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

Answer:

a) 0.3246

b) 0.0043

Step-by-step explanation:

• For player 1 ; Probability of winning = P(W) = 1/3

• Probability of loosing; P(winning) + P( Loosing) = 1

• P(L) = 1 - 1/3 = 2/3

a) To find Find P(N <= 10) = P(2)+P(3)+P(4)+P(5)+P(6)+P(7)+P(8)+P(9)+P(10)

= (1/3)^2 + (1/3)^2 x 2/3 + (1/3)^2 x (2/3)^2 + (1/3)^2x (2/3)^3 + (1/3)^2 x (2/3)^4

X (1/3)^2 x (2/3)^5 + (1/3)^2 x (2/3)^6 + (1/3)^2 x (2/3)^7 + (1/3)^2 x (2/3)^8

= 0.3246

b) Find P(N = 10) = (1/3)^2 x (2/3)^8 = 0.0043