Question

In the finals of a rugby tournament, two teams play a best of 5 series. Each team has a probability of 1/2 of winning the first game. For each subsequent game, the team that won the previous game has a 7/10 chance of winning, while the other team has a 3/10 chance of winning. If p is the probability that the series lasts exactly 4 games, what is ⌊1000p⌋?

Answer 1

I am not familiar with your notation of ⌊1000p⌋
but I would do it this way:

after a W prob(W) = 7/10 , prob(L) = 3/10
after a L, Prob(W) = 3/10, prob(L) = 7/10


To have a 4-game series in a best of 5 setup, the winning team could win in these ways
LWWW --- (1/2)(3/10)(7/10)(7/10) = 147/2000
WLWW --- (1/2)(3/10)(3/10)(7/10) = 63/2000
WWLW --- (1/2)(7/10)(3/10)(3/10) = 63/2000

Prob = (147+63+63)/2000 = 273/2000