Question

1. In Monopoly, your token is allowed to leave the “jail” cell if you roll doubles: you roll two 6-sided dice and each shows the same face. Zach hates being in jail, because it reminds him of watching “The Wire” on TV. So he invents a couple of weighted dice that are not independent. In particular, if you roll either die on its own it’s a fair die: each outcome has probability 1/6. But if you roll one die and then the other, the red die will take the same outcome as the blue die exactly half the time: all other outcomes are equally likely.

   (a) Suppose you roll a 3 on the blue die. What is the probability distribution of the red die given this outcome on the blue die?

(b) Find the full probability distribution for the value of the sum of the two faces of the dice. (c) What is the probability you roll doubles?

(c) What is the probability you roll doubles?

(d) What is the probability that you roll a 7 as the sum of the two dice?

Answer 1

** Bayes' theorem has been used to calculate the probability of getting a certain value on the blue dice AND a certain value on the red dice. Since they are dependent, we can't simply multiply the individual probabilities.

** The notion P(B=b, R= r) has been replaced with P(b,r).

C. Doubles happen when either of (1,1),(2,2),(3,3),(4,4),(5,5) or (6,6) occur. Each has a probability of 1/12. Hence, the probability of a double is 6*1/12 = 1/2.

D. From the distribution table, P(sum = 7) = 6/60