Question

The National Basketball Association (NBA) championship is a series of up to 7 games between 2 teams. The first team to win 4 games wins the series. Suppose that the probability that Team A beats Team B in any given game is p > 0.5, independent of the outcome of all other games. Let f(n, p) be the probability that Team A wins the series for a series of n games (n is odd) and a game-win probability p.

Explain why f(n, p) = P(X ≥ (n + 1)/2), where X is a binomial random variable with parameters n and p.

Answer 1

For each game there are two chances that is winning or losing let p be the probability of winning

So every game winning follows Bernoulli distribution (1,p)

N each game is independent of each other and we know that sum of independent bernoulli distribution follows binomial distribution thus if X denotes number of wins out of n game then x will follow binomial (n,p) , now for odd number of games e.g in a championship of 7 games one can only win if he will win at least (7+1)/2 games i.e at least 4 such that in worst case opposite can only win remaining 3 in that situation also player one will win

So for n(odd) number of games probability of winning championship is p(x>=(n+1)/2) where x follows binomial (n,p)