Intuition in one game may not work in another

Oct 30, 2001 3:01 AM

A friend, a fervid fan of Let It Ride, returned from an unusually unsuccessful casino visit lamenting that the cards were so bad he never once got three-of-a-kind. It set me wondering exactly what the chances of this hand were ”” the likelihood of getting or not getting it, on any particular round or during a session of reasonable duration, over a spectrum of poker-oriented games.

Triplets are interesting for several reasons. They’re not all that tough to make in some games but are uncommon, even rare, in others. Under conditions where hands are weighed against a payout schedule, they normally don’t yield a premium commensurate with their regularity of appearance, being a Peter robbed in order to pay more to the jackpots of Paul. And in situations where one hand must beat another, they occupy a middle ground ”” often good enough to win, yet not infrequently a costly second best.

Chances of finishing a round with three-of-a-kind in games apt to be offered at your friendly neighborhood casino are shown in the accompanying table. The low-probability end of the scale is 0.23 percent for three-card poker. The high is over 100 times greater, exceeding 28 percent for deuces wild video poker played according to “expert” strategy.

The 0.235 percent probability in three-card poker is equivalent to an expected frequency of one in every 425 hands. Intuitively, the reason for the low likelihood is that only three cards are selected from a deck of 52, and all must be of the same rank.

Conversely, the 28.380 percent probability in deuces wild video poker can be envisioned as slightly better than two out of every seven hands. This favorable outlook follows in part because solid citizens start with five cards, then have the opportunity to improve by taking another shot entirely or by drawing to a pair; it’s also a result of the ability to form triplets from a pair and one of four wild cards, or from two deuces and anything else.

The effect of game idiosyncrasy on prospects of making triplets is highlighted by the 1.26 percent probability difference between standard deuces wild video poker and a bonus version which pays double for four deuces. This occurs because the money to fund the bonus isn’t simply a gift from the kindly casino bosses; it comes by adjusting returns on other hands. To achieve a balance, some payoffs increase while others fall. The figures given in the table assume that, among other changes, a full house pays 4-for-1 in the bonus game and 3-for-1 on standard machines. Expert strategy then says to hold two pair with no deuces in the bonus version but to discard one of the pairs under corresponding circumstances in the standard implementation ”” the full house returning enough more in the second than the first. Holding two pair, you can’t possibly finish with triplets; drawing three to a pair, you have a good opportunity to achieve three-of-a-kind.

It’s almost inconceivable to go two hours at any video poker game without hitting triplets at least once. The probability, while not zero, is negligible. The cited card games ”” Caribbean Stud, Let-It-Ride, and 3-card poker ”” are another story altogether. In the former two, the chance of two hours without trips is almost 12 percent, and in the latter it’s 79 percent. The striking differences between the machine and table groups arise not only from the details of the games but also the rates at which they’re played. On the machines two hours might mean 1,000 rounds; at any of the table games, an equal time span would typically represent 100 hands. This illustrates the cautions needed in applying insights gained from intuition among situations that are superficially comparable but may be substantively distinctive.

The immortal Sumner A Ingmark, bard beloved of both bettors and bankers, conveyed the caveat in his cogent couplet:

Though circumstances seem alike, it may be urgent,
To verify if they’re the same or quite divergent.

GAME   Probability
3-card poker   0.235%
Caribbean Stud   2.110%
Let-It-Ride   2.110%
7-card stud   4.830%
Jacks or better/vp   7.449%
Joker Wild/vp   13.480%
Bonus Deuces/vp   27.120%
Deuces Wild/vp   28.380%