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Why Gamblers Score before the Law of Averages Says they Should

23 August 2006

You occasionally hear about a lucky duck with a few minutes and a coupla bucks to spare, who drops a token or two into a slot machine and bags a biggie right off the bat. You've been playing for years and never came close.

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 of tries.

To understand how distribution works, picture a gamble where probability is one out of 10 (10 percent). The idea is analogous to one out of a million or 10 million -- but a game with chances of one out of 10 is more intuitive and easier to visualize.

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 related common misconception is that as many aspirants fall below 10 as above. In reality, well over half -- 612,580 -- of the players should score with less than 10 tries. Another 38,742 hit precisely at 10. And the remaining 348,679 of the million hypothetical bettors are projected to require more than 10 rounds. At the far end of the spectrum, 100,000 won't have the brass ring following 22 attempts while 10,000 will still be pounding sand on their 46th go. On real slots, 22 or 46 spins don't seem like much. But this simplified model involves tenths while slots may have chances in the ten millionths, so multiply tries by a million to see where this leaves the luckless.

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,
May lead more to problems than offer solutions.

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Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.