4.10.6 Consider the set of sequences of seven letters chosen from W and L. We ma

Question

4.10.6 Consider the set of sequences of seven letters chosen from W and L. We may think of these sequences as representing the outcomes of a match of seven games where w means the first team wins the game and L means the second team wins the game. The match is won by the first team to win four games (thus, some games may never get played, but we need to include their hypothetical outcomes in the points in order th at we have a probability space of equally likely points). a) W at is the probability that a team will win the match, given that it has won the first game? l) What is the probability that a team will win the match, given that it has won the first two I c) What is the probability that a team will win the match, given th at it has won two out of the first three games?

Explanation / Answer

the sample space consists of 27 =128 equiprobable points
if team has 4 or hypothetical >4 wins in 7 matches,

P(win) =(7C4+7C5+.7C6+7C7)/27 =1/2

A.having won the 1st, "we" only need 3 wins in 6 matches,
Pr = (6C3 +6C4 +6C5 +6C6)/26 = 42/64 = 21/32

B.having won the 1st & 2nd, we only need 2 wins in 5 matches,
Pr = (5C2 + 5C3 + 5C4+ 5C5)/25 = 26/32 = 13/16

==>(1 - (5C0 + 5C1))/25 = 26/32,

C. having won 2 out of the 1st 3, we only need 2 wins in 4 matches,
Pr = (4c2+4c3+4c4)/2^4 = 11/16
for probabilities of winning other than 0.5, say, 0.6 using the binomial eg part (c) would become
4C2*0.62*0.42 + 4C3*0.63*0.4 + 0.64 = 0.8208

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