Suppose you have a die that has probability p of resulting in the outcome 6 when rolled, where p is a continuous random variable that is uniformly distributed over [0, 1/3 ]. Suppose you start rolling
this die. (The value of p does not change once you start rolling.) Give exact answers as simplified
fractions.
(a) Compute the probability that the first roll is 6.
(b) Compute the probability that the first two rolls are both 6.
(c) Let S 1 be the event that the first roll is 6 and S 2 be the event that the second roll is 6. Compute P(S2 |S1).
(d) Are the outcomes of the first two rolls independent? Justify your answer.
(e) Compute the probability that the first k rolls are each 6. Compare this with the probability if it were a fair die.