How do the gurus get video poker expert strategy and should you follow it?

Pretend you're playing jacks-or-better video poker and get 10H-9H-7H-6S-2H as a starting hand. Do you dump the deuce and go for the inside straight, or toss the six and hope to make a flush? Most solid citizens would opt intuitively for the flush, mainly because it pays more than a straight. So-called "expert strategy" for the game confirms this as the better choice.

Many situations, however, aren't as obvious. Fortunately, expert strategies are widely available in articles, books, and websites for almost every flavor of video poker you'll encounter. Enquiring minds, however, want to know whether there's a legitimate basis for these expert strategies or if some self-anointed gurus (who'll be glad to sell you their secrets for a pittance compared to what they'll supposedly earn you) are just offering opinions off the tops of their heads.

Some proffered strategies may, indeed, be garbage. But many are based on the math associated with the games and are solid. The best ways to distinguish the two types involve the reputations of the people presenting them and the adage to disbelieve claims that seem too good to be true.

The 10H-9H-7H-6S-2H starting hand offers a simple example of how legitimate expert strategies are derived. Here's how it's done for games with two alternate payout schedules, the relevant hands on which return the following for a $1 total bet: machine a) flushes - $6, straights - $4, high pairs - $1; machine b) flushes - $5, straights - $4, high pairs - $1.

Discarding the deuce, you need one of the four 8s still in the deck for the inside straight. You've seen five cards, so the deck still has 47 left. The chance of pulling one of the 8s is 4/47. If you get it, the return is $4 in either machine. So the "expected" return is (4/47) x $4 or $0.34. Sacrificing the six, you need one of the nine remaining hearts. Expected return on machine (a) is (9/47) x $6 or $1.15. On machine (b), it's (9/47) x $5 or $0.96. The possible flush beats the possible straight either way, so it's the rule codified in expert strategy.

The wicket gets stickier with JS-10H-8H-7H-2H. Discarding the deuce then opens the possibility of drawing a Jack for a high pair as well as a nine for an inside straight. As before, the chance of a nine is 4/47 and the return is $4 so expectation for this result is again $0.34. The probability of drawing one of the three remaining Jacks is 3/47 and the return is $1 so expectation for the high pair is (3/47) x $1 or $0.06. Combined, expectation for this play is $0.40. Jettisoning the Jack to fantasize about a flush is the same as in the first example – $1.15 for machine (a) and $0.96 for machine (b) – because no high pair is in the offing. The flush is better and is expert strategy.

A slightly more complex case would involve an initial QS-JH-10H-8H-2H. Ditching the deuce leaves a shot at the inside straight as well as two high pairs. Expectation for the straight is unchanged at $0.34. But six cards will yield a high pair – the remaining three Jacks and three Queens. For this, the probability is 6/47 and the return is $1 so the expectation is (6/47) x $1 or $0.13. Combined, the two winning hands have expectation of $0.47. Losing the lady leaves expectation for a flush at $1.15 or $0.96 depending on the machine, while for a pair of Jacks it's (3/47) x $1 or $0.06; together that's $1.21 or $1.02 – the better option and expert strategy.
The arithmetic can get hairy when two or more cards are to be drawn. Sometimes, it's still practical to use direct calculations. In other instances, the authorities may prefer a type of computer program known as "combinatorial analysis" in which the software cycles through every combination of available cards and identifies the one with the greatest expectation.

Remember that expert strategy can only boost payback percentage to the maximum level inherent in the probabilities of drawing cards and the returns for various winning hands. Anything more is mystical rather than statistical.

Proficient video poker aficionados occasionally find games with sets of returns that yield overall expected paybacks over 100 percent. These are usually at machines offering players a selection of multiple games. They occur because the slot machine designers erred in entering the data or writing the program to determine the expectation when they establish the returns. And they may go undetected by the bosses for a while because profits from other games in the machine mask losses on the alternative in question. But, even here, you must know and follow the expert strategy or you'll be below the peak return percentage. Moreover, the effect is a long-term phenomenon. You can lose, often badly, in individual sessions and only realize the impact of a positive expectation gradually, if you play tens or hundreds of thousands, perhaps millions, of rounds. And, of course, you can win big on a low-return game without knowing what you're doing. The bettors' bard, Sumner A Ingmark, might well have had this in mind when he mused:

What statistics say you should expect may well be technically correct,
But the figures offer little grace applied to any single case.