How do the gurus decide what's the best way to handle a video poker hand?

Everyone who plays the machines fantasizes about hitting a jackpot (name somebody who doesn’t). Most don’t actually expect it to happen, but still nurture the thought that it could.

Pretend you’re at a jacks-or-better video poker game with the following return schedule: royal – 800, straight flush – 50, quads – 25, full house – 9, flush – 5, straight – 4, triplets – 3, two pair – 2, high pair – 1. Further, imagine you start a round being dealt a hand only one card away from a royal. If the hand pays nothing as-is – for instance 4-S, 10-H, J-H, Q-H, K-H – you won’t lose anything by discarding the four. Not that you would, but in theory, you might instead dump the 10 as well as the four; this would obviate the royal while raising the likelihood of a high pair, still allowing for a flush or another straight, and opening a window for two pairs or triplets.

What about 10-S, J-H, Q-H, K-H, A-H, though? Most bettors wouldn’t hesitate to sacrifice the straight and go for the royal. After all, straights are no great shakes and won’t make or break anyone’s day. But royals return real money and are hands video poker buffs talk about for years.

Players who would mull over a pat straight in pursuit of a royal might go so far as to reason that just one card out of the 47 still available, the 10-H, will work. They may figure that one out of 47 (2.13 percent) is a longshot. But it’s a walk in the park next to the one out of about 40,000 (0.0025 percent) probability of a royal starting from scratch. Besides, if they miss the royal, they could still get a win. They could even dope out the chances of a flush, another straight, or a high pair. For reference, these prospects are eight, two, and 12 out of 47 (17.02, 4.26, and 25.53 percent, respectively), leaving 24 out of 47 (51.06 percent) for a wipe-out.

Gambling gurus, however, have not only done the math, but assert that likelihoods of alternate results, alone, aren’t enough to determine what’s “best..” Rather, they argue that payoff as well as probability must be considered. And they use “expected value” to combine these factors.

As a simplified example of what’s meant by expected value, picture a game in which you risk a buck to draw a card from a set of four. Two in the set lose, one pushes and returns your wager, and the other pays two dollars. If you don’t play, your chance of having a dollar is 100 percent; the expected – in this case, it could be regarded as the certain – value is 100 percent of a dollar, namely $1.00. If you participate, the probabilities of finishing a coup with various amounts are 2/4 with $0, 1/4 with $1, and 1/4 with $2. The expected value is the sum of the probabilities times the amounts: (2/4) x $0 + (1/4) x $1 + (1/4) x $2 = $0.75. The $1.00 certain value of taking a pass exceeds the $0.75 expected value of playing; by this criterion, it’s best to skip the game.

Here’s how expected value per dollar bet would suggest whether to stand pat with 10-S, J-H, Q-H, K-H, A-H or go for the gold. Since the straight pays 4-for-1, the certain value of the starting hand is $4.00. The expected value of dumping the 10-S is then found as follows:

• The chance of drawing the 10-H to complete a royal (800-for-1) is 1/47; the contribution to expected value is (1/47) x $800 = $800/47.

• The chance of drawing any other heart (two through nine) to complete a flush (5-for-1) is eight out of 47; the contribution to expected value is (8/47) x $5 = $40/47.

• The chance of drawing a 10-D or 10-C to complete a straight (4-for-1) is two out of 47; the contribution to expected value is (2/47) x $4 = $8/47.

• The chance of drawing a non-heart face card or ace to make a high pair worth (1-for-1) is 12 out of 47; the contribution to expected value is (12/47) x $1 = $12/47.

• The chance of drawing anything other than the above to get nothing is the remaining 24/47; the contribution to expected value is (24/47) x $0 = $0/47.

The sum of the contributions to expected value is $860/47 = $18.30. This is greater than the $4.00 certain value of standing pat, indicating that the jackpot try is the better choice.

The same math yields slightly different expectations depending on the dealt starting hand. But the contribution to the expected value of an 800-for-1 royal on this game is always $800/47 = $17.02 per dollar bet. Contributions to the expected value of the other final hands that can be achieved by drawing one to a four-card possible royal are between $1.20 and $1.30. Therefore, the expected value rule is to draw if the certain value per dollar bet is below $18.32, and to stand pat if it’s above. The only hand with a higher certain value is a straight flush (50-for-1), at $50. Even here, players can’t be faulted for chasing the royal. Expected is less than certain value in dollars and cents with this decision. But who’s to say what solid citizens think jackpots are worth, either in bragging rights or in the relative utility of $800 rather than $50 burning holes in their pockets. The celebrated songster, Sumner A Ingmark, succinctly summarized the subject as:
Although the math says you should not,
You may still want to take a shot.