could somehow play it with a real deck of cards.  The overall math is a bit more complex than figuring out the probability of a single number in Roulette or of getting a blackjack, but the concepts are the same.  Let’s start with a simple example.  Let’s say you are dealt the following:

3♥        4♦        5♣       6♠        10♥

            The play is fairly obvious.  Discard the 10 and go for the Straight.  What is the probability of drawing the Straight?  There are 8 cards that will complete it, with 47 possible cards to be drawn.  Thus, the probability is 8/47 or about 0.17.  With a payout of 4, we multiply this by the probability to arrive at the Expected Value (EV) of this hand of 0.68.

            What if we make the hand a bit more complex?  What if the 10 was another 6?  Now there are two possible plays.  We can do as we did before and go for the Straight or we can discard the 3-4-5 and hold the Low Pair.  We don’t have to guess what the right play is.  While the specific result for a single hand will be determined by the Random Number Generator of the machine, we can look at every possible outcome of each situation and determine which results in the higher Expected Value.  When we look at all the possible draws or use some combinatorial math, we find that starting from a Low Pair and drawing 3 cards (16,215 possible outcomes) will result in 45 Quads, 165 Full Houses, 1854 Trips, 2592 Two Pairs and 11,559 losing hands.  When we multiply each of these by the payouts of each hand, and divide the total by the total possibilities, we come up with our EV of a Low Pair, which is 0.82.

            This is considerably higher than the EV of the 4-Card Straight (0.68).  Thus, the proper play is to hold the Low Pair.  By looking at every possible (2,598,960), every possible way of playing each one (32) and every possible draw for each of these ways (varying depending on how many cards are drawn), we can figure out the probability of absolutely everything that can happen in video poker.  In total, we have to look at more than 675 BILLION combinations of Deals/Draws.  Fortunately, with the help of today’s computers, this really isn’t all that daunting of a task (and there are some shortcuts to help!).

            The important thing to realize is that there is no guesswork here.  There is hard, cold and very precise math based on a 52-card deck and the idea that the probability of any card appearing is the same as every other card.  A long time ago, I saw someone suggest that the way to tell if a slot machine is a ‘good one’ is to play 20 times and count the number of winners.  A machine set to pay more will have a higher win frequency than one set lower (I can’t even verify this much!), so based on how many winning hands you have in the 20 times gives you an idea of if the machine is a good payer or not!  HUH??

            Show me a video poker machine’s paytable and I’ll tell you the win frequency and the payback in a matter of minutes (okay, if it’s something new, it might take a bit longer!)  This can be done because there is nothing hidden in video poker.  The payback is known.  The hit frequency is known.  The strategy is known.  Everything is known!

            If you’d like to know more, one of the best ways to learn more about video poker is from my father’s book Video Poker: America’s National Game of Chance.  It is 200 pages of dozens of some of my father’s best articles about video poker, all geared to teaching you how to play in a more laid back way.  It retails for $19.95, but for a limited time, I’m making it available for ONLY $6 each or 2 for $10, which includes 1st class shipping and handling.  Send a check or money order to Compu-Flyers, P.O. Box 132, Bogota, NJ 07603.