Video poker: what is expected value?

The term expected value, or EV for short, appears frequently in articles and game strategies. Video poker is a game of pure mathematics, making it possible to precisely determine the EV for every play. The concept of EV is cardinal to a successful strategy for any game. So just what is meant by EV?

To put it in as simple terms as possible, the EV of any chance event is the average of all possible outcomes. When most people hear the word average they usually think of what mathematicians call the mean; that is, you add up a bunch of numbers and divide by the number of numbers. To calculate the EV, however, we must use a weighted average, not just the mean.

For example, suppose you are playing common 9/6 jacks or better video poker, and you hold an ace and a jack of the same suit. The possible payoffs are a royal flush, which pays 800-for-1, four of a kind at 25-for-1, full house at 9-for-1, flush at 6-for-1, straight at 4-for-1, three of a kind at 3-for-1, two pair at 2-for-1, a high pair at 1-for-1, and zilch (no payoff) at 0-for-1.

The mean of these possible payoffs, which is (800+25+9+6+4+3+2+1+0)/9 = 94.44, tells us absolutely nothing of any useful value. To get the EV, we must take into account the probability of each payoff, as I show below.

If you multiply the amount of your bet by the EV of the play, the result is the expected return, which is the average that you can expect to get back if you made that same play many times. For example, an EV of 1.023 would mean that you have a 2.3% advantage, and you can expect to get back an average of $10.23 for each $10.00 bet. Conversely, an EV of 0.982 would mean that you could expect to get back an average of only $9.82 for each $10.00 bet, which translates to a 1.8% disadvantage. Obviously, we would like to place bets only on events with an EV greater than one.

Now let’s apply this to video poker. For any dealt five-card hand there are thirty two possible ways to play it, and for each of these thirty two ways of playing a hand there may be many possible outcomes. Assuming an honest video poker game, each unseen card has equal probability of appearing at any time. Assuming that you’re playing a consistent strategy, this makes it possible to calculate the probability of each possible payoff. The EV of any play can then be calculated by multiplying the probability of each possible outcome by its payoff and summing the products.

Another way to look at it is to multiply the number of ways each final hand can occur by its payoff, sum those products, and divide by the total number of possible outcomes. Using that method, let’s determine the EV of one video poker play.

Suppose we’re playing 9/6 jacks or better, and we want to know the best play when dealt: Kh Qs Jh 10h 4h.

Draw 1, hold K-Q-J-10, hoping for a straight. We are drawing one card from the remaining 47 unseen cards, so there are 47 equally likely possible outcomes, as follows. Any ace or nine will make a straight, which pays 4-for-1, and there are eight such cards in the deck. Also, any king, queen or jack will make a high pair, which pays 1-for-1, and there are a total of nine such cards remaining in the deck. Any of the other 30 remaining cards will yield zero payoff, so the expected value can be computed as:

(8×4+9 x1+30×0)/47 = 0.8723

Of course, the “30 x 0” term doesn’t contribute anything to the EV, it is purely for demonstration purposes.

Draw 2: Hold K-J-10-4 (all hearts), hoping for a flush. Any heart will make a flush, which pays 6-for-1, and there are nine such cards remaining in the deck. Also, any king or jack will make a high pair, which pays 1-for-1, and there are six such cards available. None of the other 32 cards yield a payoff, so this play has a total expected value of: (9×6+6×1)/47 = 1.2766

Draw 3: Hold K-J-10 (all hearts), hoping for a royal flush. This is more complicated because we are drawing two cards instead of one. Since there are (47*46)/2 = 1,081 possible outcomes instead of just 47, I’ll skip the mathematics and tell you that the expected value is 1.3506.

Therefore, if you were dealt this same hand many times and always drew for the straight, you could expect an average return of only a little over 87 cents for each dollar bet, so you would be losing money on the play. However, if you always drew for the flush you could expect an average return of $1.27, and if you always drew for the royal you could expect an average return of $1.35 for each dollar bet. Which way would you play the hand?

Note that in each case we are usually hoping for one particular final hand, which I call the primary payoff, but part of the expected value (sometimes most of it) comes from other possible outcomes, which I call secondary payoffs. In draw number three above, for example, nearly half of the expected value comes from secondary payoffs (i.e. final hands other than a royal flush). This is good because it reduces the volatility of the game. That is, it reduces our likely bankroll fluctuations.

There are 29 other ways to play this hand, but these three give the highest expected values. If we were building a hand rank table, we now know that a three-card royal flush (abbreviated RF 3) is generally higher than a four card flush (flush 4), which in turn is higher than an open-ended four card straight draw (straight 4).

By analyzing hundreds of different hands in this way, we can build a complete hand rank table, which can then be used as a playing strategy. Once you have an accurate hand rank table, it is no longer necessary to consider specific expected values. It is the job of analysts like myself to provide such tables, so that anyone can play optimum strategy without having to understand the math. For any given dealt hand, you simply scan down the table and hold the first available combination.