Two tennis players A and B will play against each other. Player A has probabilit

Question

Two tennis players A and B will play against each other. Player A has probability p of winning a set against B.

a) If the match is made of up to 5 sets and the winner is the first who wins 3 sets (as in every Grand Slam tournament), what is the probability that Player A will win the match?

b) If the match is made of up to 3 sets and the winner is the first who wins 2 sets (as in most tennis tournaments), what is the probability that Player A will win the match?

c) Which of the previous situations is better for Player A if p < 1 2 ?

d) Which of the previous situations is better for Player A if p = 1 2 ?

Explanation / Answer

Probability of player A winning a set = p

so Probability of player B winning a set = 1-p

a)

for player A to win the 5 set match the favourable patterns are AAA, AABA, ABAA, BAAA, ABABA, ABBAA, AABBA, BABAA, BBAAA, BAABA

probability that Player A will win the 5 set match = p3 + p*p*(1-p)*p + p*(1-p)*p*p + (1-p)*p*p*p + p*(1-p)*p*(1-p)*p + p*(1-p)*(1-p)*p*p +(1-p)*p*(1-p)*p*p + (1-p)*p*p*p+ (1-p)*p*p*(1-p)*p

= p3 +3*p3(1-p) +6*p3(1-p)2  

b)

for player A to win the 3 set match the favourable patterns are AA, ABA, BAA

probability that Player A will win the 3 set match = p2+p*(1-p)*p+(1-p)*p*p

= p2+2p2(1-p)

c)

let us take any value of p such that p < 1/2 and compare the values

let p = 1/4

probability that Player A will win the 5 set match =  p3 +3*p3(1-p) +6*p3(1-p)2

= (1/4)3+3*(1/4)3(1-1/4)+6*(1/4)3(1-1/4)2 = 0.1035

probability that Player A will win the 3 set match =  p2+2p2(1-p) = (1/4)2 + 2(1/4)2(1 -1/4) = 0.1562

so if p<1/2 then the probability for A to win 3 set match is better than 5 set match

d)

if p = 1/2

probability that Player A will win the 5 set match =  p3 +3*p3(1-p) +6*p3(1-p)2  

= (1/2)3+3*(1/2)3(1-1/2)+6*(1/2)3(1-1/2)2 = 0.5

probability that Player A will win the 3 set match =  p2+2p2(1-p) = (1/2)2 + 2(1/2)2(1 -1/2) = 0.5

so if p = 1/2 then the probability for A to win 3 set match is same as probability to win 5 set match

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