Question

Suppose three players go on multiple rounds of kart race. In each round, every player has a winning probability of 1/3, independent of other rounds. Let N denote the number of rounds until player 1 has two consecutive wins.

a) Find P(N <= 10)

b) Find P(N = 10)

Answer 1

For Player 1 ,P(W) = 1/3,P(L) = 2/3

A. Now , there should be atleast two rounds for two consecutive wins. Now, if player 1 plays 3 rounds , he has to win in the last 2 rounds and lose in the first round. Similarly, if player 1 plays 4 rounds , he has to win in the last 2 rounds and lose in the first & second round.

Probability of two consecutive wins in 2 rounds is given by, P(2) = (1/3*1/3) = 1/9

Probability of two consecutive wins in 3 rounds is given by, P(3) = (1/3*1/3*2/3) = 2/27

Probability of two consecutive wins in 4 rounds is given by, P(4) = (1/3*1/3*2/3*2/3) = 4/81

Probability of two consecutive wins in 5 rounds is given by, P(5) = (1/3*1/3*2/3*2/3*2/3) = 8/243

Probability of two consecutive wins in 6 rounds is given by, P(6) = (1/3*1/3*2/3*2/3*2/3*2/3) = 16/729

Probability of two consecutive wins in 7 rounds is given by, P(7) = (1/3*1/3*2/3*2/3*2/3*2/3*2/3) = 32/2187

Probability of two consecutive wins in 8 rounds is given by, P(8) = (1/3*1/3*2/3*2/3*2/3*2/3*2/3*2/3) = 64/6561

Probability of two consecutive wins in 9 rounds is given by, P(9) = (1/3*1/3*2/3*2/3*2/3*2/3*2/3*2/3*2/3) = 128/19683

Probability of two consecutive wins in 10 rounds is given by, P(10) = (1/3*1/3*2/3*2/3*2/3*2/3*2/3*2/3*2/3*2/3) = 256/59049

Therefore, P(N<=10) = P(2)+P(3)+P(4)+P(5)+P(6)+P(7)+P(8)+P(9)+P(10) = 32.5% = 0.3251

B. P(N = 10) = 256/59049 = 0.43% = 0.0043

Hope I clarfied your query