You have two D₆ dice. Die 1 is all white, whereas die 2 is all black. Games master picks one of them. Let A_k be that he picked die k.

(a) What are the prior probabilities P₀(A₁) and P₀(A₂)?

(b) What is the probability that his first toss lands white, p = P(white)?

(c) What is the probability that his tosses of the die are white first, then black?

(d) Let die 3 be “the average die” between the all-white die 1 and the all black die 2, that is, a die with 3 white sides and 3 black. What is p = P(W) for this die, and what is P(WB)?

(e) Is repeated tosses with die 3 equivalent to randomly picking either die 1 or die 2, and then tossing the picked die repeatedly?