A fair die is being tossed repeatedly Alice bets on 2 at eve

A fair die is being tossed repeatedly. Alice bets on 2 at every toss, while Ben bets on 3 and 4 (that is, that either 3 or 4 would be the outcome). (a) What is the probability that Alice is the first one to win a bet? (b) What is the probability that Ben is? (c) What is the expected number of tosses until Alice wins a bet? (d) What is the expected number of tosses until either Alice or Ben win a bet?

Solution

P(favourable event for A) = 1/6

P(favourable event for A) = 2/6

P(A wins) = 1/6 + (3/6)*(1/6) + (3/6)*(3/6)*(1/6) + (3/6)*(3/6)*(3/6)*(1/6) + ............

Infinite GP sum = 1/3

P(B wins) = 1 - 1/3 = 2/3

P(A wins in 1 toss) = 1/6

P(A wins in 2 toss) = (3/6)*(1/6)

P(A wins in 3 toss) = (3/6)*(3/6)*(1/6)

E[No. of tosses A will win in] = 1*1/6 + 2*(3/6)*(1/6) + 3*(3/6)*(3/6)*(1/6)

Can be found using AGP method

P(either wins) = 1/2 + 1/2 *1/2 + ...

E[No. of tosses either will win ] = 1*1/2 + 2 *1/4 +3*1/8