Question

#5. Suppose that two teams (A and B) play a series of games that ends when one of them has won 3 games. Suppose that each game played is, independently, won by team A with probability p. Find the probability distribution of number of games.

Answer #1

Let probability that team A win is p and probability that team B win is 1-p.

Let A shows the event team A win and B shows the event that team B win.

Let X is a random variable shows the number of games. Here X can take values 3, 4, and 5.

When X=3, then two possible outcomes are

AAA, BBB

So the probability for X=3 will be

----------------------

When X=4, then one team win one game and other team three games. The possible outcomes are

BAAA, ABAA, AABA, ABBB, BABB, BBAB

So the probability for X=4 will be

-------------------

When X=5, then one team win two games out of first four games. Other team win two games out of first two games and must win last game.

For team A: Number of ways selecting 2 games out of 4 is C(4,2) = 6

Likewise for team B number of ways selecting 2 games out of 4 is C(4,2) = 6

So the probability for X=5 will be

Hence, the probability distribution of number of games is:

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