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Best of Alan Krigman

Gaming Guru

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Playing it smart - Gamblers should repeal, or at least forget, the law of averages

4 June 2007

More than a few solid citizens believe that if they're losing and continue to play, the law of averages will eventually turn the situation around. They assume that when random tests are run repeatedly, results get closer and closer to what's predicted by multiplying theoretical probability times number of trials.

As an example, make believe you're drawing marbles from an urn containing 49 reds and 51 blues. You win on a red, with a probability of 49 percent. By the law of averages, after 100 trials you'd be close to 49 successes. After 1,000 trials, the count should be closer yet to 490 wins. After a million trials, you'd anticipate being within a whisker of 490,000 red draws.

Mathematicians call this concept the "Empirical Law of Averages." "Empirical" meaning "derived from experiment or observation." But it's not a mathematical or physical law. In fact, it isn't corroborated by valid experiments or observations. It's a popular mis-rendering of something else that does meet rigorous scrutiny.

That "something else" is a form of the "Law of Large Numbers." This verifiable principle casts the effect in the realm of percentages, not numbers, of events. Percentages being frequencies and probabilities instances of success divided by numbers of trials. The Law of Large Numbers says that the chance of a big difference between the observed frequency and the actual probability tends to decrease as the number of trials increases.

Consider the urn of marbles. After 100 trials, you could have drawn 46 reds and be at 46 percent; 10,000 rounds might have you at 4,870 reds, 48.7 percent; and a million draws could bring you to 489,700 reds, 48.97 percent. Extending play would have cut the deficit from the expected 49 percent. It started at 3 percent then dropped to 0.3 and 0.03 percent. But, shrinking percentage offsets still had numerical disparities growing from 3 to 30 to 300 marbles. In an even-money game with $10 bets, every marble less than the statistically correct value costs you $20.

The figures used for illustration are arbitrary. But the conclusions reflect what happens in practice. The frequency of events tends to approach the theoretical probability. The gap between the actual and expected numbers dollars when you're talking about gambling tends to get larger. And it grows at a rate related to the square root of the number of trials.

The operative words in the Law of Large Numbers are "chance of a big difference" and "tends to decrease." Nothing precludes sequences of trials during which the difference between observed and predicted percentages increases.

On the other hand, the Law of Large Numbers doesn't require actual frequencies to be under predicted levels. In the marbles game, players are as apt to be 3, 0.3, and 0.03 percentage points above as below expected results. However, exceeding expectations isn't the same as overcoming edge and earning a profit.

The monetary impact of edge increases in proportion to the length of a series. But the moderating influence of a positive offset consistent with the Law of Large Numbers goes up with the square root of this length. And square roots are smaller and grow more slowly than their base numbers. For instance, the square roots of 100, 10,000, and 1,000,000 are 10, 100, and 1,000 respectively.

Here's how this works with the marbles model. After 100 rounds, 3 percent favorable offset above 49 percent is 52 wins and 48 losses, a $520 - $480 or $40 profit. After 10,000 rounds, 0.3 percent over 49 percent is 4,930 wins and 5,070 losses, so a bankroll would be $50,700 - $49,300 or $1,400 in the hole. And, after a million rounds, 0.03 percent excess is 490,300 wins and 509,700 losses, a $5,097,000 - $4,903,000 or $194,000 shortage.

All of which is why you don't have to be a mathematician to open a casino any more than you have to be a baker to buy a Dunkin Donuts franchise. Or, as the poet, Sumner A. Ingmark, put it:

On nature's laws an innovation,
Will rarely yield remuneration.

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.