How Do the Experts Know the Best Way to Play Video Poker?

1 April 2003

By Alan Krigman

Many video poker decisions are no-brainers. The choices will be intuitively obvious to anyone whose card career has passed old maid, go fish, and cut the deck for a dime. For instance, it doesn't take a poker pro or a math meister to know how to handle four sundry hearts and a club. Or three fours, a six, and a nine.

For the enigmas, there's "expert strategy." As an example, what should you do with a six, a suited queen-ace, and a pair of fours in a jacks-or-better game? Hold the fours hoping they'll mature into two pair, triplets, quads, or a full house? Keep the queen and ace looking for anything from a high pair to a royal? With most pay schedules, the authorities say to hold the pair.

How do the gurus know? Contrary to common belief, not by practice or observation but arithmetic. There's room for opinion, although not in the way many think. Instead, opinion whether an exception that complicates a rule matters enough to merit memorizing. Or whether it's better to guarantee winning at least something on a hand, rather than maximize the expected or average payback.

As an illustration, say you're in a game with nothing wild. Your starting hand is five hearts -- two, four, five, six, and seven. Straights return 4-to-1, flushes 6-to-1, and straight flushes a fat 50-to-1. Stand, and you've got a "made" flush. Discard the deuce and you may be blessed with the straight flush, end with another flush or a straight, or finish in the sewer.

To find the expert strategy based on expectation for this situation, compare the 6-unit certain return by standing, with the statistical average payoff you obtain by discarding the deuce and drawing a replacement. Here's how it would be figured.

You've seen five cards so 47 remain in the deck. Of these, two (the three or eight of hearts) give you the straight flush; your chance is 2/47 of a 50-unit return, so the corresponding component of expectation is (2/47)x50 or 2.127 units. The six remaining hearts (ace, nine, 10, J, Q, or K) recover a flush; your chance is 6/47 of a 6-unit return so the expectation term is (6/47)x6 or 0.766 units. Any of six non-heart threes and sixes form a straight; this yields a 6/47 probability of 4 units, for a contribution to expectation of 0.511 units. Adding all the expectation terms comes to 3.404 units. This is less than the 6 units you get by standing. The experts therefore say to stand.

How would everything work if, in lieu of the low-card flush, you started with nine, jack, queen, king, and ace of hearts. You could still stand and take your 6 units. Or you could discard the nine on a hunch that the 800-unit royal was in there waiting.

Find the expectation for all the alternatives. One card, the 10 of hearts, makes the royal for (1/47)x800 or 17.021 units. A flush is (7/47)x6 or 0.894 units. A straight is (3/47)x4 or 0.255 units. And a high pair is (12/47)x1, another 0.255 units. The expectation results add up to 18.425 units. Much better than the 6 units secured by standing. So expert strategy is to go for it.

Other factors may be more important to an individual, always or at a particular moment, than maximum expectation. Few solid citizens gamble for the greatest long-term profits, anyway. That's what casinos do. Some players are more interested in a shot at a payoff that will change their lives. They might break a pair of jacks to hold a 10-jack-queen of diamonds looking for the royal. Others prefer to stay in the game long enough to earn a comp for the all-you-can-eat buffet, without crawling to the bank machines in the lobby for more money. They might stand with a high-card flush as opposed to going for the royal, taking the sure 6 units rather than risking a 46-to-1 chance at an 800-unit return. These options are what "utility" theory is all about.

So, should you follow expert strategy or flout it? Ultimately, it boils down to knowledge. Knowing what's supposed to be "right" and why it is. And knowing when something else may be right for you. As the lauded laureate, Sumner A Ingmark, cleverly quipped:

Perceiving when to buck the rules,
Sets geniuses apart from fools.