Calculating expected value in video poker

Last week, I discussed how every strategy decision made in the casino is based on the expected value. In the game of Blackjack, there are usually only two, maybe three realistic ways to play a hand.

Many hands are fairly obvious if you get even the most basic strategy down. Not much to choose from when you’re dealt a 3-4. You can’t split them. You’re not going to double and hopefully you won’t stick on it, even if the Dealer has a 6.

Video Poker is a bit more complex, sort of. You’re dealt five cards. There are now 32 distinct ways, theoretically, that you can play the hand. But, like Blackjack, most of the ways get eliminated with just the slightest amount of strategy.

You’re not going to throw away a pair to hold two High Cards. If you have a 4-card Straight, you’re not going to keep a random 5 along with a King. In reality, there are usually just two or three realistic ways you’re going to play the hand, with a handful of hands that might stretch this out to four or five.

To hopefully help you understand how we calculate the expected value, I’m going to start with a simple example. Let’s say you are dealt the following:

5 of Diamonds - 5 of Spades - 5 of Clubs - 8 of Clubs - 10 of Hearts

I think we can all figure out how to play this hand. I purposefully designed it so that there is no question at all. There is not even a 3-Card Straight Flush for you to think about.

How do we calculate the expected value for this hand? Before I even start, let me be clear to the newbies out there. You do not have to learn how to calculate this on your own, nor do it on the fly while sitting there. I am simply demonstrating how it is calculated so that you see that there is real math behind the expected value and the strategy overall.

As most versions of video poker use a 52-card deck, there are 47 cards remaining in the deck after the 5-card deal. We are going to start with the obvious strategy here and hold the three 5’s. That leaves us drawing two cards. There are 1,081 ways to draw two cards from the remaining 47 cards. This is called ‘47 choose 2’ as the order of the two cards is meaningless. Simplified the math comes down to 47 times 46 divided by 2. This is 1,081.

So, we use a computer program to figure out what the final hand will be if we were to play out all 1,081 hands. 46 of these hands will result in a Four of a Kind. Sixty-six will result in a Full House. The remaining 969 will stay Three of a Kinds.

We now add up the coins we would have won. Assuming a basic jacks or better full-pay machine, we have:

(46 x 25)+(66 x 9)+(969 x 3)=4,651

We divide this by the 1,081 possible hands and get a value of 4.30. This is the expected value of a Three of a Kind. If we were playing Bonus Poker, the Full House would pay only 8 and the expected value would be only 4.24.

When we build our strategy, we take nothing for granted. So, the computer program would actually go out and calculate the expected value of the other 31 ways to play the hand, like keeping just one 5 or just a 5-8. Obviously, the calculations for all of these will result in a lot of non-winning hands and the expected value will be much less than 4.30.

Next week, I’ll dissect a hand with a bit more decision making involved.