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3. Poker Dice and Chuck-a-Luck

Poker Dice

The game of poker dice is a bit like standard poker, but played with dice instead of cards. In poker dice, 5 fair dice are rolled. We will record the outcome of our random experiment as the (ordered) sequence of scores:

X = (X₁, X₂, X₃, X₄, X₅) where X_i in {1, 2, 3, 4, 5, 6} is the score on the i'th die.

Thus, the sample space is S = {1, 2, 3, 4, 5, 6}⁵. Since the dice are fair, our basic modeling assumption is that the random variables X₁, X₂, X₃, X₄, X₅ are independent, and each is uniformly distributed on {1, 2, 3, 4, 5}.

1. Show that the random poker dice hand X is uniformly distributed on S:

P(X in A) = #(A) / #(S) for A S.

In statistical terms, a poker dice hand is a random sample of size 5 drawn with replacement and with regard to order from the population D = {1, 2, 3, 4, 5, 6}. For more on this topic, see the chapter on Finite Sampling Models. In particular, in this chapter you will learn that the result of Exercise 1 would not be true if we recorded the outcome of the poker dice experiment as an unordered set instead of an ordered sequence.

The Value of the Hand

The value V of the poker dice hand is the random variable defined as follows:

• V = 0: None alike. Five distinct scores occur.
• V = 1: One pair. Four distinct scores occur; one score occurs twice and the other three scores occur once each.
• V = 2: Two pair. Three distinct scores occur; two of the scores occur twice each and the other score occurs once.
• V = 3: Three of a kind. Three distinct scores occur; one score occurs three times and the other two scores occur once each.
• V = 4. Full house. Two distinct scores occur; one score occurs three times and the other occurs twice.
• V = 5. Four of a kind. Two distinct scores occur; one score occurs four times and the other score occurs once.
• V = 6. Five of a kind. One score occurs five times.

2. Run the poker dice experiment 10 times in single-step mode. For each outcome, note that the value of the random variable corresponds to the type of hand, as given above.

The Density Function

Computing the density function of V is a good exercise in combinatorial probability.

3. Show that the number of different poker dice hands is #(S) = 6⁵ = 7776.

In the following exercises, you will frequently need to use the multiplication rule of combinatorics to count the number of poker dice hands of a given type. In each case, try to construct an algorithm for generating the hands of the given type, and then count the number of ways of performing each step in the algorithm.

4. Show that P(V = 0) = 720 / 7776 = 0.09259.

5. Show that P(V = 1) = 3600 / 7776 = 0.46396.

6. Show that P(V = 2) = 1800 / 7776 = 0.23148.

7. Show that P(V = 3) = 1200 / 7776 = 0.15432.

8. Show that P(V = 4) = 300 / 7776 = 0.03858.

9. Show that P(V = 5) = 150 / 7776 = 0.01929.

10. Show that P(V = 6) = 6 / 7776 = 0.00077.

11. Run the poker dice experiment 1000 times with an update frequency of 10. Note the apparent convergence of the relative frequency function to the density function.

12. Find the probability of rolling a hand that has 3 of a kind or better.

13. In the poker dice experiment, set the update frequency to 100 and set the stop criterion to the value of V given below. Note the number of hands required.

1. V = 3
2. V = 4
3. V = 5
4. V = 6

Chuck-a-Luck

Chuck-a-luck is a popular carnival game, played with three dice. According to Richard Epstein, the original name was Sweat Cloth, and in British pubs, the game is known as Crown and Anchor (because the six sides of the dice are inscribed clubs, diamonds, hearts, spades, crown and anchor). The dice are oversized and are kept in an hourglass-shaped cage known as the birdcage. The dice are rolled by spinning the birdcage.

Chuck-a-luck is very simple. The gambler selects an integer from 1 to 6, and then the three dice are rolled. If exactly k dice show the gambler's number, the payoff is k:1. As with poker dice, our basic mathematical assumption is that the dice are fair, and therefore the outcome vector is uniformly distributed on {1, 2, 3, 4, 5, 6}³:

X = (X₁, X₂, X₃) where X_i in {1, 2, 3, 4, 5, 6} is the score on die i.

14. Let Y denote the number of dice that show the gambler's number. Show that Y has the binomial distribution with parameters n = 3 and p = 1 / 6:

P(X = k) = C(3, k) (1 / 6)^k(5 / 6)^{3 - k}, for k = 0, 1, 2, 3.

15. Let W denote the net winnings for a unit bet. Show that

W = -1 if Y = 0; W = Y if Y > 0.

16. Show that

1. P(W = -1) = 125 / 216
2. P(W = 1) = 75 / 216
3. P(W = 2) = 15 / 216
4. P(W = 3) = 1 / 216

17. Run the chuck-a-luck experiment 1000 times, updating every 10 runs. Note the apparent convergence of the empirical density of W to the true density.

18. Show that

1. E(W) = -0.0787
2. var(W) = 1.239

19. Run the chuck-a-luck experiment 1000 times, updating every 10 runs. Note the apparent convergence of the empirical moments of W to the true moments. Suppose you had bet $1 on each of the 1000 games. What would your net winnings be?

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