Two tennis players Mel and Nat face each other in a best of

Two tennis players, Mel and Nat, face each other in a best of 5 (i.e. a player must win 3). Mel has a 75% of winning each game and we assume the games are independent. Determine the probability mass function (p.m.f) and cumulative distribution function (c.d.f) for how many games the series will last. Determine the expected number of games they will play

Solution

There are two tennis players Mel and Nat face each other in a best of 5.

i.e. there are 5 games.

Mel has a 0.75 probability of winning each game and we assume that the games are independent.

Let X be a Binomial random variable with parameter n=5 and p=0.75.

The p.m.f. of X is,

P(X=x) = (n C x) * p^x * (1-p)^(n-x) (C is used for combination)

P(X=x) = (5 C x) * (0.75^x) * (0.25^(5-x))

And the C.D.F of X is denoted by F(x).

F(x) = P(X x)

= (5 C x) * (0.75^x) * (0.25^(5-x)) (X is from 0 to x)

Expected number of games they will play,

We know that in Binomial distribution E(X) = np

E(X) = 5 * 0.75 = 3.75

Approximately games they will play.