Ali and Bala take turns to roll a fair six-sided die, with Ali rolling first. The first person to ro

Question

Ali and Bala take turns to roll a fair six-sided die, with Ali rolling first. The first person to roll a six wins, and the game stops. - Let p be the probability that Ali wins. Explain why, if neither player rolls a six in the first round of one roll each, the game is in the same position as at the start. Hence show that p= 16+ 2536p, and find p. - In a different game of the same kind, each roll succeeds with probability q, where 0<q<1. Show that the probability that the first player wins is 12-q. - Explain why the first player always has the better chance of winning, and describe what happens to this advantage as q becomes very small.

Accepted Answer

(a) $p=\dfrac{6}{11}$ (b) shown (c) $\dfrac{1}{2}\lt\dfrac{1}{2-q}\lt 1$ for $0\lt q\lt 1$; the advantage tends to zero as $q\to0$

First player to roll a six

First player to roll a six

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