Do You Sweat the Odds in Shooting for Big Jackpots?

I recently discussed table games with jackpots. And, yes, it happens, I goofed in deriving the figures. I said, at "Pai Gow poker, the odds of forming a royal from five out of seven cards, irrespective of the other two, is one out of 30,939... Adding the condition that the two-card hand be a pair of aces changes the odds to one out of 11,148,712." The numbers are wrong.

My calculations were OK. But they were based on a 52-card deck. Pai Gow poker uses a joker so it's played with 53 cards. The extra card lowers the probabilities. The chance of a "natural" royal (no joker) drops to one out of 34,163 when the sixth and seventh cards are irrelevant. And the chance of a royal and a pair of aces -- both natural -- worsens to one out of 12,845,257.

Correcting this error offers an opportunity to explore two issues. First, how much of a difference is caused by a minor change such as the addition of a joker to a deck of cards? Second, do -- or should -- players sweat the odds of winning when taking extreme longshots at big jackpots?

The change caused by adding the joker would affect the edge on the bet. Depending on the payout schedule for various results, it might even shift advantage between house and player. But solid citizens who take longshots at big returns rarely worry about how much the casino rakes from the pot. Whether this is wise or foolish is a vast topic unto itself and won't be examined here.

Independently of edge, some players trying for substantial jackpots think only about the size of the prize. They know the chance of winning is small. How small doesn't seem to matter.

Other gamblers fit the decision theory model. They balance the perceived worth to them of a jackpot against what they have to risk, using the odds of success as the lever. In the casino, bets are usually made repeatedly, rather than once-and-for-all. The trade-offs therefore make the most sense when couched in terms of the overall possible investment and chance of winning during one or a series of casino visits rather than on a single wager.

For this purpose, it's necessary to estimate how many times an individual is apt to make the bet in question. This depends both on the duration of play and the speed of the game. The period may range from several hours for someone who goes to a casino once a year or a lifetime, to hundreds or thousands of hours for a "regular." Within any such period, the number of tries will depend on the specifics of a game -- anywhere from 30, 50, 100, or more decisions per hour at the tables to 500 or 1,000 spins per hour on the slots. For practical purposes, with propositions having very low probabilities, the chance of winning during a session can then be approximated by multiplying the probability per round by the number of rounds.

I'll give you more precisely calculated figures for the Pai Gow poker jackpot. This game is slow. A player might get 100 hands in three hours. For this much action, the odds against receiving at least one royal with any other two cards would be 308.8-to-1 without the joker, and 341.1-to-1 with the extra card in the deck. Likewise, the odds against a royal with pair of aces would be 111,486.6-to-1 without the joker and 128,452.1-to-1 with it.

Frequent players might get 1,000 hands in a month. Now the odds are 30.4-to-1 and 33.7-to-1 for the royal-plus-anything in 52- and 53-card games, respectively. They'd be 11,148.2-to-1 and 12,844.7-to-1 for the royal-plus-aces in the same games. In a year, say 10,000 hands, odds improve to 2.6-to-1 versus 2.9-to-1 and 1,114.4-to-1 and 1,284.0-to-1 for the various cases.

With these session odds and the corresponding total bets, players can make better decisions whether to go for the gold. After all, gambling's more than "Yeah, I know chances ain't good, but somebody's gonna win. And it might as well be me." Er, isn't it? The poet, Sumner A Ingmark, adroitly addressed this attitude:

Rather than work the ne'er-do-well schemes,
On grandiose shots at American dreams.