Two players T im and Jessica alternately and independently

. Two players, T im and Jessica, alternately and independently flip a coin and the first player to obtain a head wins. Let P(head) = p and assume Tim flips first. a) What is the probability that Tim wins? b) If P(Tim wins) = p/(1 ? (1 ? p) 2 ), then show that for all p, (0 < p < 1), P(Tim wins) > 1/2.

Solution

probability for getting heads = p

probability for getting tails = 1-p

tim starts the game

so first he should get the heads.

this will be either in first round itself or second or third and so on

so probability for tim to win is

p + (1-p)^2 * p + (1-p)^4 * p + ..

= p( 1 + (1-p)^2 + (1-p)^4 + ....)

we have a sum of infinite series

so this will be p ( 1/(1-(1-p)^2))

(1-p)^2 =1 + p^2 -2p

therefore the answer is p/(2p-p^2) = 1/(2-p)