Question

You have two fair, 6-sided dice. Die 1 has 4 white faces and 2 black faces. Die 2 has 2 white faces and 4 black faces. You roll Die 1. If it comes up white, then Die 1 is the “chosen die” and you put Die 2 away, but if it comes up black, then Die 2 is the “chosen die” and you put Die 1 away. You now roll the chosen die twice, noting the color that comes up in each of these rolls.

Let W1 be the event that the first of these 2 rolls of the chosen die comes up white, and let W2 be the event that the second of these 2 rolls of the chosen die comes up white. Give exact answers as simplified fractions and provide a brief 1-2 sentence explaining your reasoning.

(a) Calculate P(W1) and P(W2).

(b) Calculate P(W2 | W1) (Hint: you cannot assume without proof that W1 and W2 are independent.)

(c) Are W1 and W2 independent? Justify your answer.

(d) If the 2 rolls of the chosen die both come up white, what is the probability that Die 1 is being used?

Answer 1