# The Straight Flush Theory

Recently, a friend came over to me and announced that earlier that week he got his first ever Straight Flush.  Now, he's not an avid regular, but he does go and play from time to time.  I also have to admit that this is the first time I've ever had a friend 'brag' about his first Straight Flush as opposed to his first Royal Flush.  Somewhat jokingly, I remarked to my friend that Straight Flushes are almost as rare as Royal Flushes.  He looked at me rather oddly and replied how Straight Flushes were of course far more common.

Royal Flushes are just a particular type of Straight Flush.  So, in reality there are 10 types of Straight Flushes (A High down to 5 High).  So, it would seem that Straight Flushes should occur nine times as often as a Royal Flush as there are 9 Straight Flushes to 1 Royal.  When we look at the actual distribution of final hands for a jacks or better game, however, we find that Straight Flushes are only about 4.5 times as common as a Royal.  Why is this?

This occurs because of the payouts of the respective hands.  A Royal pays 800 for 1 vs. only 50 for 1 for a Straight Flush.  Adding to this is the fact that High Cards are worth more than Low Cards, so a 10-J-Q has a higher expected value than a 5-6-7 also because the former can turn into a variety of High Pairs.    But does this fully explain why my friend got his first Straight Flush and found it intriguing?

Quite frankly, no.  My friend is not an Expert Player.  He's never taken the time to learn the right strategy and simply plays by his own strategy.  He's a very smart guy, so he might figure out some of the strategy on his own, but no one can perform the complex calculations on the fly.  As a result, I'm 99% certain the he doesn't properly play all partial Straight Flushes.  The net is that he winds up with less Straight Flushes than he should.

Partial Royals are easy to spot because everyone is looking for them.  More than likely, most Players overplay these hands.  A 2-Card Royal is playable, but not over 4-Card Flushes and 4-Card Straights.  I wonder how many people hold a suited J-K while discarding an offsuit 10-Q that went with them.  This is a very bad move as the 2-Card Royal has an expected value of about 0.6 while the 4-Card Straight has one of 0.87.  Now, if you can pull in a 3-Card Royal, you would play this over a 4-Card Straight or a 4-Card Flush, but not a 4-Card Straight Flush (or 4-Card Inside Straight Flush).

An 800 unit top prize is quite enticing.  To get the 800 you have to have 5 coins in, so you're talking a 4000 unit payout.  Even at nickels, this is \$200.  For quarters, it is a cool \$1000 and for dollars a whopping \$4000.  Not that 50 is something to sneeze at, but it is far less impressive.  As a result, I'm guessing that most Players pick up on 4-Card Straight Flushes, but less so on a 3-Card Straight Flush unless the rest of the hand is really bad.  So, what do you do if you're dealt 3-4-5 (suited) and off-suit J-Q?    The 3-Card Straight Flush has an expected value of 0.63 while the 2 High Cards is 0.49.  Not much of a decision really.  What if the J-Q is suited?  It gets closer, but the 2-Card Royal is again at 0.60 which is below the 3-Card Straight Flush.  Be honest, how many of you play the hand this way?  How many of you go for the 2-Card Royal?  You're not killing yourself on overall payback, but you will reduce your chances of getting that Straight Flush.

For those that are not Expert Players but have picked up some strategy points, you may have heard that in jacks or better you don't play 3-Card Straights or 3-Card Flushes.  Some of these Players may have extended this to 3-Card Straight Flushes.  But this is a big mistake.  EVERY 3-Card Straight Flush is playable.   If we change our prior example to 3-4-6 (suited) and off-suit JQ, how does this change our strategy?  It doesn't.  The 3-Card Inside Straight Flush is STILL HIGHER than the 2 High Cards (0.53 to 0.49).  However, this is now enough to keep the 2-Card Royal if the 2 High Cards are suited.

What if it is 3-5-7 suited?  This is the bottom of the barrel, but it is still above a Razgu.  So, if the neither of the other cards are High Cards, you'd still hold the 3-Card Double Inside Straight Flush (with an expected value of 0.44).  But, this is below a single High Card expected value so even if you have just a Jack through Ace, you would hold this singleton over the 3-Card Straight Flush.

Partial Straight Flushes account for only about 2% of our playable hands.  But playing them right is what will increase the number of Straight Flushes you wind up with and in turn, increase your payback closer to the theoretical amount.