Playing it Smart - It helps to know what you're doing

Jill and Phil were discussing jacks-or-better video poker. They disagreed about starting hands of two pair when one is jacks or higher. Jill remarked that all the books tell you to hold both pairs and dump the unrelated card. Phil insisted that those books were written by casino stooges who wanted you to lose, or to win as little as possible, and you should hold only the high pair.

Here's Phil's logic. Two pair returns 2-for-1 if you don't improve. And the only possible improvement is to a full house worth 9-for-1 on a good machine, 8-for-1 or 7-for-1 otherwise. Keeping only the high pair still guarantees 1-for-1. And you can recover to two pair, the full house remains a possibility, and trips at 3-for-1 or quads paying 25-for-1 become achievable.

Phil's key points were a) 2-for-1 on two pair isn't that much better than the 1-for-1 on the high pair by itself, b) two pair have only one way to improve while the high pair has four, two of which pay over 2-for-1. "This is one of them there secrets the casino bosses don't want nobody to know," he claimed.

A solid citizen certainly could luck out and earn 25-to-1 with quads when drawing three to a high pair. But the strategies expounded by every gambling guru meritorious of the mantle imply playing the percentages rather than praying for providence. The problem with Phil's reasoning is in confusing the enumeration of possible results with the number of ways or, equivalently and more conveniently the probability these outcomes can occur.

The accompanying table gives the chances for the two alternate approaches, along with the corresponding returns and expected values. The returns shown are for "9-6" games, on which a full house pays 9-for-1. The probabilities are inherent in the structure of video poker how many cards are left in the deck and how many make one or another winning hand. The returns for each outcome are displayed on the machines. The component expected values are the probabilities multiplied by the corresponding returns. The overall expected values are the sums of the component contributions.

Expected value of a video poker hand dealt as a high pair and a low pair, played as a high pair and as two pair per dollar bet

	probability	return	expected
	(fraction)	($)	value ($)
high pair
no improvement	0.7130	1	0.7130
two pair	0.1600	2	0.3200
three of a kind	0.1140	3	0.3420
full house	0.0102	9	0.0918
four of a kind	0.0028	25	0.0700
overall	1.0000		1.5368
two pair
no improvement	0.9150	2	1.8300
full house	0.0850	9	0.7650
overall	1.0000		2.5950

The tabulated figures reveal that in a 9-6 game, playing the hand as two pair by drawing a single card has an expected value (rounded to the penny) of $2.60 for every dollar bet. Dumping the low pair and drawing three cards has an expected value of $1.53 on the dollar. Were the return for a full house 8-for-1, the respective figures would be $1.54 and $2.51. At 7-for-1 for a full house, the expected values would be $1.53 and $2.42. In all cases, holding two pair is two-thirds again better than taking a shot at a bigger payoff from the viewpoint of expected value.

It's OK if big boys and girls use a chance for higher payoff rather than expected value as their primary decision criterion in gambling for money. However, even with this basis for a strategy, discarding the low pair in the situation at issue gives cause for pause. The likelihood of an intermediate return with a full house drops from 8.50 to 1.02 percent, from about one in 12 to one in 100. And the prospects of quads with the high pair is a scant 0.28 percent, approximately one in 357. Think in terms of having a sure extra dollar versus one chance in 357 of picking up $25.

Should you play by the book or chart your own course? It's your choice. But a technique that suits your personal preferences can still be predicated on the percentages of the game, not on a belief that having attracted you to a casino in the first place the rogues who run the joints try to dupe you into playing poorly. Sumner A Ingmark, the celebrated songster of sporting sophisticates everywhere, said it like this:

If you think a casino needs tricks or a ploy, ya
May be looking the wrong way for what can destroy ya,
Or perhaps have a case of acute paranoia.