# Getting Ugly

For the past few weeks, I've been slowly walking through the strategy table for full-pay jacks or better video poker.

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# Distinguished Royals

I've spent the past few weeks walking through the strategy table for full-pay jacks or better video poker.

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# Stroll Through the Strategy Table

This week we continue our walk through a video poker strategy table.

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# Get the Inside Scoop

Last week I began reviewing the strategy table for full-pay jacks or better video poker.  I got about 20% of the way through the table by volume, but not very far in terms of useful information.  The top 8 hands were mostly of the no-brainer category as they were the pat hands with the exception of the 4-Card Royal.  This week, I'll keep moving down the table and provide some insight into the nuances of video poker strategy.  Please remember that this particular strategy is applicable ONLY to full-pay jacks or better.

After a Straight, we find the following entries on our table:

·       4-Card Straight Flush
·       Two Pair
·       4-Card Inside Straight Flush
·       High Pair
·       3-Card Royal Flush
·       4-Card Flush

The first thing you might notice about the above entries is that we have two entries for a 4-Card Straight Flush and a 4-Card Inside Straight Flush.  There is a big difference between the expected values for Straights that are open and those that are Inside (or Double Inside).  The common definition of Inside Straight is when the opening is in the middle and not on the ends (i.e. 5-6-7-9).  However, this leaves off some Inside Straights.  It is more accurate to define a 4-Card  Inside Straight as one that can only be filled ONE WAY.  So, an A-2-3-4 can only be filled with a 5 and thus is an Inside Straight.   With this definition you can see that an Inside Straight can be completed with only 4 cards while a regular Straight can be completed with 8 cards.  Straight Flushes are no different - except they have the possibility of being turned into Flushes as well.

In this particular case, there is really no benefit to splitting out the 4-Card Straight Flushes.  The one hand that lies between them can't possibly be a 4-Card Straight Flush (Inside or not).  We show them separately because in some version of video poker, the hands that appear in between may be able to overlap with them and we will find that in some cases we will want to keep a 4-Card Straight Flush ONLY if it is not an Inside Straight Flush.  Also, as we will see as we move down the table, this distinction becomes very important as we take a closer look at 4-Card Straights.

The 4th entry on the table is a critical one - High Pair.  It is the 4th most common hand.  Thus, playing it correctly is very important.  Looking at the entries above it and below it what we learn is that a High Pair is played OVER any 4-Card Straights and 4-Card Flushes.  We will, however, play all 4-Card Straight Flushes over a High Pair.  But, we will NOT play a 3-Card Royal over the High Pair.  So, if you have a suited J-Q-K along with another Queen, you stick with the sure winner - the Pair of Queens.

Below High Pair, we have a 3-Card Royal Flush and a 4-Card Flush.  There is much to learn here as well.  The most obvious is that if you have a 3-Card Royal and a 4-Card Flush, we hold the 3-Card Royal.  This can be a tough choice because the likelihood of hitting the Royal is still relatively small.  But, by holding a 3-Card Royal we give ourselves more chances for a Straight.  We might still hit a Flush and we have the longshot at the Royal.  Also, with a 3-Card Royal, we leave ourselves 2-3 cards that can be matched up for a High Pair.  The expected values are not really all that close with a 1.41 for the 3-Card Royal and 1.22 for the 4-Card Flush.  The decision is relatively clear.

From these entries we also learn that if the Player has a 3-Card Royal that is also a 4-Card Straight Flush (8-10-J-Q), we hold the 4-Card Straight Flush.  With the 4-Card Straight Flush, we still have many chances for Straights and Flushes so we don't throw away the extra card even if it gives us a chance to get the Royal.

I've stopped at this particular point in the Strategy Table because the 14 hands I've listed (over the past 2 weeks) are the only ones with an expected value greater than 1.0.  That means these hands are net winners in the long run.  Some will be winners 100% of the time.  Some will not.  But in the long run, we can expect to get more back than we wagered.  These hands make up about 40% of the table and about 25% of the total hands dealt.  Beginning next week, we'll review the hands with an expected value below 1.0.  Even though these are losers in the long run, it doesn't make them less important.  In fact, they may be more important because they account for a larger percentage of hands dealt.

# Strategize

Every casino game that is more than pure luck has some strategy associated with it.  This goes beyond the basic strategy that simply says you're better off not playing at all.  For many games, the strategy can be summed up with a simple sentence or two.  For Three Card Poker, it is Play Q-6-4 or better.  Four Card Poker has a two sentence strategy that tells you when to fold and when to Raise.  Let It Ride's strategy takes a few sentences telling you when to pull down the 1 and 2 wagers.

As strategy gets more complex, it is helpful to try and put it into as easy as a format as possible to help a mere mortal to utilize it.  It is relatively easy to program a computer to play a game perfectly.   Very few humans can take every game to this level.  Also, expending that much energy on memorizing a very complex strategy can pretty much sap the fun right out of the game.  Blackjack utilizes a relatively simple matrix that crosses the Player's hand with the Dealer's upcard.

Creating a strategy for video poker is quite a challenge.  As said earlier, telling a computer which one of the 32 ways to play a hand is relatively easy.  But, there are 2,598,960 unique 5-card deals from a standard deck.  Coming up with a way to group these together in a way that a Player can use is a whole different story.  I believe it was my father, Lenny Frome, who was the first person who accomplished this.  He grouped hands together in a way that Players could easily understand and hopefully memorize.

A video poker strategy table consists of only two columns.  The first contains the hand rank as it was categorized by my father.  The second contains the expected value of the hand.  Ironically, this second column isn't even needed to play video poker properly.  It is there just for reference.  So, that means the video poker strategy table consists of a single column - usually with about 30-40 rows/entries in it.  To play video poker the correct way, you have to memorize the order of these entries.  This is not nearly as daunting as it seems.  About 10-15 of these entries are more than a little obvious.  So, you're left with about 25 hand types that you need to learn.

Let's start at the top of the strategy table which contain the most obvious hands:

·       Royal Flush
·       Straight Flush
·       Four of a Kind
·       4- Card Royal
·       Full House
·       Flush
·       Three of a Kind
·       Straight

We'd be having a great night at video poker if these were the only hands we were dealt.  These are all big winners, all with expected values of 4.00 or better.  In fact, only one of these hands is not a sure winner - the 4-Card Royal.  This is also the only hand that might overlap with any of the others, creating the only strategy decision in the bunch.  What do you do if you are dealt a Straight (or a Flush) that is also a 4-card Royal?  Well, now you know the answer.  You have to throw away the sure winner to go for the big winner.  The good news is that if you have a 4-Card Royal, you have a very good chance of still winding up a winner.  There are 47 possible draws, 1 of which will result in the Royal.  Another will give you a Straight Flush.  6 or 7 more (depending on whether you threw away a Straight or Flush) will result in a Flush.  5 or 6 will result in a Straight and a host more will give you at least a High Pair which will seem like small consolation.

While this decision might be agonizing, mathematically, it is very clearly the proper play.  The expected value of the 4-Card Royal is 18.66.  The expected value of the Flush is 6 and the Straight is 4.  Of course, don't expect to see this hand every hour.  A 4-Card Royal will show up once in about 2700 hands and only about a third of these will be a Straight or a Flush.   One other key point to note.  Do NOT throw away a Straight Flush to go for the Royal.  That Straight Flush has an expected value of 50 which far exceeds the 18+ of the 4-Card Royal.

Next week, I'll move down the strategy table to the hands that require a bit more thought.

# Table your Hunches

Last week, I described how all casino game strategy is based on expected values.   You hit or stick in blackjack not because you hope the next card is of a certain value, but because there are certain probabilities as to what the next card will be and how it will affect your hand and your chances of winning or losing.   If you're dealt two face cards, you don't give much thought to strategy.  Hopefully, you're not one of those Players who even thinks about splitting 10's!

But, if you are dealt a 16 and the Dealer has a 7, you start giving thought to the strategy.  With a 16, you have 5 cards that will help you and 8 that will bust you.  The odds don't look to good and this is why a lot of people stick on this hand, albeit incorrectly.  You can stay put, but with a 16, the only way you can win is if the Dealer busts, which will happen only 26% of the time.  So, your choices are a 61% chance of busting right away or sticking and having a 74% chance of losing that way.  Of course, by hitting you also have an opportunity improve your hand.  All of the 5 possibilities improve your hand.  If you pick up an Ace, you'll be most likely to push.  Pick up a 5 and you'll win more than 92% of the time.  Don't get me wrong, it is not a strong hand and the decision to hit is not an overwhelming one, but it is still the right move.  In the simplest form, if you face this situation enough times - which you will if you play for a few hundred hours, you'll find that you do better by hitting than by sticking.

In blackjack, you don't have to memorize all of the math behind the game.  You don't have to figure out how many cards will bust you or bust the Dealer.  To learn to play blackjack, many Players use a simple strategy table.  It is a simple little chart that shows every possible Player hand and each possible dealer upcard.  It then shows what to do - hit, stick, double, split, surrender, etc..  Guys like me have already done all the number crunching for you.

Video poker is no different than blackjack except the decision making process is far more complex.  In blackjack, the result is essentially binary - you win or you lose (okay, you can tie also, so it is not really binary).  In video poker, you can have 1 of many results - ranging from a Royal Flush down to a High Pair or you can lose.  Since each of the different winning hands pays a different amount, the specific result must be taken into account.  If someone invented a game of video poker in which all hands above a certain rank paid a fixed amount, then we'd be able to lump all the hands into win or lose.  But, we need to know the probability of each final outcome with a different payout in order to appropriately determine the value of getting that hand.  Surely, it is more valuable to wind up with a Straight Flush than just a Straight.

Video poker is also more complex than blackjack in that there is more than just a handful of different possibilities for each hand.  The Player can hold all 5 cards or discard all 5 cards or anything in between for 32 different possible plays.  Yes, most of these possibilities will be quickly discarded, but they still must be considered from a mathematical perspective.  They are only discarded because the human mind can quickly recognize possible draws that would clearly not be the best strategy.

Despite the extra complexity of video poker, the similarities are still stronger than the differences.  In the end the decision still comes down to the expected value.  Like in blackjack, you don't have sit there trying to figure out how many cards you need to complete a Straight or the like.  Again, guys like me have already done the job.  We have looked at every possible deal, every possible draw for every possible deal and summed up all of the final hands.  Using this distribution, each possible draw is assigned an expected value.  Whichever draw has the highest expected value is deemed the right play.  The last step in the process is too try and categorize the way each hand is played into a format that a human can use to play the hands.  We call this a strategy table.

Unlike blackjack where the strategy is a matrix that crosses Player hands with Dealer hands and tells you what to do, a video poker strategy chart lists all the possible playable hands in order in a simple table.  The table usually contains the expected value of each hand too, but this is just for information.  To use the strategy table, you basically work from the top and find the first hand that your dealt hand can make and that is the way to play the hand.  So, if you are dealt a hand that is a 4-Card Straight and a Low Pair, you start at the top of the table and work downward.  If a 4-Card Straight appears first, you play that.  If a Low Pair appears first, you play the hand that way.  If you can't find any hand that matches the hand you were dealt, then you fall to the bottom of the table and find a RAZGU which means throw all five cards.

Next week, we'll begin breaking down a strategy table for full-pay jacks or better.  You'll be on your way to becoming an Expert Player.

# The Advantage of Expert Play

This morning, I had a discussion with a friend of mine about a game he is developing.  I explained that playing 'perfect' strategy would be nearly impossible due to some subtle complexities of the way the game is played.  As a result of this, the game would not likely play anywhere near its 'theoretical' payback.  Many games have this 'problem'.  Blackjack pays 99.5%, but very few players play anywhere near this.  Ultimate Texas Hold'em has a payback well into the 99% range too, but stats from the casinos make it clear that very few Players, if any, can manage this high of a payback.

My friend stated that he thought that he would be able to play the game close to the theoretical because he is an accomplished Poker Player.   I asked him if he was an accomplished video poker Player and he said that he wasn't.  I told him that any table game against a Dealer was really nothing more than playing video poker and had no resemblance to poker even if the game resembles poker.   Poker is about reading Players, understanding their betting patterns and their tells.  Video Poker is about one thing - math.   There is no one to bluff.  All that matters is what is the probability of all final hands given what I choose to discard.   Let's take a look at a simple example:

5♠        5♦        6♣       7♥        8♦

In theory, there are 32 ways to play this hand, but I think we can quickly rule out 29 of them.  I don't think anyone is seriously going to consider holding only the off-suit 6-8 or holding all 5 cards (which would result in an immediate loss).    There are really on 3 possibilities, 2 of which are identical.  The Player can either hold the Pair of 5's or the 4-Card Straight (hence, the 2 identical possibilities as it doesn't matter which 5 the Player keeps.)

If the Player keeps the 4-Card Straight, 8 cards will result in a Straight and the rest will result in a loss.  So, if we add up the total payout, we'd have 8 Straights at 4 units each for a total of 32 units.  There are 47 possible draws.  We divide the 32 by 47 to get 0.68.  This is called the Expected Value (or EV) of this hand using this possible discard strategy.

Calculating the Expected Value of holding the Pair is a bit more complex, but easy enough to calculate using a computer.  There are 16,215 possible draws if the Player holds 2 cards.  We look at these possible draws and look at the final hands.  The Player can wind up with a Four of a Kind, Full House, Three of a Kind or Two Pair.  We add up the total payout of all of these winning hands and divide by 16,215.  The result is an EV of 0.82.

This Expected Value is greater than that of the 4-Card Straight, so the proper play is to hold the Low Pair.  When Playing video poker (and virtually every other casino game), the proper play is to follow the one with the highest EV.  You don't go with a 'hunch' that a 5 is coming up or that you just feel a 4 or a 9 is going to fill out that Straight.  There is a distinct probability of each of these events occurring and we use those probabilities to our advantage.  This is what allows a Player go achieve the theoretical playback of a game.

It is an 'advantage' because most Players don't play this way.  Because of this, the casinos can off the games with a relatively high payback, knowing that they can rely on human error to pad their profits.  For the Players who play according to the math, they have the advantage of being able to play to the theoretical payback over the long run.

Mastering video poker takes some significant effort.  The strategy is a complex one and learning whether to hold the Low Pair or the 4-Card Straight is merely one example of where a strategy where you play by what you think is right may in fact be quite wrong.  The good news is that thanks to guys like me, the toughest party of learning the strategy (creating it) has already be done for you.  The next step is learning that strategy and putting it to practical use.  We'll save more of that for next week.

# Video Poker Primer

It was just over 10 years ago that I started writing for Gaming Today.  I have to be honest, that really blows me away.  That means I've written roughly 500 columns when I take into account off weeks and the fact that for the first 6 or 9 months, my column was bi-weekly.  I remember when I wrote my first few columns, I would wax poetic about how my father (Lenny Frome) had written nearly 1000 columns for a variety of different publications.  I remember when I hit column number 100, I remarked how far behind I was.  Now, my total count is probably about 600-700 columns and I can almost see myself someday surpassing my dad's total.  That said, I definitely don't plan on taking steroids or PEDs to get me there.

Part of what is so amazing about having written 500+ articles is that I have somehow managed to come up with that many things to write about.  I'm not really sure that there are 500 unique subjects to write about.  I have to remember that if I borrow a subject from 2005 that there is a strong likelihood that if someone reads it today, they didn't read that article from 8 years ago.  So, in that spirit, I'm going to start back at the beginning today and discuss some basics about video poker.

Video poker is truly a unique game in the casino.   Far too often it is lumped together with Slots, but there is little in common except for the technology.  I don't think of a video blackjack machine as a slot machine and the same is true for video poker.   As the world starts turning more to online gambling, the separation will no longer be about the technology.  Instead it should be about the essence of the game.  Video Poker is a game that is based more on skill than almost any other game in the casino.  This doesn't mean that luck doesn't play a part, especially in the short run.  But, if I were to challenge a random Player to a slot competition, there would be no way to gain an advantage.  If I were to challenge a random Player to a video poker competition, I'd like to believe that I would have a distinct advantage.  The longer the competition runs, the more strategy and skill will rule the day and the less that luck will impact the results.

How is video poker a game of skill?  Because the Player must make a decision that will clearly impact his results.  This decision is frequently NOT of the 'no-brainer' variety.  Technically, in the game of Casino War, the Player must make a decision to - whether or not to go to War when the Player and Dealer tie.  But, the proper decision is the same all the time - to go to War.  So, while technically, there is 'strategy', I doubt very many people get this one wrong.  In Three Card Poker, there is one strategy decision - to Play or Fold.  The decision is also relatively simple.  If the Player has Q-6-4 or better, he should Play.  As simple as this sounds, many Players don't follow this rule (and I don't mean that they go with Q-6 or Q or better), and as a result, they give up a larger portion of their bankroll to the casino than they need to.

Video Poker strategy is far more complex than this.  First of all, the decision is not one of Fold or Play, but rather which cards to Discard.  There are 32 ways that a Player can make each of these decisions, ranging from keeping them all to discarding them all.  Granted many of these possibilities will fall into the brainless category.  If you are dealt Three of a Kind and two off-suit kickers, which cards to discard is pretty obvious.  If you are dealt a Straight, then you don't have to discard at all.  Oh wait, what if it is also a 4-card Straight Flush or a 4-Card Royal, then what is the proper play?

If you are dealt the following:

4♦        4♠        5♠        6♠        7♣

the decisions get a bit more complex.  You might keep the Pair of 4's, or the 4-Card Straight or maybe the 3-Card Straight Flush.  This is 3 of the 32 ways the hand can be played.  The other 29 are quickly discarded, so there isn't a need to go through 32 possible decisions for each hand.  Obviously, you're not going to keep the off-suit 4-7 in this case.

Unlike table poker (which involves even a higher level of skill), the strategy in video poker is based strictly on math.  You don't play hunches and you're not trying to beat another Player.  You don't have to worry that you might pull your Straight and he might come up with a Flush.  All that matters is the likelihood (aka probability) of each final hand and how much that hand pays.  But, I'll leave that for next week.  For now, I'll be happy if I've convinced you just a little bit that video poker is not slots.

# What is the Allure of Progressives

There is a theory in physics that goes for every action there is an equal and opposite reaction.  In gambling, there is a similar theory.  For every table game there will eventually be a sidebet.  And, for every sidebet there will be a Progressive version of the sidebet.  The math behind Progressives is probably the least understood math of any type of gambling.  It really isn't that hard once it is explained properly, but I've worked with a lot of inventors on a lot of Progressives, and it is fairly obvious to me that few people, even in the industry, understand how a Progressive works mathematically.

Generally speaking there are 3 components of a sidebet - the fixed pays, the seed and the contribution rate.  Normally when we calculate the payback of a sidebet, we simply multiply the fixed pays by the frequency of each winning hand and sum up these values.  For a Progressive, we have to alter one step slightly and add one.  For the jackpot event, we use the seed amount as the equivalent of the fixed pay for that event.  Each time it is hit, the casino is on the hook to put that money back on the meter, so it is similar to a fixed pay in that regard.  We then need to add the contribution rate - which is the amount of each dollar wagered that goes on the meter - to the total payback calculated.    I'll save more details for another day, as this is not the point of today's column.  What is the point is to discuss how a Progressive differs from other wagers.

While the top pay for most sidebets are pretty large, the amount they contribute to the overall payback is usually pretty small.  If you pay 1000 for a 1 in a million even, the contribution rate is a meager 0.1%.   In video poker the Royal Flush contributes only 2% to the payback of the game.  If we were to look at most table game sidebets, we'd probably find that most top pays contribute about 1-2% (or less) to the overall payback.  But, when we switch to a Progressive, we find that the top pay frequently contributes 15-20% to the payback when we take into account both the seed and contribution rate.  What does this mean for the Player?

As I said, the Royal Flush accounts for 2% of the payback of video poker.  What this usually means is that until you hit one, you're only playing at about 97.5% which can be a bit rough.  When you hit one - and if you are a regular player, you WILL hit one, you bring the theoretical payback back to 99.5%.  Hit the Royal more frequently than 'normal' and you're likely up money as you will be above 100%.  With Progressives, it doesn't quite work the same way.  That top hand is either more rare or you'll be playing a game that deals much more slowly than video poker, meaning that there are no guarantees that you will EVER hit it.  So, even if the sidebet were paying 99.5% like video poker, ONE PLAYER is going to wind up winning 15-20% of that payback and everyone else will be playing at 77.5% - 82.5%.

When you consider the fact that many sidebets have paybacks far lower than 99.5%, you realize that the picture for those that don't hit the jackpot is even more bleak.  So, why do people play Progressives?  There are two main reasons.  One is a bit emotional and the other a bit more practical.

First, Players have always been willing to accept low paybacks for a chance to win a life-changing amount of money.  The Lotto has made a lot of money for a lot of states.  Most states payout only 70% on their lotteries.  This is lower than the legal minimum of any casino game here in Nevada.  But, for the chance, however slim, of winning millions of dollars, Players are willing to throw a few dollars in for the hope of getting struck by lightning.

The second reason deals with the way Progressives work and makes far more mathematical sense.  To the casinos, the payback of a game is the long-term payback, which is calculated as I described earlier.  You'll note that what I described completely ignores the specific value on the meter at any point in time.   This money is merely an accumulation of the contribution rate over time.  It really doesn't matter to them (mathematically), if a jackpot that is supposed to hit about once a year, doesn't get hit for 3 years.  However, to the Player, the payback of ANY wager is dependent upon the specific payouts for each winning hand at the point in which you make the wager.  It doesn't really matter if the contribution rate is 10% of 20%.  If a Jackpot which is supposed to average \$250,000, goes all the way up to \$600,000 then the payback at that point in time is WELL above the theoretical payback.

It is possible that at a particular point in time that the payback of a wager could be over 100%.  At this point, it makes sense to play the game mathematically.  The problem is, however, that it will be one person that will benefit from this occurrence and it may not be you.  Then again, it might!

# Why Play Max Coins?

Generally speaking, I advise players to play max-coins when playing video poker.  For most versions, this means 5 coins.  The penny Player puts up 5 cents, the nickel player 25 cents, the quarter player puts up \$1.25 and the dollar player has to put up \$5 per hand.  This is done for one simple purpose.  On most video poker machines, the top payout - the Royal Flush - changes from 250 for 1 to 800 for 1 when that 5th coins is put in.  If you are playing a Progressive, the only way to win that jackpot is to play 5 coins.

A payout of 800 for 1 on the Royal is worth approximately 2% of the total payback of the machine.  A payout of only 250 reduces this down to about 0.65%.  So, the Player is giving up more than 1.25% of payback if he plays below max-coin.  In similar fashion, if the machine is offering a Progressive, which should push the Royal payout to above 800, then the Player would be surrendering even more payback by playing below the max-coin level.

The notion of playing max-coin does NOT mean you should wager 5 times the amount you feel comfortable wagering.  Instead it means you should consider lowering your denomination to the next lower level and then play 5 coins.  So, rather than playing 1 quarter, you should play 5 nickels.  This, of course, assumes that all things are otherwise equal.  It is certainly possible that when you go to a nickel machine (or change to the nickel option on a multi-denominational machine) that the paybacks may change as well and you may find that the payback on the nickel machine is well below that of the quarter.   This makes things a bit more complicated.  If the quarter machines pays 99.5% at max-coin, then it will be closer to 98% if you play 1 quarter.  If the nickel machines pays 98.5% at max-coin, then you'll still be better off playing max-coin nickels.

There are a few times when you may want to play less than max-coin.  The first is when you are first leaning how to play.  As you are more apt to make mistakes at this point, you might be better off simply playing 1 nickel at a time.  Yes, you will be playing at a lower payback, but at this point, your goal is to become a better player while playing on a real machine.  Ideally, you'd spend most of your 'learning' time playing on your computer (or phone or tablet) at home for free ,but I realize that playing for free may be a lot less exciting than even playing for a single nickel.

Another reason that you may not want to play max-coin is your bankroll.  If your bankroll is not large enough to support playing max-coin then you might be better off playing single-coin.   Once your bankroll is gone, you're done and you need to make sure you have enough money available to ride out the cold streaks.  Of course, one solution to this issue is again to simply drop down in denomination.  So, this advice really only applies if machines of a lower denomination are not available.  Since the advent of the multi-denominational machine, finding machines that play the denomination you want to play has become much easier, however.   So, this second reason may have limited practical applications.  But, if you find yourself in a situation where your bankroll will support 5 nickel play, but you only have quarter machines available, you may want to consider playing a single quarter as opposed to five quarters.

One critical point to consider.  Just because you switch a machine from quarter play to nickel play, do NOT assume that the paytable is the same even if you are switching to the same variety of video poker.  There are no requirements that state that a machine must use the same paytable when you move from one denomination to another.   In similar fashion, don't assume that a bank of similar (or identical) looking machines all have the same paytable.  Casinos frequently and presumably purposefully mix the machines up, making sure to sprinkle higher paying machines in with lower paying ones.  I dare say that you may find no rhyme or reason to the pattern of machines on the casino floor.

# Better is Better than Best

A long time ago, I remember reading an article about product marketing in which it stated that you'll never (rarely?) see a company say that there product is better than a similar product.  Why?  Because if you say something is better you have to prove that it is better.  If you, on the other hand, say your product is the 'best' product then it is possible for the other product to be 'best' too.  Best, in marketing parlance, simply means as good as, whereas, better means superior.  So much for those Run, Spot, Run books which talk about good, better and best.  In marketing it is more like good, best and better.

As bad as it is in the English language, I think it gets even worse when it comes to math.  You can make numbers say just about anything you want them to.  In the past couple of years, much has been made about the 1% (or is it 2%) of the country that is the wealthiest members of our society.   Then there are those that focus on the remaining 99% (or is it 98%).  I don't get political in this column, so please understand I am making no political commentaries here - only mathematical ones.  A friend of mine posted up on Facebook the other day some statistics about what happens if you look at the top 1% of the world instead of just the U.S.A.  All of a sudden a very significant portion of the country is in the top 1%.   Which statistic is more relevant?  I know people who are millionaires who think they are doing 'ok' and I know people who make very modest salaries who are as happy as can be.   The relevance is more likely in the message that someone is trying to send rather than anything absolute.

In the case I just discussed, I'll assume that all data presented was reasonably accurate.  A such, no lies were told.  No misinformation was disseminated.   Data was simply presented in a way to try and get some particular message across.  A few months ago, I wrote a column in which I asked if the U.S. has had a Democrat or Republican President more.  In the past 5 years, it has been a Democrat.  But in the last 8 out of 13 years, we had a Republican.  But, in 13 of past 21 years, a Democrat.  In 20 of Past 33 years, a Republican.  I can keep going backwards through the 20th century.  Someone attempting to make a political point is likely to use whichever statistic backs his point the best, even though it may have only minor relevance to the point.

When it comes to gambling, the numbers can be manipulated just as much, if not more so.  If I had to take a guess, I'd say that the average video poker machine in Las Vegas probably has a payback of 97%.   Now, if I'm a casino whose average video poker payback is only 96%, I might put together some advertising that simply says 'come to Vegas where the video pokers pay an average of 97%!' .  The implication is that the casino pays this as well, but that's not what they said.  On the other hand, a locals casino that likely has a significantly higher average payback is much more likely to say 'come to the XXX casino, where OUR video poker machines payback 98.5%'.  Of course, we don't really know how they calculate these averages.   With slot machines they simply can present how they paid in the prior month because there is no human error involved.  With video poker machines do they simply take a straight average of all their video poker paytables?  Do they weight \$1 machines more than nickel machines?  Maybe there is a uniform method for doing this or maybe they have enough wiggle room to give you whatever number they feel will send the right message.

Of course the 'average' payback has very limited value to an Expert Player.  In the simplest example, let's assume we have 2 casinos with 2 video poker machines each.  One casino boasts a 98% average and the other a 97% average.  So, should you head over to the one with the higher average?  What if their 2 machines each have a 98% payback.  But the other casino has one paying 94.5% and the other paying 99.5%.   The only machine out of the bunch worth playing is this last one, even though the casino's average is lower than the others.

One of the local casinos here boasts of having the most machines paying over 100%.  I'll assume that they are telling the truth with this, but that doesn't mean that their machines don't pay 100.01% and that all of the machines below 100% pay 96%.  Now, to the experienced Player you will still happily seek out the 100.01% machine, but this doesn't mean that you can just show up and sit down at any machine and know that you are playing 100.01%.  At the same time, you would be better off finding a 101% machine at another casino that may have only 3 machines of this type and whose average machine pays 96%!

So, what's my point?  Don't be sucked in by numbers that can be made to say anything they want.  The payback of a machine is an absolute number, calculated with mathematical certainty.  It doesn't mean that every time you play (even if it is for a few hours), that you will experience this payback.  It does mean that over time, your experience should begin to approximate this theoretical payback, assuming you are using the right strategy.  When you look at the paytable on a machine, that tells you the payback of the machine.  This is all that matters.  Marketing numbers that the casinos produce are just that - numbers they produce.  After all, why trust the group that has actually made better, better than best?

# Think Loss Rate

How much of a difference is there in terms of payback from one casino game to another?  Most table games have a payback between 97 and 99.5%.  Video Poker can range from about 95% to 101%.  Slot machines probably range from about 85% up to 95%.  Sidebets, quite frankly are all over the place, ranging from just over the legal limit of 75% and going up to the low-mid 90%.  While there is a lot of overlap, one of the largest determining factors is strategy.  More complex strategy means a combination of more human error and/or Players not even trying to follow it.  Simple strategy is much easier to learn and follow.  Three Card Poker has one simple strategy rule.  Follow it and you should approach the theoretical payback of about 98%.  Don't follow it and you can only do worse.

Video Poker has paybacks considerably higher.  Not all of the versions, but you can still find plenty of them well above 98%.  Video Poker's strategy, however, is far more complex than Three Card Poker's strategy.  The average Video Poker machine has more than 30 different strategy items that need to be memorized and in the appropriate order so that you know how to play the hand.  So, first you need to review the hand and determine the realistic ways the hand can be played and then you have to know which of these ways has the highest expected value, which tells us which way the hand should be played.

In most games, many of the hands are pretty obvious even if you knew little.  If you're dealt a 6-7-8 in Three Card Poker, I don't think you need to have read a book to know what to do.  What if you are dealt K-3-2?  What about Q-8-2?  What about Q-3-2?   For each hand, the Player is really asking himself if he is better off Playing or Folding.  Those are the only two options in Three Card Poker.  The answer is pretty obvious for the Straight and a good deal less obvious for the other three hands.  The strategy is determined by the math behind the question of whether the Player is better off Folding or Playing.  By Folding, the Player forfeits his original wager (one unit).  By Playing, he wagers an additional unit.  If Playing can return at least that additional unit (on average), then the hand is worth Playing.   The Player does not have to perform some complex calculation on each hand.  The decision is to Play or Fold and the math works out very neatly.  For every hand stronger or equal to Q-6-4 the Player is better off Playing.  For Q-6-3 or less, he is better of Folding.  You've just become an expert at Three Card Poker strategy.

Video Poker is not nearly this simple.  First of all, there is no folding and no additional wagers.  You make an original wager and your only goal is to maximize the amount of money you get back on average for each hand.  If you're dealt a Straight off the deal, there isn't much to think about - unless of course it is also a 4-Card Straight Flush or a 4-Card Royal - then what?  What if you're dealt Three of a Kind and 3-Card Royal?  How about a Pair and a 4-Card Flush?  Does it matter if it is a High Pair or a Low Pair?  (Yes, it does!)

In Video Poker, the hands are categorized into about 30-40 different hand ranks and partial hand ranks.   Each of these is assigned an expected value.  This expected value is calculated by looking at ALL the possible draws for that hand and tabulating the total units won for each final winning hand.  We then divide this total by the number of possible draws so that we can compare apples to apples.  So, to look at a simple example.  Suppose you are dealt the following hand:

4♥        5♥        6♥        7♥        8♦

The decision here should NOT be driven by your favorite Clint Eastwood line ("are you feeling lucky, punk?").  It should be driven by the math.   The straight has an expected value ("EV") of 4.00.  There is no draw in this case and the EV is simply the payout of the hand.  If you decide to discard the 8, there are 47 possible draws.  2 will result in a Straight Flush, 5 will result in a Straight (remember that you would have discarded a card that could also have made it a Straight) and 7 that will result in a Flush.  All other cards result in a losing hand.  So, do you throw away the sure 4 units to go for the Straight Flush?  When we add up the payouts of the winning hands, we get 162 units (2 x 50, 5 x 4, 7 x 6).  We divide this by 47 (the number of possible draws) and get 3.45.  As this is less than the EV of the Straight, we keep the Straight.  In the long run, this will be the better move.

While most Player would play this correctly (I guess?), the simple reality is that except for those that learn the right strategy, there will be a significant number of Players who will NOT play this correctly.  Throw in the roughly 25% of hands that require a real decision and the casinos can count on Player error to help pad their winnings.  This is why they can offer the 99.5% paybacks on so many full-pay jacks or better Video Poker.   Someone like myself might sit down and get the 99.5%, but the vast majority of Players will play well below this level.   They are likely to play in the 97-98% range if they have some idea of what is going on and perhaps as little as 95% if they just 'wing it'.   The difference between 99.5% and 96% may not seem like a lot, but I always suggest you turn that around to the loss rate - 0.5% vs 4%.  Now there is a 700% increase from one to the other.  The impact to your bankroll could be staggering.

# Frustration

I consider myself to be a very competitive person.  Anybody who has ever played against me in a board game or on a sports field is pretty aware of this.   I play fair and hard.  I'll never cheat and don't throw tantrums.  But I really hate to lose.  So, you can only imagine what I feel like when I'm having 'one of those nights' while playing video poker.   Gambling isn't exactly the type of thing one does if they hate to lose.  Even if you're playing video poker or blackjack, games that are near 100%, you're still going to lose more than 50% of the time over short sessions.  Not a bad record if you're the Marlins, but I prefer to win, well, closer to 100% of the time.

When I'm on the sports field, I have a significant amount of control in the outcome.  If I'm playing tennis, well, it is just about all on me.  If I'm playing softball, I can do my best to get on base when I'm at bat and make all the plays that come to me.  I can't help my right fielder catch the ball, however.  In this regard, gambling is more of a team sport than a single Player sport.  I'm an expert at just about any game in the casino that I will sit down to play.  So, I can make sure that I'll play each hand the way I should to maximize my overall payback.

Unfortunately, luck still plays a significant portion of casino gambling (kind of like my right fielder catching the flyball?).  I can't control which hands I'm dealt.  In the long run, I know I will get my fair share of each type of hand.  In a given night, the difference between winning and losing is about getting your fair share of key hands.  You're not going to make money off of 4-Card Straights, so you don't usually keep track of how many you got.

When we look at the final paying hands of video poker, it should be no surprise that most of the payback comes from the bottom 3 hands.  Jacks or better gives us about 21-22% of our payback.  Two Pair gives us 26%, and Three of a Kind gives us another 20-21%.  This is almost 70% of a total of 99.5% payback.  Straights give us over 4%, Flushes over 6% and and Full Houses around 10%.  That brings us to 90%.   Four of a Kinds give us about 6%, Straight Flushes a mere 0.5% and Royal Flushes the remaining 2%.

The more common a hand is, the more likely no matter how weird your session is going that at the end of it, you're going to have very close to the number of those hand that you are supposed to.   So, if you play 3000 hands and the average shows that you should have about 650 High Pairs, you're not going to find out that you only had 500 of them.  Maybe you have 630 on a bad night and 670 on a good night, but you'll get very close to the 21-22% payback you are supposed to.

On the other end of the spectrum is the Royal Flush.  If you play 3000 hands, you're well below the roughly 40,000 hands it takes to play to catch a Royal.  If you play a session and miss the Royal, you're inherently playing at 97.5%.  If you hit one then, well, you're assuredly playing well over 100%.   As a result, there really isn't a lot to discuss where the Royal is concerned.  It is literally hit or miss.  Straight Flushes simply don't add enough to the mix and are also so rare that you can't really look to them for a good or bad night.

The critical hand is the Four of a Kinds.  Earlier I said that they make up 6% of the payback.  That is on a jacks or better game.  Move to Bonus or Double Bonus or Double Double Bonus and these number goes way up.   You win or lose in these games based on two key factors.  Do you get your fair share of Quads and which Quads do you get (when playing the bonus games)?   If you play 3000 hands, you can 'expect' to hit about 7 Four of a Kinds.  It would not be uncommon to play this many hands and get only 2 or 3.  If you have one of these nights, you're not likely to walk out a winner.  Quite frankly, you may not walk out with any of your bankroll left.  Fortunately, it is just as common to get 10 or 11 of them.  In these cases, you are very likely to walk out a winner.  If you're playing Double Double, you'll also want to hit some of the bonus Quads and/or the 'double' bonus quad with one of the kickers.

Playing the right strategy is, of course, a critical component of getting your fair share of Four of a Kinds.  But, the right strategy does only so much to make the 5th card in Quad 3's also be a 2, 4 or Ace.  Sometimes it just takes luck to have that good night.  Sometimes my right fielder actually catches the ball.  All I can do is hope.

# Psychological Warfare - How they 'rig' slot machines

A few years ago, I wrote a column about a story I read in The Economist magazine.  It described a study done testing the impact of near misses on a slot machine on the human brain.  What the researchers found out was that near misses generated almost an identical reaction in the brain as an actual win.  So, if bar-bar-plum (a loser) can make the Player feel almost as good as bar-bar-bar (as winner), all the manufacturers have to do is figure out how make near misses show up a lot and Players will feel like their winning almost all the time.  Fortunately, the regulations and the technology do not make this much of a challenge.  Slot machines can legally be programmed to generate a disproportionate number of near misses relative to what might be considered random.   So, while they might throw in some fruit salad once in a while as an ugly loser, most of your losses will appear to be 'oh so close' to winners.

Now, a new study was released this week that says the bells and whistles used on slot machines makes the Player feel like he is winning even when he isn't.  The days of coins dropping out of the slot are virtually gone, so the casinos added sound effects to the machine.  When you used to hit a cherry and get 2 coins back and heard klink-klink, this was simply not the same as hearing 20 or 100 coins going klink-klink-klink.  But, in the digital age, no one says the sound effects has to mimic the actual win.  So, the casinos can have a simple 2-coin win sound a lot like a 10-coin win.   To prove the theory, the researchers had slot Players play with sound and without sound.  Those with sound had a stronger impression that they were winning, even when they weren't.

While this latter concept can be used for video poker, it holds a little less water because in most varieties of video poker there is no such thing as winning but really losing.  While many hands in video poker result in a push - which may FEEL like winning because your original wager is returned (i.e. Jacks or Better), there is generally no hand that returns only a portion of your original wager.  With the new generation of slot machines it is not uncommon to wager dozens of coins.  Frequently, a 'win' will result in getting only a fraction of your wager back.  Did you really win?  If you wager 20 coins and get back 5, is this a win or a loss.  Admittedly, I am the first to argue that once you wager the money it is lost and any money you get back is a 'victory'.  This seems much more applicable to table games where you play 30-40 hands/hour rather than a slot or video poker machine where you can play hundreds of hands per hour and repeatedly wagering 20 and returning 5 can quickly wipe out your bankroll.

So, what is a Player to do when faced with all of this psychological warfare used by the casinos?  Ironically, you have to use your own type of science against them.  The science of math.  Yes, with the exceptions of some varieties of video poker, the math says that in the long run you will lose.  I've written many times that you need to look at casino games as a form of entertainment.  The question is do you want your night of entertainment to cost \$20-\$40 or to cost \$100-\$200?  I'm guessing that you'll get a lot more value for your money if you spend less money.  Most of the games in the casino are built to allow the Player to win about a third of the time over a 3 hour session.  This assumes that you learn to play each game correctly and try to pick the right games/paytables to play.
While I strongly advocate for playing video poker, if you wind up playing a jacks or better that pays 6-5 (Full House/Flush), you'll be playing a game that has a payback below 96% and your chances of winning will decrease considerably.   In similar fashion, playing a full-pay game has limited value if you don't learn the right strategy.  Casinos rely on these two factors for games like video poker.  Slots have no strategy and inherently have lower paybacks, so they need to come up with ways to essentially fool the Player into thinking he is doing better than he actually is.  Video Poker doesn't need to create artificial near misses.  A deck of cards and a dealt hand do an amazing job of creating these in its natural random fashion.

To combat the near misses and the bells and whistles of the casino requires doing a little bit of homework to learn which games to play and to learn the right strategy for those games.    It requires some discipline to stick to those strategies and to seek out the right games.  Math can be your rock to the casino's 'psychological warfare' scissors.

# The Nature of the Game

My elder son has finished up his year in college and came home the other day.  As we do our best to keep him entertained while in Vegas, we went to the Laugh Factory at the Tropicana the other night.  Invariably, when comedians are in Las Vegas, they will tell jokes about the dry heat and about losing money while gambling.  I think I've been very honest about the odds of long-term winning while gambling.  With the rare exception of some tough to find video poker games and/or the ability to count in blackjack, you're simply not going to win in the long term.  But, this doesn't mean that you have to 'lose your shirt' either.

A few weeks ago, I showed how playing blackjack for an hour, a \$5 Player should expect to lose only a little over \$1/hour.  This, of course, assumes playing properly.  If you are too timid to double down on soft hands, or don't like splitting 2's looking into a 7, then, well, all bets are off as to what your payback will really be.  The comedian was hopefully joking when he talked about struggling to add up his cards while playing blackjack.  If you're really struggling with this, maybe you should try Casino War or Three Card Poker.

In that same column where I talked about the average you can expect to lose while playing blackjack, I also spoke of the average you can expect to lose while playing full-pay jacks or better video poker.  As the two games have similar paybacks, the only real difference is the average amount you wager in an hour of each game.  Much to many Player's surprise, a max-coin quarter video poker actually wagers more in an hour than a \$5 blackjack Player.  That said, however, the game of video poker is far more volatile and while the average loss rate by only be a couple of bucks an hour (depending on speed of play), actual results will wind up all over the place.  Blackjack is a much less volatile game and we will find that our actual results will really tend to be very close to the theoretical amount.

To help illustrate this point, I ran 100,000 multi-hour sessions of blackjack, each consisting of 100 hands.  I then tabulated the amount won or lost, rounding to the nearest dollar.   First of all, the Player had a winning session nearly 46% of the time.  He lost 49% of the time, with the remainder being breaking even.  Around 32% of the time, the Player will wind up within \$20 of his starting point, with only a slight slant towards the losing side.    He will wind up within \$40 of his original bankroll more than 55% of the time.   He will wind up losing \$100 or more only 5% of the time.  To be clear, this is NOT the same as saying that if he starts with \$100, he will go 'bust' only 5% of the time.  The simulation I ran does NOT take into account a Player who may have at some point been down more than \$100 and then came back to lose less than \$100.  This will not be a huge number, but it will add to the total.

I'm not downplaying the impact of losing \$100.  This is not a small amount and could be considered to be a high cost for 2+ hours of entertainment.  At the same time, we are only talking about a 1 in 20 chance.  At the same time, the Player has a 4.4% chance of WINNING \$100 or more.   That's why it is called gambling.

But, the overall point is that the notion that everytime you gamble you're going to lose your shirt is simply not accurate.  If we assume that 'paying' up to \$25 is a fair price for the 2-3 hours of entertainment value, then we find that the Player will meet this goal 62% of the time.  In fact of this 62%, he will actually wind up winning money nearly 75% of the time.

As stated earlier, this all assumes playing properly.  This tends to be what trips up Players far more often that the basic nature of the game.  Blackjack has a payback of about 99.5% when played properly.  Played improperly, the payback could drop dramatically,  If you drop it to 98%, which is still a respectable payback for most table games, this may not seem like a lot.  However, turned around, it means the casino advantage increases fourfold.  If I were to simulate such a strategy, we would find that the numbers are not so generous to the Player, and the likelihood of losing one's shirt will go up considerably.

Thus, while the nature of the game it still one where the Player will lose in the long run, the Player can still greatly control (within reason), just how much will be lost by learning to play using the right strategy.

# 3-Card Straight Flushes and Deuces Wild

I love getting together with friends.  Invariably, I get asked questions about casino games that becomes fodder for a column.  This past week, I got several possible topics based on some questions I was peppered with.  One dealt with 3-Card Straight Flushes.  Another dealt with which High Cards to hold and when.  Lastly, I was 'informed' that Deuces Wild is a bad game to play because of poor payback.  Admittedly, these came from friends who will openly state that they are not expert Players.  One of these friends also asked me what benefit there was to the casino to offer games with a Player advantage.  The simple answer to this is that so few people play these games correctly, that they're willing to allow 1% or 0.1% or maybe 0.01% of the people to make a few dollars at the expense of the other 99+%.  The rest of the questions I was asked only proves the point.  If you don't know when to hold a 3-Card Straight Flush or which High Cards to hold, the odds are (no pun intended) is that you're not going to play the 100+% game at 100+%.

So, I'll use this week's column to answer their questions and hopefully educate some of you on how to move a few steps closer to Expert Strategy.  The first question dealt with 3-Card Straight Flushes.  Normally, Straight Flushes are the black sheep of hands.   They should occur every 9000+ hands, but because so many people ignore the 3-Card Straight Flushes, they tend to be even more rare.  The person who asked the question actually spoke of how often he hit them, which was surprising.  But, not after he told me that he tends to throw Low Pairs in favor of 3-Card Straight Flushes.   This is not such a good idea.

While 3-Card Straight Flushes tend to be forgotten, you don't want to over value them either.  A Low Pair outranks EVERY 3-Card Straight Flush (NOT 3-Card Royals, however).  When you consider that in jacks or better, every 3-Card Straight Flush, even Inside and Double Inside ones are playable, this can lead to a lot of mistakes if you throw away the Low Pair.  When you consider that a large percentage of 3-Card Straight Flushes are also Low Pairs, this error will prove to be very costly to your bankroll.  I don't have the space here to list out all the strategy for 3-Card Straight Flush, but for now, let's go with, they are below a Low Pair and every one of them is playable.  You also play 4-Card Straights (not Inside) over the comparable 3-Card Straight Flush.  The critical part is that you play even the most awful looking 3-Card Straight Flush over a single High Card, except for the Double Inside with 0 High Cards, which only outranks the Razgu.

Next up in the question bin was how to play High Cards.  Generally speaking, the goal is to keep the suited High Cards.  So, if dealt 3 High Cards and 2 are the same suit, those are the two we play.  If all 3 cards are of a different suit, but one is an Ace, we play the 2 High Cards that are not an Ace.  If it is J-Q-K all of different suits, we hold all 3 cards.   If you have 2 unsuited High Cards and one is an Ace, then you keep both.   That describes which ones to keep.  As for when to keep them, see the earlier part of this article and you learn that we frequently keep a 3-Card Straight Flush made up of 3 Low Cards over a High Card,  So, if dealt 3-4-5 suited and JQ (off suit), we hold the 3-4-5.

That brings us to the last question regarding Deuces Wild.  Should it be avoided due to low payback.  I'm not sure where my friend got this notion.  Perhaps he some bad experiences playing it.  If he tried to use his jacks or better strategy on a Deuces Wild game, things would not be pretty.  In reality, Deuces Wild can offer some of the highest paybacks in the casino.  However, like all version of video poker, you have to check the paytable.  Sometimes, different variations of video poker are given different unique names and sometimes, they just scramble the paytable a bit and still call it Deuces Wild.  So, pay attention to the paytable and make sure you are using the right strategy for that paytable.

While I do my best to give tips out in my column, the only way you're really going to become an expert Player is by learning the complete strategy for the games.  I continue to offer our 3 best selling video poker books for \$5 each (includes shipping and handling) to my loyal Gaming Today readers.  You can choose from Expert Video Poker for Las Vegas, which explains everything about the game from start to finish, Winning Strategies for Video Poker, which contains the strategy tables for 60+ video poker variations or Video Poker: America's National Game of Chance, which contains over 200 pages of my father's best columns, stories and quizzes.  If you like to learn from anecdotal stories, this is the book for you.  Just send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89134.

# Win Frequency Overrated

Why does a good blackjack Player stick on bustable hands aginst a Dealer 6?  The quick answer is that with a 6 upcard, the Dealer is likely to bust.  Of course, this is not completely accurate.  The Dealer's bust rate with a 6 is 'only' 42%, which means 58% of the time, he won't bust.  So, first he is not 'likely' to bust.  He is just more likely to bust with a 6 than with any other card.  58% of the time, he will wind up with a 17 through 21 and will beat your hand.  So, why stick?  Well, we need to take into account how often the Player will bust if he takes a hit.  If the Player busts, it doesn't matter what the Dealer does.  This is all a wordy way of saying that the Player is more likely to win if he sticks than if he hits.  Or, in other words, his expected value is higher if sticks than if he hits.  Depending on his specific hand, it might be a relatively small difference between these expected values or it might be a big difference.  But, the difference doesn't matter.  The correct play is the one that has the highest expected value.  This is the key thing to learn for EVERY casino game.

Blackjack is essentially a binary game.  You either win or lose your base wager.  With the exception of blackjacks itself and Doubles and Splits, the wager is a single unit and the outcome is either even money or the Player loses.  Thus, the critical factor becomes win frequency because for the most part, one win is worth as much as any other win.  In video poker, the outcomes are a bit more varied and thus the analysis is actually a good deal more complex.  If we define 'winning' as any hand that is Jacks or Better, that leaves us with a win frequency of 45% (roughly), but not all wins are created equal.  There are essentially 9 different levels of winning, ranging from Royal Flush down to a High Pair.  The payouts range from 800 for 1 down to a push (which is all you get paid when you have a High Pair).

This explains why when playing video poker the win frequency is not very relevant.  Take the following hand as an example:

8♣       9♣       10♣     Q♣      Q♥

There are two ways to play this hand.  A Player can keep the pair of Queens and have a sure winner.  He'll still have a chance to improve to Two Pair, Trips, Full House or Quads.  But, his win frequency will be 100%.  His other choice is to go for the 4-Card Inside Straight Flush.  If he chooses to go this route, his win frequency will be around 30%.  Of the 47 draws, 8 will result in a Flush, 3 in a Straight, 2 in a High Pair and 1 as a Straight Flush.  The other 34 will result in a loss.  If you're motivated by win percentage, then the right play is to stick with the pair of Queens.  If you're motivated to use the proper strategy, you use expected value to guide you.  When the math is all done, we find that the 4-Card Inside Straight Flush has an expected value of 2.39.  The Pair of Queens has an expected value of 1.54.  It's not really much of a choice.  The 4-Card Inside Straight Flush is by far the superior play.

Decisions for casino games are made based on the criteria of expected value.  This is not a concept unique to any particular game.  The same methodology that developed blackjack strategy is essentially the same one used for video poker or Three Card Poker or Ultimate Texas Hold'em.  Some of the toughest decisions are of the type I just described where the Player might have to give up a sure winner to go for a hand that in the long run will pay more, but will have a significantly lower win frequency.  The example I gave here is probably not all that hard to follow.  Since the sure win is only a single unit, it won't feel like you are giving up much.

But, you may have to make a similar decision if you are dealt a Flush that is also a 4-Card Royal.  If you're playing max-coin quarters, you'll be giving up a sure \$7.50 to go for that big payout of \$1000.   IF you're a dollar player, you'll be risking \$30 to win \$4000.  Definitely worth it, but it might just be a little harder to walk away from that sure \$30.

# The Cheap Cost of Entertainment

I received an e-mail this week from a loyal reader who was questioning some of the numbers from my recent column.  The column was discussing the definition of payback and had the amount of the buy-in is irrelevant to the discussion of the payback.  The example I cited was discussing someone who sat down to play 100 hands of \$5 blackjack.  With a payback of 99.5%, a Player can expect to lose \$2.80.  The point of the column was to discuss how this \$2.80 will not change no matter how much the Player buys in for.  If he buys in for \$20 or \$100 he will still lose the same \$2.80.  All that changes is the percent of the Player's bankroll that he will lose.  The \$2.80 is a fixed amount.

My reader questioned this calculation.  Not so much for its pure math, but because I 'ignored' the situation where the Player might lose his first 4 hands be 'bankrupt.'  My reader is quite correct.  The situation I described ignored the numerous circumstances in which the Player will actually lose his entire buy-in before reaching 100 hands.  With a buy-in of only \$20, this is fairly likely to occur.  Roughly 1 in 16 times, he will lose the first 4 hands and be done right then and there.  This doesn't even include the times he may double or split in the first couple of hands and go broke before even 4 hands.

That said, this was not really the purpose behind my calculation.  Since the point was to show how the expected loss rate does not change based on the buy-in, I could have just as easily used a \$100 and \$500 buy-in in my examples.  With a \$100 buy-in, it is far less likely that the Player will go broke before 100 hands.  However, my reader does bring up a very, very important point about the importance of being properly bankrolled for any game.  The amount will vary greatly from game to game, mostly dependent on the volatility of the game.  Blackjack is a relatively low volatility game so \$100 would be good enough most of the time.

The second part that the reader questioned was my math regarding the anticipated loss while playing 1000 hands of full-pay jacks or better video poker at max-coin quarters.  I said that it would be \$6.25.  My reader wished that his expected loss was only \$6.25 and that this would make it 'cheap entertainment'.  Well, I stand by this number.  On a max-coin machine, the Player will wager \$1.25 per hand.  Over 1000 hands, he will wager \$1250.  A full-pay jacks or better machine pays about 99.5%.  Losing just 0.5% of his total wager brings us back to \$6.25.

Of course, this is the long term average.  Unlike blackjack, video poker has a much higher volatility.  Blackjack is a lot like a coin toss.  You win about half the hands.  You lose about half the hands.  Except for actual blackjacks, splits and double downs, all  payouts are even money to the original wager.  There tends not to be huge swings in how you will do.  After 1000 hands, you'd probably be very close to the theoretical 99.5% for blackjack.

Video poker is quite different.  You 'win' about 45% of your hands, but an overwhelming majority of these are really pushes (High Pair).  The rest of the payouts range from even money all the way up to 800 for 1 for a Royal Flush.  That Royal accounts for about 2% of the total payback.  This means that until you hit the Royal, you're only playing a 97.5% game which means the loss rate over 1000 hands would be closer to \$20.  Eventually, you will hit that Royal and for that 1000 hands, you will have a significant win.  When you add up the TOTAL amount you wager and multiply it by 0.5% (the loss rate), the total amount you've lost should be very close to this number.  At the same time, if you hit more Royals than 'average', you'll probably be up significantly.  If you hit less than average, your loss rate is likely to be quite a bit more.

When we tie together the two thoughts that my reader brought to me, we realize the importance of being properly bankrolled when playing video poker.  Given the volatility of the game, it becomes even more important to make sure you are in the game until you get to one of the big hands.  In jacks or better, this mostly means the Royal.  In double double bonus video poker, you have the luxury of a few of the Quad payouts AND the Royal.

I had an opportunity to experience this first hand twice this past week.  I ventured out on 2 separate occasions to play video poker.  In one case, I was down about \$40-\$50 when I hit two solid hands and came all the way back and left even.  In the other case, I hung around even most of the night.  I was down about \$5 when I hit I was dealt 3 Aces on a five-play double double machines.  Short of being dealt quads, this is about all you can hope for.  Now all you have to do is hit the Quads.  On the fifth hand, I was dealt an Ace and a 3.  Not only did I hit the 4 Aces, I hit the bonus 4 Aces.  About 5 hands later, I left up with a nice victory.  In the case of my first night, if I had brought only \$40 with me, my bankroll would've been gone and I never would've made it to the big hands.  Also, if I weren't using proper strategy, my losses up to that point would have been that much larger, and even a \$60 or \$80 bankroll might not have lasted as long as it needed to.

Proper strategy and proper bankrolling are keys to achieving the theoretical paybacks of a casino game.  In turn, this is what can lead you have 'only' that much of an expected loss rate and get a cheap night of entertainment.

# Customer Power

There were two different articles that appeared today that on the surface appeared only marginally related.  Yet, to someone like me, I found that they were far more important to one another than meets the eye.  The first article discussed the upcoming building boom here in Las Vegas.  Several major casino building projects are in the planning stages and Las Vegas may, in a few years, welcome its first major new casinos in several years.  One of the sub-plots of this article was some local columnists discussing what they felt was needed to build a perfect casino.

In reading these suggestions, I can't say that I have a lot of hope that many (any?) of them would be implemented.  One of the suggestions dealt with the idea of putting the attractions near the front of the casino and the casino way in the back.  Another dealt with moving the restaurants closer to the self-parking garages.  Yet another suggested that casinos go smoke-free (I'm all for this one!).  The one that got my attention was the one that requested that casinos do away with blackjack that pays only 6 to 5 (instead of the traditional 3 to 2).  This one also mentioned better paying slots, but the focus was on blackjack.

For those who have read my column over the years, you know I'm all for 3 to 2 blackjack and do my best to warn people about playing 6 to 5.  Roughly, 1 in 21 hands will be a blackajck.  That's about 2 hands per hour.  If you're a \$5 player, this will cost you about \$3/hour.  This may not seem like a lot, but it will increase your loss rate by about 300%!  A 99.5% game quickly becomes a 98% game and now you're playing a game that requires a great deal of strategy with a payback that is in the same range as many of the table games with little strategy.

That brings me to the 2nd article I read today.  It talked about how gaming revenue in Las Vegas was WAY UP compared to last year, for the month of February.  A significant portion of this was due to the Chinese New Year occurring in February of this year vs. January of last year.  But, even when this is accounted for, revenue was still up.  Revenue on the strip was up even more than the rest of the city.  It is on the strip that we find virtually all of the 6 to 5 blackjack tables.

6 to 5 blackjack was created because over time blackjack Players were getting better and better and the hold at blackjack tables was dropping.  Casinos have a lot of overhead to cover - from Dealer salaries to the massive electricity bills.  While every business should run efficiently, this is not exactly the case of they should keep the customer happy at all costs even if it means eking out a small profit.  Casinos are expected to make huge profits at the tables and slots to help offset many of the things they provide at low cost or free.  I have no idea what the cost is to present the pirate ship battle at the Treasure Island, but they've been doing for about 20 years for free every night.  The money to do these performances comes from the gambling side of things.

So, the casino decided to come up with a way to greatly increase the house edge on blackjack.  They could have tweaked the rules a bit - use larger shoes, limit when the Player can double, etc..  But, these have limited practical impact to the house edge.  One of the most common Player errors is not doubling on soft hands when they should.  So, eliminating this as an option doesn't really help the casino at all.  So, they chose to pay blackjack at 6 to 5 and take a bit out of the bankroll of the good and bad player alike.

Now, if you've been going to a buffet on the strip that give you free drinks included with the price of the buffet and all of a sudden they tell you that they're going to start charging you for your sodas, you might think twice about where to eat.  Yet, for some reason, paying 6 to 5 didn't have much of an impact to the amount of money people wagered on blackjack.

However, if we look at the report about Las Vegas gaming revenue for February, we find that the biggest spike occurred on the Strip.  If you head out to the casinos in the 'suburbs' where you find more local Players, you'll find almost NO 6 to 5 blackjack.  Local players tend to be better players (or they don't stay local very long) and the better player knows that playing 6 to 5 blackjack is very hard on your wallet.   You need to learn a complex strategy just to be able to earn a 98% payback?  A Player can sit and play a relatively simple game like Three Card Poker and earn the same payback and have a chance for a larger single payoff (with Trips of a 3-Card Straight Flush).  And, if you're not an accomplished blackjack Player, your real payback could easily drop to 95-96% which leaves a Player with very little chance of having a winning session.

But, if nobody complains about having to pay for the soda, AND they have the same number of customers this month as they did last month (or more!), then there is little reason for the casino to go back and give out free drinks at the buffet.  This is even more true at the tables.  If a Player is just going to sit and take his 6 to 5 payout with little regard to the impact to his wallet, who can blame the casinos for making this their basic offering.

# The Definition of Payback

I never get through a holiday without a serious discussion of what I do for a living with someone I've never met before.  Family (and friend) functions tend to bring together people for a large meal leaving them with loads of time to discuss all sorts of things.  As I have one of the more unique jobs around, my vocation tends to take up a larger than proportionate amount of the time we spend together.  This past holiday season was no different.

First I listened to one person tell me how he has a system for roulette.  Admittedly, he didn't get a chance to explain it to me in much details when I had to tell him that it doesn't work.  No system does.  He told me how each time he came to Vegas, he would use this system and invariably walk away with a few hundred dollars.  Of course, his sample size was about 6-12 sessions, which isn't exactly statistically significant.  Based on what he told me, I commend my new found friend for his discipline which can be an important part of any successful gambling story.  Know when to get out when you are ahead.  But, that said, if you really have a system that nets you \$400 in an hour or two, it is forever repeatable, which means you do it every night and then you send out a team of people to repeat your system.  No 'real' system could work only if you use it once every few weeks.

Next up in the discussion came my favorite topic (ha!) - slot machines.  The system here was to attempt to outguess when the machine was going to pay off by altering the amount wagered for each 'pull'.  It was hard to keep a straight face when we got to this point.  I've heard of people varying their bet when playing blackjack in an attempt to guess the next cards.  If you do this well, it is card counting.  If you simply try to outsmart the shoe, you're just guessing.  If you try it with a slot machine, you are definitely guessing.

We've all seen the disclaimer that says 'past performance is not an indication of future returns'.  Nothing could be more true with slot machines.  What happened in the last spin has absolutely no bearing on what happens in the next one.  A slot machine is programmed to have a winning spin some percent of the time.  Every time you spin the wheels, the chance of winning is this exact percent.  With some combinatorial math we can also say that the probability of having X winning hands in Y spins will be some percent (assuming we know the probability of winning in any given spin).  But that is only true for the next Y spins.  We absolutely, positively CANNOT use any of the past spins in our calculation.  If the past 100 spins were losers, the probability of winning on the next spin is still whatever it is.  If the past 100 spins were winners, the probability of winning on the next spin is the same percent.

When I suggested to my new friend that he might want to avoid slot machines due to their 92+% payback, which makes them some of the worst payers in the casino.  Of course, when you look at the machine you have no way of knowing if it is programmed at 98% of 85%, which is as much as part of the problem as the average of 92+%.  My friend wanted to know how this payback was calculated especially when taking into account the way he plays - altering his wager from spin to spin.

I explained that the payback used for any game is the highest payback that can be obtained by a Player assuming he plays using the best possible strategy he can.  For a game like video poker this means he uses perfect strategy to play each hand and that he plays max-coin in order to get the benefit of the 800 for 1 payout for Royal Flushes.  For slot machines, there is no strategy, so that does not impact the payback.  With slots, the impact of max-coin can frequently be even greater than with video poker.  Not only do you buy additional lines with additional wagers, you sometimes also buy additional combinations of winning hands.   As a result, playing less than max-coin can be even more punishing to your bankroll.  The payback of a slot machine thus assumes a max-coin play on each spin.

Payback (for any game) is the amount that a Player can expect to have returned out of the TOTAL amount wagered.    The amount you buy-in for is completely irrelevant to this definition.  If you sit down at a blackjack table for \$20 and play 100 hands of a \$5 table, you'll wind up wagering about \$565 (when you account for splits and double downs).  With a 99.5% payback, you can expect to lose about \$2.80.  This works out to be 14% of your buy-in, but if you had bought in for \$100 it would've been 2.8%.  Just further proof that the buy-in is not relevant to the payback discussion.

If you play 1000 hands of video poker (quarter machine, max-coin), you'll wager \$1250.  If you're playing full pay jacks or better with a 99.5% payback, you can expect to get back \$1243.75.  No matter how much you put into the machine, you should expect to have lost \$6.25.  If you play less than max-coin, your expected loss will be higher.

Slot machines are no different.  If you spin the wheels 1000 times on a nickel machine with 27 lines, you'll wager \$1800.  You won't know the exact payback of that machine, but if we use a generous 95% payback, you can expect to get back \$1710 of that \$1800 wager and sustain a \$90 loss.   If you choose to vary your bet from spin to spin, your payback might be even lower, raising your expected loss.

In all these cases, the paybacks are expected 'long-term' paybacks.  Long-term can mean different things to different games.  In a 2-3 hour session of playing, your results can and will greatly vary from the examples shown here.