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Why Gamblers Score before the Law of Averages Says they Should23 August 2006
Doesn't the law of averages say this shouldn't be? Or, shouldn't be expected? Doesn't it say that, if a probability is something like one out of a million, it should take about a million tries before the event happens? Alternately, do the statistics mean that after a million or so attempts by solid citizens in general, the right person will be in the right place at the right time?
None of the above is strictly true. The law of averages simply says that if
the probability of a discrete event is one out of a million, the average number
of attempts between occurrences will be roughly a million. It's mute on how
many tries each individual must make before winning, or how participants are
bunched -- more technically, "distributed" -- among those numbers
Pretend that a million hopefuls play until they prevail, then quit. With a one out of 10 shot each time, approximately 100,000 (10 percent of the million) will triumph immediately. This leaves 900,000 to bet again, of whom 90,000 (10 percent of 900,000) will win on the second round. Now, 810,000 are left, with 81,000 (10 percent of 810,000) scoring on the third round. Continuing, the fourth through 10th tries will see 72,900, 65,610, 59,049, 53,144, 47,830, 43,049, and 38,742 hitters, respectively. In one computer simulation, the longest run was 125 tries without a hit.
Across all players, the average coup on which the hit occurs is the 10th, just as the one out of 10 probability implies. But it's frequently assumed that triumphs are concentrated around the average -- in this case, 10 -- and that figures taper off on either side of this value along the famous "bell-shaped curve." This assumption is wrong.
The math and the simulation show, instead, that the greatest number of happy
campers -- 100,000 out of the million -- is predicted to win on the first try.
And the number of successes on each subsequent round declines steadily from
there. The curve is not symmetrical about a peak at 10. Rather, it's shaped
more like a ski jump with the high-point at a single try.
A conclusion to be drawn from considering distribution is that a disproportionately large number of players will hit probabilistic outcomes in random games earlier than might be expected based on averages alone. For their bonanzas, these folks can thank the small number of people who come up empty-handed for lopsidedly longer periods than the average would suggest. Great for those gamblers who grab the greenbacks and run. Cool for the casinos, who clear their cash on the averages and can honestly advertise having lots of winners. Bad for the blokes who miss the mark continually and end up buying everyone else's lunch. This sad fact of punting life is reminiscent of that unforgettable utterance by the bettors' bard, Sumner A Ingmark:
Accounting for averages, not distributions,
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