Expected value’s ticket to success

Jan 11, 2005 4:11 AM

Long before we put our money in a video poker machine we should have a pretty good idea about how we can successfully play these very liberal machines. Like scuba diving, it is best to study and get into condition rather than exposing ourselves to getting hurt. The first step is to learn how we define an expert player and decide if we want to become one.

Every expert player has a strong desire to play well, not only for the extra jackpots that may result, which at best will be far short of a living wage, but primarily for the satisfaction derived from having mastered the challenge of a fascinating game. The expert player knows that there is a solid mathematical foundation underlying expert play. It is simply to try to maximize the "expected value" of the hands the machine deals out to us.

Following this approach makes the expert player discriminating in both selecting what versions to play and then how to play each hand dealt.

With this background, we’ll assume you are still with us because you want to understand the mathematical basis of the game. Now, don’t get worried that we will be throwing a lot of math at you — we promise we won’t.

Intuitively, we can believe that all card games are analyzable. Given the composition of the deck plus a table of pay values, we would expect that one could figure out what kind, and how many of every playable hand could come out of the deck. Without a computer, one could reason that there are four natural royal flushes in a deck; remember we ignore the order of the cards so any hand containing the 10 through the ace of just one suit is a natural royal flush. Furthermore, the inclusion of wild cards does not change this figure.

Similarly, without wild cards there would be 624 four-of-a-kinds; each set of quads (13 in all) could be accompanied by one of the 48 other cards. But wait, that’s only true if there are no wild cards. Make "Deuces Wild" and we can find 528 without any deuce, 8,448 with one deuce and 19,008 with two deuces. How about with three deuces? ”¦ there are none! That’s because any hand with three deuces would not be played as four of a kind ”¦ its expected value is nearly 14.9 (times our bet) if we hold the bare three deuces. We arrive at that figure by looking at all the possible draws and averaging the payouts they generate. That’s where the pay schedule of the machine first enters the analysis.

Of course, given three deuces and a matching set of royal cards we’d hold all five. Simply because the pay table rewards a deuce royal at 25 for 1. If deuce royals paid less than 15 for 1 we would alter our strategy by discarding the high ones.

Following this procedure, a table of "playable" hands ranked in order of their expected values is created and the 2,598,960 hands which can be dealt out are distributed into this table. The expert player will always examine every hand and will make the hold/discard decision in the way it offers the highest expected value.

As an example, in full-pay Jacks or Better a straight is paid 4 for 1 on all machines. If dealt the hand: 6C-7D-8H-9S-JS, we can see (even without a computer) that by discarding the jack, we have 47 possible cards we might draw; eight of these complete the straight and 39 are losers. This gives us an EV of 32/47 or .68; we would get the same EV for any four-card straight which is open on both ends and has no high cards jack or higher in this version). If we checked the EV of this hand played differently, we would find no other EV to be as large. Holding the jack alone gives an EV of about .47 (but we need a computer here). If we discard the six, the 47 draws would include three pairs of jacks but only four straights. The EV would be found to be 19/47 or .40, making this the poorest possible play. Thus, we play all such hands as a four-card outside straight. Now, back to the method of analysis.

With the ranking table completed and every combination of five cards distributed to highest ranking type it matches in the table, we then multiply the EV’s by the number of corresponding hands; the sum is the theoretical maximum total payouts we would expect assuming an average run of cards.

As an illustration, in Jacks or Better, there are 36 kinds of playable hands. The best is the natural royal with its EV of 800, the worst are complete throwaways (RAZGU) with an EV of .36. Add up all the expected payouts and you find a 99.6 percent return.