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# How long do you have to wait for a hit?

7 March 2011

Some casino aficionados believe that if they keep betting enough to cover previous losses, they'll do OK in the end. This philosophy works sufficiently often to tempt those who've tried it and triumphed, or heard of folks who have. But, realistically, what's the chance that chasing losses will be a path out of a hole, and what are the associated dangers and the penalties for failure?

A well-known, praised and reviled, implementation of this notion is Martingale betting on even-money propositions. The idea is to double a wager after every loss and either quit or start a new series at the minimum upon a win. If the plan succeeds, the profit is one unit.

Two principal problems arise with this strategy. First, bet sizes escalate rapidly. If you start as low as \$1, a string of losses would have you betting \$2, \$4, \$8, \$16, \$32, \$64, \$128, and so on. It doesn't take a record cold streak before you have to put big bucks on the line and have run through a large bankroll. And since each bet is a dollar more than the sum of your previous losses, all you stand to earn is \$1. Second, if you've got the green to keep going and the will to wager the kind of dough necessary, the likelihood of success approaches but never reaches 100 percent.

Pretend you try this scheme on Red at single-zero roulette – an even-money proposition. The chance of winning on the first spin is 18 out of 37, or 48.65 percent. That of losing on the first spin and winning on the second is 19 out of 37 times 18 out of 37, or 24.98 percent. Likewise, losing on the first two then winning on the third has a probability of 12.83 percent. Each coup is still a 48.65 percent shot. But, as you proceed, the prospect for success of the system as an entity equals the sum of the chances of winning on all previous and the current rounds. For the first three spins, this would be 48.65 plus 24.98 plus 12.83, for a total of 86.46 percent.

If you're set to ride this pony for up to 10 spins, your outlook for ultimately winning that dollar is 99.87 percent. Good, but not quite certain. Starting at \$1, though, you had to bet \$512 on that 10th spin. And you needed a \$1,023 bankroll to do it. Hopefully, you won't have to go this far. But the chance is 0.13 percent chance, one out of 784, that you will, and come up short, losing \$1,023.

Another way to keep betting so you cover previous losses presents itself in games that offer longshots. An example would be a "straight-up" bet, also at single-zero roulette. The chance of winning on any spin is one out of 37 or 2.70 percent; the payoff is 35-to-1.

Imagine you bet \$1 on a single spot. A win pays you \$35. Say it loses, and you bet \$1 on the next round. If this hits, you're paid \$35, but you lost \$1 on the previous spin so your profit is \$34. At round three, you're looking to net \$33. The probability of joy on each spin is 2.70 percent. In terms of a series, however, individually, the chance of winning on the first round to get \$35 is 2.70 percent; that of losing on the first and winning on the second to earn \$34 is 97.30 times 2.70, or 2.63 percent; and the likelihood of losing the first two and winning the third to net \$33 is 97.3 times 97.3 times 2.70 or 2.56 percent. More critically, combined, the chance of hazarding from \$1 to \$3 to grab \$33, \$34, or \$35 is 2.70 plus 2.63 plus 2.56 or 7.89 percent.
A comparison of these two approaches is useful, intrinsically, to illustrate the effect of short versus long odds. It also serves as a reminder that gambling offers players a wide array of opportunities to make tradeoffs. For this purpose, forget the nonsense of having to invest over \$1,000 to win a dollar. Instead, pretend you saunter up to a table with \$31 burning a hole in your fanny pack. Your idea is to playing until you either win on a spin or exhaust your \$31.

Martingale betting on Red gives you up to five tries (\$1, \$2, \$4, \$8, \$16) for the total of \$31. The probability you'd win before busting out is good at 96.43 percent – actually, not much less than the 99.87 percent chance you get with 10 shots and a bankroll of \$1,023 . The downside is that the potential profit for your 3.57 percent chance of losing \$31 is only \$1. Betting \$1 straight-up for as many as 31 spins – if necessary – will earn you as much as \$35 or as little as \$35 - \$30 or \$5. The probability of success somewhere in this range is 57.23 percent. The chance you'll lose your \$31 with 31 misses in a row is the complimentary 42.77 percent.

Which appeals to you more? With the same \$31 loss as the downside, a strong 96.43 chance of winning \$1 or a middling 57.23 percent shot at grabbing somewhere between \$5 and \$35? As is often the case in deciding how to gamble, it's a matter of personal preference and an opportunity to balance risk and reward. Something surprisingly few bettors consider before putting their hard-earned bread at risk. Which caused the remarkable rhymesmith, Sumner A Ingmark, to ruminate:
Superficial expectations,
Are results of aspirations,
Formed ignoring implications.