The Calculus Diaries - Jennifer Ouellette [31]
Pascal’s bank account may have suffered during this period of his life, but his health was never better. Ironically, the mental exertions of his correspondence with Fermat triggered a “trance” a few weeks later, from which Pascal never fully recovered. He became deeply religious, eschewing his former “corrupt” ways, and died of a brain hemorrhage at thirty-six. Maybe he should have stuck with gambling.
RISK AND REWARD
What happens in Vegas stays in Vegas, or so the advertising tagline goes—and more often than not, your money stays too. For craps, the probabilities are fairly straightforward because there are only two dice with six sides each, so there are only 36 possible combinations: six possibilities for each of the two dice (6 × 6 = 36). Yet not all outcomes are created equal, and therein lies the secret of the house advantage. There are more ways to roll a 7 than a 2, for example. To roll a 2, you would need to roll snake eyes (1 + 1). In contrast, there are three different combinations that total 7: 1 + 6, 2 + 5, and 3 + 4. Furthermore, because each die is distinct, probability also includes the combinations 4 + 3, 5 + 2, and 6 + 1. Ergo, 7 is the most likely number to be rolled. It’s no accident that the losing roll (craps) is 7 once the game gets under way.
How does this play out when one takes into account the odds and payoffs for the various bets in craps? For the pass and don’t-pass bets, the payoff odds are one to one: Winners receive one dollar for each dollar they bet. This does not mean you have a 50/50 chance (0.5) of winning a pass-line bet; the actual probability is exactly 0.49293—just slightly less than a 50/50 chance, giving the house an edge of about 1.42 percent.
Once the point has been established, the next most favorable side bet to further narrow the house’s edge is called a “freeodds” bet. For example, before the next roll, S-Money would add between one and three extra chips behind his original pass bet. While line bets have a one-to-one payoff, the payoff for a free-odds bet is determined by the exact mathematical odds against winning the bet. If the point is either a 4 or a 10, the odds against winning are 2 to 1; ergo, the payoff if the shooter rolls the point before a 7 is 2 to 1. So if S-Money bet $10 as a free-odds bet, he would win $20. If the established point is a 5 or a 9, the odds against winning are 3 to 2, so the same $10 free-odds bet would win $15. And if the point is a 6 or an 8, the odds against winning are 6 to 5, so a $10 free-odds bet would bring in $12.
Dominic assures us that the free-odds bet gives the house the smallest possible advantage, and thus is an excellent way to maximize one’s winnings. But there’s still just a single number (the point) by which you can win. You could increase your chances of winning if there were more possible winning numbers. That’s where the “come” bets and “point” bets come in. For a come bet, you place a chip in the come section of the table before the next roll, and whatever number the shooter rolls becomes a new point for that particular chip. So now you can win on this point or on the original point established on the come-out roll. If the shooter rolls a 7, of course, you lose outright—because the game is over.
There are some disadvantages to the come bet. First, it is a “contract bet,” meaning it remains in place until the end of the game, just like the pass and don’t-pass bets. (You can also place don’t-come bets, once again betting against the shooter.) Second, you are at the mercy of a roll of the dice to determine the new point. If you want to be able to pick your own number to be the new point, and have the freedom to add or remove chips at will, you should make a point bet. The payoff odds aren’t quite as good as the come bets, but you have more control over the board. There are plenty of other, higher-risk bets, but these are really the only bets you can make in craps with reasonable odds. You’ll still lose money in the long run, but