Denote the probability that a run of j successive

Reference no: EM131005700

1. A coin having probability p of coming up heads is continually ?ipped. Let Pj(n) denote the probability that a run of j successive heads occurs within the ?rst n ?ips.

(a) Argue that Pj(n) = Pj(n - 1) + pj(1 - p)[1 - Pj(n - j - 1)]

(b) By conditioning on the ?rst non-head to appear, derive another equation relating Pj (n) to the quantities Pj(n - k), k = 1, ... , j.

2. A and B roll a pair of dice in turn, with A rolling ?rst. A's objective is to obtain a sum of 6, and B's is to obtain a sum of 7. The game ends when either player reaches his or her objective, and that player is declared the winner.

a. Find the probability that A is the winner.

b. Find the expected number of rolls of the dice.

c. Find the variance of the number of rolls of the dice.

Reference no: EM131005700

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