How Often Should You Get What You Expect?

You don't have to be a Nikolai Ivanovitch Lobatchevskii to grasp some vital facts about events whose chance of occurring is "one out of some number." Namely, in that many trials, the average, expected, and most likely number of instances all equal one.

Take a coin flip. The chance of heads is one out of two. If enough solid citizens each flip twice, they'll average one heads each. For each pair of flips, they'd expect one heads. And in each trial, the most likely number of heads would be one. More rigorously stated, the probability of one heads in two flips is 50 percent, while that of zero or two heads is 25 percent each.

What about the outlook for rolling a seven at craps? Dice doyens know it's one out of six. And the compu-cops would claim that you should average, expect, and find the most likely number of sevens in six rolls to be one. But, what do the statisticians cite as the precise prospects of various instances of sevens in six rolls? The figures are 33.4898 percent for no sevens, 40.1878 percent for one, and 26.3224 percent for two or more.

Now think about the chance of being dealt a blackjack. In an eight deck game, it's 4.7451 percent - close to one out of 21. In actuality, the probability of no blackjacks in 21 hands is 36.0272 percent. The likelihood of one is somewhat greater, at 37.6887 percent. That of two or more is 26.2841 percent.

How do these figures change as you go to more exceptional events? A straight or better at Caribbean Stud or Let It Ride is just under a one-out-of-132 shot. The true probability of receiving no such hands in 132 rounds is 36.8874 percent. That of a single straight or better is 36.9283 percent. And two or more hands this good come in at 26.1943 percent.

Getting more extreme, a royal at seven-card stud poker has one chance out of 30,940. Playing weekly on the kitchen table for about four or five hours at a stretch, you might receive 30,940 hands yourself in five or six years, and will see this many in under a year. If you enjoy casino poker, you could easily be in a room where over 30,904 hands are dealt in a month. You'd therefore expect to see one royal in these time spans. Precise probabilities are 36.7873 percent for none, a marginally greater 36.7885 percent for one, and 26.4241 percent for two or more.

High-value slot machine jackpots are rarer yet, often in the range of one chance in 500,000 or worse. The average, expected, and most likely frequency of successes at this level is (you guessed it) once every 500,000 pulls. The true probabilities are 36.7879 percent for no jackpots, 37.7880 percent for one hit, and 26.4241 percent for two or more fulfillments of the premonition you had in your reverie on the casino bus that very morning.

There's a clear trend here. You can see it in the accompanying table, which summarizes the data for the previous examples.
Chance of zero, one, or two or more successes in the number
of trials for which the "expected" instance of hits is unity

trials	no successes	one success	two or more successes
2	25.0000%	50.0000%	25.0000%
6	33.4898%	40.1878%	26.3224%
21	36.0272%	37.6887%	26.2841%
132	36.8774%	36.9283%	26.1943%
30,940	36.7873%	36.7885%	26.4241%
500,000	36.7879%	36.7880%	26.4241%

For cases where one success is expected in every two trials, the probability of that solitary hit is 50 percent. As the number of tries for an expected single success increases, the actual chance of a lone score quickly approaches a steady value around 36.8 percent. The prospect of no wins starts at 25 percent for a one out of two situation; it rises rapidly to a final value also near 36.8 percent, but always remains less than that of one hit. And the chance of multiple successes in the requisite number of trials starts at 25 percent and soon settles around 26.4 percent.

Of course, pragmatic people know they don't always get what they expect. But, now you know how often to expect the expected. And are expected to know what the poet, Sumner A Ingmark, meant by:

Those who understand statistics,
Know the games aren't ruled by mystics.