Why Probability Doesn't Tell You What'll Probably Happen Next

Every gambler should have a haruspex. Then we wouldn't need probabilities to tell us what'll probably happen next. Which is just as well, because that's not what probabilities do.

Probabilities are good for predicting long-run averages. In the casino, for instance, the house's edge on a row of slot machines -- found by combining possible payouts and their probabilities of occurrence -- will forecast its monthly profitability dependably enough to take to the bank. This, because a typical installation gets tens of millions of pulls, all with roughly comparable bets, during any meaningful accounting period. It says next to nothing about how Max and Mabel will fare in today's three-hour session.

Most players spend their casino careers in the "short run." They get nowhere near the amount of action necessary for results to average out. And for extreme longshots such as jackpots, numbers of trials undergone are inevitably far less than what's confusingly often referred to as a "cycle" or "statistically correct cycle" so they shouldn't "expect" to hit at all.

Of course, individual solid citizens are well advised to tailor their play according to the laws of probability. For instance, by following Expert Strategy at video poker or Basic Strategy at blackjack. But doing so won't help anticipate where they'll be at the end of any particular session. In fact, they'd really like to finish someplace where probability alone says they won't.

To get an idea why probability has more to do with the uncertain than the likely, picture a three-spin parlay on a single number at roulette. The chance of hitting any number on a spin in a double-zero game is one out of 38. That of hitting any number in three successive spins is one out of 38 multiplied by itself three times. In case you don't have your abacus or slide rule handy, that's one out of 54,872.

Would 54,872 starts give you confidence in hitting the parlay? Is it realistic to figure on good luck bringing home the bacon earlier? Should you be prepared to persevere a lot longer to reap your reward? Lacking a personal haruspex, you might turn to computer simulation to help ponder puzzles like these.

I ran two such simulations for this purpose. In each, virtual players bet on the three-spin parlay until it hit, then started over and repeated the process until they completed 1,000 winning series. Outcomes showed numbers of tries to achieve success. The separate simulations yielded somewhat different results, as would be expected because 1,000 repeats are ample to suggest trends but still exhibit a moderate degree of volatility.

In the first simulation, the average number of tries required to hit the parlay was 55,255, with the smallest being 61 and the largest 501,299. The second battery showed an average of 56,470 tries, ranging from 1 to 533,777. The distribution of results is summarized in the accompanying table.

In both simulations, the greatest concentrations of hits were from 25,000 to 75,000 tries: 324 of the 1,000 sequences in the first run, 371 in the second. However, 94 and 86 blissful bettors won within only 1,000 attempts, while 76 and 75 sad souls needed over 150,000 rounds. None of the series took the theoretically "expected" 54,872 bids. The closest on either side were 54,843 and 54,917 in one case and 54,759 and 54,950 in the other.

So, what does "one out of 54,872" really tell the person who steps up to take a few shots? Not much! Here's how that veritable versemonger, Sumner A Ingmark, characterized the conundrum: