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Best of Alan Krigman
You Can Trust Your Intuition at Video Poker, but Just So far15 February 2005
Most solid citizens would go for the flush, reasoning intuitively that it would pay better. The gurus would use more rigorous logic but, in this case, reach the same conclusion. It doesn't always work so neatly. Intuition can sometimes lead you astray. It therefore helps to understand how the experts arrive at the rules, so you'll know they're not just made up by casino bosses to get your money, or one pedant's opinion versus another's.
The optimum strategy for any particular situation is the decision that yields the greatest theoretical or "expected" value. To find this, you have to multiply the probability associated with each result times the corresponding monetary return. The arithmetic is normally figured "per dollar" bet. The hand used as an example is fairly simple, and offers a good way to illustrate the method.
If you try for the flush, you'll win by drawing one of the nine remaining hearts. Anything else returns nothing. Since you've seen five out of the 52 cards in the deck, the hearts must come from the remaining 47, so the probability of a heart is 9/47. Say your game returns 6-for-1 on a flush. Expected value is (9/47) x $6, which equals $1.15. If the flush pays 5-for-1, expected value would be (9/47) x $5, or $0.96. What about the expected value for the straight? Here, eight cards will be winners ?? any of four sixes or four jacks. Almost all jacks-or-better games pay 4-for-1 on straights. So the expected value is (8/47) x $4 or $0.68.
Note that trying for the straight is a "negative expectation" endeavor, since starting with this hand often enough will leave you with an average of only $0.68 for every dollar you bet. Aiming for the flush has positive expectation with a 6-for-1 payoff ($1.15 on the dollar); it's slightly negative with 5-for-1 ($0.96 on the dollar). Either way, the flush has a greater expected value than the straight so it's the optimum option.
Your decision gets harder if you begin with 3-H 8-H 9-H 10-H J-D. Now, discarding
the jack and going for the flush, expected value is the same as before. But,
if you get rid of the three, you have not only eight ways to make the straight,
but also three (the rest of the jacks) to get a high pair. This is 11 total
ways to win ?? more than the nine for the flush. But is it a superior play?
The expected value in this instance, with a 1-for-1 return on the jacks, is
(8/47) x $4 plus (3/47) x $1. This equals $0.74. So it's still worse, in principle,
than the try for the flush. But, intuition is not as helpful in making the choice.
The entries in the table show that what may seem intuitively like a close call, for instance the flush with no high cards and straight with one, are actually quite divergent. The difference is $0.22 on the dollar in a 5-for-1 game and $0.41 on the dollar when flushes pay 6-to-1. Sure, luck has a role and anything can happen on a particular hand. Then, too, the $0.22 or $0.41 are theoretical amounts rather than cash in your fanny pack. But play enough rounds and these kinds of differences add up to a strong incentive to learn the statistically optimum decisions. It's as the studious scribe, Sumner A Ingmark, so succinctly said:
While fools and their money reach quick separation,
Best of Alan Krigman