The Sum of Its Parts

The purpose of the last several columns has been to show that math drives everything in the casino. While the casinos have a macro approach, the Player has a more micro approach, generally looking at things in shorter sessions. Also, the Player doesn’t really care how any other Player does, only himself. When you are looking at it this way, it all begins with each hand. In the end, the total experience is nothing more than the sum of its parts.

For the Player, this means learning how to play each hand correctly. Last week, I mentioned the first leg of Expert Strategy which is to know which games to play. The second leg is knowing how to play that game using the right strategy. That is this week’s topic. It doesn’t matter what game you are playing – except slots – which has no strategy. At the other end of the spectrum is a game like Video Poker which is just full of relatively complex strategy. For those not familiar, Video Poker looks like a slot machine, but it really is essentially a table game played in video version. You are playing 5-card Draw Poker against a PAYTABLE, not against the Dealer or any other Players. This means there is no bluffing or tells. All that matters is hard cold math.

In Video Poker, to begin play, you make a wager. You are dealt five cards. You decide which to hold and which to discard. You are dealt replacement cards (called the Draw). If your final hand is a Pair of Jacks or Better (for most versions of the game), you win. The stronger your hand, the more you win. Every imaginable statistic about Video Poker can be calculated with absolute precision. There is no guessing and no estimating. There are exactly 2,598,960 unique possible hands that can be dealt from a 52-card deck. There are exactly 32 ways you can play each hand, ranging from holding all five cards to discarding all five cards. For each of the 32 ways, there is a finite number of draws – the exact number depends on how many cards are drawn.

Thanks to modern computers, we can crunch all these numbers to know the probability of any final hand given the deal and which cards you choose to discard. Let’s start with an easy example. Let’s say you are dealt the following:

5C 5D 8D 8S KH

I think it is fairly obvious how you would play this hand (as Two Pair). There are 47 possible draws if you do this as there are 47 remaining cards in the deck. Four will result in a Full House and the rest will leave the hand as a Two Pair. For a full-pay jacks or better machines, the Two Pair pays two and the Full House pays nine. We multiply each winning hand by the number of times that hand happens and sum up the values. So we get ((4 x 9) + (43 x 2)) = 122. We divide this by the number of draws (122/47) to get our expected value of 2.60. In theory there are 31 other ways to play this hand, but in the interest of saving space, you’ll have to believe me that none provide an expected value even CLOSE to this one. So, this confirms that the right way to play the hand is a Two Pair.

Many hands are easy like the previous one. Others are a bit more tricky. How would you play the following hand:

6C 6D 7D 8D 9H

So, should you play the Low Pair, the 3-Card Straight Flush or the 4-Card Straight? With the 4-Card Straight, there are again 47 possible draws. Eight will result in a Straight, which pays 4. The remaining will result in losing hands. So, we have a total payout of 32, which when divided by 47 gives us an expected value of 0.68. It gets more complex for the other two possible hands. When the numbers are crunched, we find that the 3-Card Straight Flush has an expected value of 0.63 and the Low Pair has an expected value of 0.82. The proper play is the Low Pair. How many of you have been playing this hand wrong? And, while 0.14 might not seem like a lot, it is actually HUGE given how often this hand occurs.