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# Why the law of the maturity of chances seems to work but doesn't

27 February 2007

Gamblers faring poorly sometimes turn to the thought – or the hope – that the more rounds they lose, the more they're due for a win. This idea gets an aura of authenticity with a technical name: the law of the maturity of chances. The principle is that whatever the probability of an event, the likelihood it'll occur repeatedly shrinks as the run stretches. For instance, if the chance of one loss is 60 percent, two in a row come in at 36 percent, three at 21.6 percent, and so on.

The principle is true, but it's falsely applied to these situations. One way to see the flaw in the logic is to examine the likelihood not only of multiple adverse outcomes, but of strings of routs followed by an eventual victory.

Consider for example, bets on Red at single-zero roulette. Red has 18 ways out of 37 to win — a probability of 48.6 percent, and the complementary 51.3 percent to lose. The probability of two losses in a row is 26.37 percent. However, the likelihood of a loss followed by a win is 24.98 percent. The good news is the chance of being disappointed twice, back to back, is less than that of losing just once. The bad news is the chance of a loss followed by a win is less than that of two successive losses.

The effect strengthens with the length of the string. The chance of 25 successive losses is indeed small, one out of about 17 million. The chance of 24 losses followed by a win, though, is even smaller – one out of 18 million.

A strategy for betting on Red, based on the decreasing chance of adverse series of increasing length, is dangerous. The amounts needed to recover past losses escalate steeply. Starting with \$5, on the 10th spin you'd have to bet \$2,560. The chance is slim of getting to nine losses in a row, putting you \$2,555 in the hole. Only one out of 402. But breaking the streak anywhere along the way would still give you a paltry \$5 profit.

A maturity of chances strategy is less financially punishing on low-probability high-payoff propositions. Here, an eventual win can overcome a series of losses without raising wagers on successive rounds. For instance, say you bet \$5 Straight Up at single-zero roulette. A win pays \$175. You can therefore lose 35 successive spins, getting into a \$175 hole gradually, and recovering by risking only \$5 on the 36th. The probability of losing on a single spin is a high 97.3 percent, while prospects of 36 losses in a row are 37.3 percent. But the likelihood of 35 losses followed by a win is far less yet, roughly 1.0 percent.

Lay bets, having greater likelihood of winning than losing but paying less than the amount at risk, are a third category. An illustration in single-zero roulette might be \$5 on each of two columns. This is \$10 at risk, with a 64.9 percent probability of winning versus 35.1 percent of losing. But earning only \$5. The likelihood of six losses in a row is a mere one out of 532. The chance of five straight losses followed by a win is actually better although still low at one out of 288. The price of the improvement is that bets needed for recovery grow more sharply than for even-money wagers. If you bet \$10 on the first spin and lose, you have to put up \$20 simply to break even on the second. A second loss drops you \$30 into the hole, calling for a bet of \$60 to break even. The sixth bet will have to be \$1,620. A lot of money considering your plan to risk \$10 to earn an easy \$5.

Losses that persist until a bankroll is busted don't necessarily convince believers that the law of maturity of chances is junk math. Positive reinforcements when it's worked are supported by the notion that they simply didn't have quite enough stake to see it through. The reason it seems to work in many cases has nothing to do with chances of success changing because of past events. It's that the less you're ready to accept as profit relative to a loss limit, the greater your prospects for success. The philosopher, George Santayana, said "Those who cannot learn from history are doomed to repeat it." The maxim applies to losing coups at the casino. As does this rhyme by Sumner A Ingmark:

You send money into the hands of the bosses,