Question

Two teams A and B play a series of games until one team wins four games. We assume that the games are played independently and that the probability that A wins any game is p and B wins (1-p). What is the probability that the series ends after...

a) 5 games
b) 6 games
c) 7 games
d) n games

Answer 1

a)

For team A to win the series of 5 games it has to win any 3 games of the first 4 games and then win the last game too.

So there are four ways for this to happen

i) AAABA

ii) AABAA

iii) ABAAA

iv) BAAAA

which is ways. The probability that A win will 4 times is and team B wins once is .

So probability that team A wins 4 times and game ends after 5 games =

Similarly probability that team B wins 4 times and game ends after 5 games =

So the total probability that game ends after 5 games =

b)

For team A to win the series of 6 games it has to win any 3 games of the first 5 games and then win the last game too.

So there are 10 ways for this to happen

which is ways. The probability that A win will 4 times is and team B wins twice is .

So probability that team A wins 4 times and game ends after 6 games =

Similarly probability that team B wins 4 times and game ends after 6 games =

So the total probability that game ends after 6 games =

c)

For team A to win the series of 7 games it has to win any 3 games of the first 6 games and then win the last game too.

So there are 20 ways for this to happen

which is ways. The probability that A win will 4 times is and team B wins thrice is .

So probability that team A wins 4 times and game ends after 7 games =

Similarly probability that team B wins 4 times and game ends after 7 games =

So the total probability that game ends after 7 games =

d)

For n=4

Then probability that game will end after 4 games

= Probability that teams A wins all 4 games + Probability that teams B wins all 4 games

=

For n=8

then game will definitely end.

So probability than game will end in 8 games = 1

For n>8

No game will stop at 8th trail.

So for n>8, Probability that game will end = 0.