Fabricantes Actuales

The American dice game Craps evolved from the English dice game Hazard:

“According to tradition, blacks living around New Orleans tried their hand at Hazard. . . In the course of time they modifed the rules and playing procedures so greatly that they ended up in- venting the game of Craps (in the U.S. idiom known as Crap- shooting or Shooting Craps and here identified as Private Craps to distinguish it from Open Craps and the more formalized vari- ants offered in gambling casinos). . . . The popularity of the pri- vate game of Craps with the U.S. military personnel during World Wars I and II helped to spread that game to many parts of the world.”5

Craps is played with two fair dice, each marked in a specific way. According to Hoyle,

Each face of [each] die is marked with one to six dots, opposite faces representing. . . numbers adding to seven; if the vertical face toward you is 5, and the horizontal face on top of the die is 6, [then] the 3 should be on the vertical face to your right.”6

The shooter rolls the pair of dice, resulting in one of 6_{× 6 = 36 possible} outcomes. Of interest is the combined number of dots on the horizontal faces atop the two dice, a number that we denote by the random variable X. The possible values of X are displayed in Figure 3.13.

Let x denote the value of X produced by the first roll. The game ends immediately if x _{∈ {2, 3, 7, 11, 12}. If x ∈ {7, 11}, then x is a natural and} the shooter wins; if x _{∈ {2, 3, 12}, then x is craps and the shooter loses;} otherwise, x becomes the shooter’s point. If the first roll is not decisive, then the shooter continues to roll until he either (a) again rolls x (makes

5

“Dice and dice games,” The New Encyclopædia Britannica in 30 Volumes, Macropæ- dia, Volume 5, 1974, pp. 702–706.

6

1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

Figure 3.13: The possible outcomes of rolling two standard dice.

his point), in which case he wins, or (b) rolls 7 (craps out), in which case he loses.

A game of craps is fair when each of the 36 outcomes in Figure 3.13 is equally likely. Fairness is usually ensured by tossing the dice from a cup, or, more crudely, by tossing them against a wall. In a fair game of craps, we have the following probabilities:

P (X = 7) = 6/36 P (X = 6) = P (X = 8) = 5/36 P (X = 5) = P (X = 9) = 4/36 P (X = 4) = P (X = 10) = 3/36 P (X = 3) = P (X = 11) = 2/36 P (X = 2) = P (X = 12) = 1/36

Let us begin by calculating the probability that the shooter wins a fair game of craps.

There are several ways for the shooter to win. We will calculate the probability of each, then sum these probabilities.

• Roll a natural.

P (X_{∈ {7, 11}) =} 6 + 2

36 =

2 32.

• Roll x = 6 or x = 8, then make point. First,

P (X ∈ {6, 8}) = 5 + 5

36 =

5 18.

Then, the shooter must roll x before rolling 7. Other outcomes are ignored. There are 5 ways to roll x versus 6 ways to roll 7, so the conditional probability of making point is 5/11. Hence, the probability of the shooter winning in this way is

5 18· 5 11 = 25 2· 32_{· 11}.

• Roll x = 5 or x = 9, then make point. First,

P (X _{∈ {5, 9}) =} 4 + 4

36 =

2 9.

Then, the shooter must roll x before rolling 7. Other outcomes are ignored. There are 4 ways to roll x versus 6 ways to roll 7, so the conditional probability of making point is 4/10. Hence, the probability of the shooter winning in this way is

2 9· 4 10 = 4 32_{· 5}.

• Roll x = 4 or x = 10, then make point. First,

P (X ∈ {4, 10}) = 3 + 3

36 =

1 6.

Then, the shooter must roll x before rolling 7. Other outcomes are ignored. There are 3 ways to roll x versus 6 ways to roll 7, so the conditional probability of making point is 3/9. Hence, the probability of the shooter winning in this way is

1 6 · 3 9 = 1 2· 32.

The probability that the shooter wins is 2 32 + 25 2· 32_{· 11}+ 4 32_{· 5} + 1 2· 32 = 244 495 . = 0.4929.

Thus, the shooter is slightly more likely to lose than to win a fair game of craps.

Milton Murayama’s 1959 novel, All I asking for is my body, is a brilliant evocation of nisei (second-generation Japanese American) life on Hawaiian

sugar plantations in the 1930s.7 One of its central concerns is the concept of Japanese honor and its implicatons for the young protagonist/narrator, Kiyoshi, and his siblings. Years earlier, Kiyoshi’s parents had sacrificed their future to pay Kiyoshi’s grandfather’s debts; now they owe the impossible sum of $6000 and they expect their children to do likewise. Toward the novel’s end, Japan attacks Pearl Harbor and Kiyoshi subsequently volunteers for an all-nisei regiment that will fight in Europe. In the final chapter, he contrives to win $6000 by playing Craps.

Kiyoshi had watched a former classmate, Hiroshi Sakai, play Craps at the Citizens’ Quarters in Kahana.

“It was weird the way he kept winning. Whenever he rolled, the dice rolled in unison like the wheels of a cart, and even when one die rolled ahead of the other, neither flipped on its side. The Kahana players finally refused to fade [bet against] him, and he stopped coming.”

We subsequently learn that Hiroshi’s technique is called padrolling. In the Army,

“Everybody had money and every third guy was a crapshooter. The sight of all that money drove me mad. There was $25,000 at least floating around in the crap games.. . . Most of the games were played on blankets on barrack floors, the dice rolled by hand. There were a few guys who rolled the dice the way Hiroshi did at the Citizens’ Quarters in Kahana. The dice didn’t bounce but rolled out in unison like the wheels of a cart. There had to be an advantage to that.”

Kiyoshi buys a pair of dice and examines them carefully. He realizes that, by rolling the dice “like the wheels of a cart,” he can keep the sides of the dice that form the axis of the wheels from appearing. Then, by combining certain numbers to form the axis, he can improve his chance of winning.

Kiyoshi teaches himself to padroll and develops the following system for choosing the axis:

1. For the initial roll, use the 1-6 axis for each die.

Padrolling this axis has the effect of eliminating the first and sixth rows and columns in Figure 3.13, resulting in the following set of possible

7

outcomes: 2 3 4 5 2 4 5 6 7 3 5 6 7 8 4 6 7 8 9 5 7 8 9 10

Notice that this choice eliminates the possibility of crapping out! Fur- thermore, assuming that the 16 remaining outcomes are equally likely, it also improves the chance of rolling a natural from 4/18 to 4/16. 2. If x_{∈ {6, 8}, then use the 1-6 axis on one die and the 2-5 axis on the}

other.

Padrolling this axis results in the following set of possible outcomes:

1 3 4 6

2 3 5 6 8

3 4 6 7 9

4 5 7 8 10 5 6 8 9 11

With this choice, there are 3 ways to roll x versus 2 ways to roll 7. Again assuming that the 16 remaining outcomes are equally likely, this choice improves the conditional probability of making point from 5/11 to 3/5.

3. If x∈ {4, 5, 9, 10}, then use the 1-6 axis on one die and the 3-4 axis on the other.

Padrolling this axis results in the following set of possible outcomes:

1 2 5 6

2 3 4 7 8

3 4 5 8 9

4 5 6 9 10

5 6 7 10 11

With this choice, there are 2 ways to roll x versus 2 ways to roll 7. Again, assume that the 16 remaining outcomes are equally likely. If x_∈ {5, 9}, then this choice improves the conditional probability of making point from 4/10 to 2/4. If x ∈ {4, 10}, then this choice improves the conditional probability of making point from 3/9 to 2/4.

If a shooter padrolls successfully, then the probability that he will win using Kiyoshi’s system is

4 16 + 6 16· 3 5 + 4 16 · 2 4 + 2 16 · 2 4 = 53 80 = 0.6625,

a substantial improvement on his chance of winning a fair game. “And,” Kiyoshi rationalizes, “it wasn’t really cheating. The others had the option of stopping any of your rolls, or they could play with a cup, or have the roller bang the dice against the wall, or use a canvas or the bare floor instead of a blanket.” So, Kiyoshi padrolls. I leave to my readers the pleasure of discovering whether or not he succeeds in winning the $6000 his family needs.