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# Don't Make the Mistake of Actually Expecting the Expected

6 November 2001

Gambling gurus glibly expound on expected incidents of events. They're using a narrowly defined statistical concept. It involves averages, not something you should literally expect to happen.

As an example, say you flip a fair coin 10 times. Statistically, expectation is to get five heads and five tails. But the precise chance of this result is only 24.6 percent. Even allowing the final set to be off by one either way, the chance of having from four to six heads is 65.6 percent - good but by no means a lock.

What about 100 instead of 10 flips? Does the law of averages kick in because of the larger number of trials, raising the likelihood you'll get the expected 50 heads and 50 tails? The probability doesn't rise, it drops from 24.6 to 8 percent. The chance you'll be within one flip either way also falls, from 65.6 to 23.6 percent. What increases is the potential of finishing within some proportional band of the expected value. The range from 40 to 60 around 50 is in proportion to that from four to six around five, but prospects of being there are 96.5 rather than 65.6 percent.

Solid citizens are especially astonished at how low the odds can be of seeing the upshots said to be expected in so-called "cycles" of possibilities. And the surprise is more than intellectual. It can be financial for anyone who acts on the assumption that occurrences are somehow ordained to align themselves with the expected frequencies.

Take roulette. A theoretical cycle in a double-zero game is 38 spins, in which each number is statistically expected to occur once. The chance of a series of 38 spins showing each and every number, in any order, is roughly one out of 2 quadrillion.

In craps, what's the chance that a set of 36 rolls will yield the statistically expected distribution shown in the accompanying table? It's one out of 706,154. Far too remote to warrant tracking 35 rolls then betting on what's missing on the 36th.

 Expected Distribution of 36 Rolls at Craps Result Expected Instances 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

The chance that a series of 36 rolls will yield exactly six sevens, the expected number, regardless of the other 30 outcomes, is also less than players might imagine. It's a bit under 17.6 percent. And the range from five to seven appearances of seven in 36 rolls doesn't quite reach the 50 percent level.

Blackjack offers yet another illustration of the disparity between statistical expectation and money in the bank. The laws of probability say a blackjack is expected once in every 21 hands. The actual chances of blackjacks in this many hands are 36.0 percent for none, 37.7 percent for one, 18.8 percent for two, 5.9 percent for three, and 1.3 percent for four. In 42 hands, expectation is to see two blackjacks. Actual chances are 13.0 percent for none, 27.1 percent for one, 27.7 percent for two, 18.4 percent for three, and 8.9 percent for four.

Cycles and expected instances of results, whether applied to the millions of combinations in giant jackpot games or the vastly fewer outcomes in lower?payoff situations, are mathematical artifices. They're valuable to characterize strategies according to long term trends and transitory fluctuations. But, bettors and bosses alike commonly err by imputing colloquial meanings to these specialized constructs, and acting as if cause and effect - as opposed to randomness and chance - control the casino.

The beloved balladeer, Sumner A Ingmark, conceivably was contemplating just such confusion when he composed this couplet:

No forces of nature loom ready to correct,
The difference between what you get and you expect.

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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.