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Born on Friday the Thirteenth

Luck, Chance and the Meaning of Everything
also available: Hardcover
tagged : applied
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Our Days Are Numbered

Our Days Are Numbered

How Mathematics Orders Our Lives
also available: Hardcover
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Some probabilities are easy to determine. Tossing a fair coin is a 50—50 proposition: half the time it comes up heads and half the time it comes up tails. Many games of chance rely on the fact that each possible outcome is equally likely. In poker, each hand of five cards is equally likely to be dealt. In roulette, the ball is equally likely to drop into any one of the 38 pockets. When all outcomes are equally likely, the probability of any one outcome occurring is simply 1 divided by the number of possible outcomes.

The mathematics of such games is well understood. In all casino games the probability of winning is always less than the probability of losing, that is, of the house winning. The odds of winning are the probability of winning compared to losing, written with a colon instead of a division sign. The odds of losing are the reverse. Mathematically speaking, if my odds of winning are m:n then my chances of winning are m/(m+n) while my probability of losing is n/(m+n) (and so the odds of losing are n:m). So odds of 2:3 indicate that my chances of winning are only 2/(2+3)=2/5, or 40%, while my chances of losing are 3/5, or 60%.

Now the odds offered for winning are not the same as the odds for losing, though they would be in a fair game. Craps is a popular game in which one person, the “shooter,” rolls a pair of dice repeatedly. There are a variety of different outcomes to bet on. Let’s examine one of these: For rolling “craps,” that is, a 2, 3, or 12, on a single roll of the dice (but not the first opening roll of the dice), there is one way to roll 2 (namely ones on each of the two dice) and one way to roll a 12 (two sixes). There are two waysto roll a 3 (a 2 on the first die, a 1 on the second, or vice versa). So in total there are four ways to roll craps. Since there are 36 different, equally likely outcomes for rolling two dice (six choices for the first die and six for the second), the probability of rolling craps on a single roll of two dice is 4/36, which is 1/9, or about 11%. The probability of not rolling craps on a single roll of two dice is therefore 1—1/9=8/9, or about 89%. The ratio of not throwing craps to rolling craps is (8/9)/(9/1) = 8/9 × 9/1 = 8/1=8. (Remember that to divide by a fraction you invert and multiply.)

Thus the odds of not throwing a 2, 3, or 12 in a single roll are 8:1. In a fair game your payment if craps came up would be 8:1 as well, that is, for every dollar wagered, you would win $8. But what you will find typically in a casino is that, if you bet on craps coming up in a single roll of the dice, your winnings are 7:1, meaning that for every dollar you wagered you could win $7. Over the long term, repeatedly making the same bet, what you can expect to win is $7, 1/9th or about 11% of the time (when a 2, 3, or 12 does come up) and you can expect to lose $1, 8/9ths or about 89% of the time. Letting amounts we win be positive, and amounts we lose be negative, on average, I’d earn 7× 1/9 +(—1)× 8/9 = 7/9 — 8/9=—1/9 , which is about —0.11. (In mathematics, this calculation — adding up the winnings and losses multiplied by their respective probabilities — is called expectation.) In the long haul, I can expect to lose about 11 cents on each dollar bet. On any one $1 bet, I might lose the dollar or win $7, but over, say, 100 such $1 bets, I would expect to lose about $11.

It is this spread that keeps casinos in the black. On any individual bet, the casino might win or lose. The probability of this happening is not usually crucial to the casino; what is vital is the payouts that are paid for winning. Casinos make sure that on every type of bet, there is enough of a spread (that is, enough of a difference) between the odds of losing and the payout odds so that, in the end, they make money. That’s all there is to it.

Of course, the lower the payout odds are, compared with the odds of losing, the more money a casino would expect to make. So why don’t casinos give, say, 5:1 payouts for rolling craps? The key is that casinos want the odds to be in their favour, but only slightly, so that individuals will lose their money more slowly, and feel they have a chance at winning. If the payout odds are 5:1 instead of 7:1 for rolling craps, on average, I’d lose about 33 cents per each dollar bet, rather than 11 cents per bet. I’d run through my money faster, and get less enjoyment from gambling. And the gambling establishments want me to keep coming back.

Is there a way to beat the house? In reality, no. The only way not to lose is not to play!

From the Hardcover edition.

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