Question

Suppose that Serena has a .7 probability of defeating Venus in a set of tennis, independently from set to set. For questions 1 – 3, suppose that they play a best-of-three-set match, meaning that the first player to win two sets wins the match.

1. Determine the probability that Serena wins the match by winning the first two sets.

2. Determine the probability that the match requires three sets to be played (meaning that each player wins one of the first two sets).

3. Determine the probability that Serena wins the best-of-three-set match.

For questions 4 – 5, suppose that they play a total of three sets, regardless of who wins the sets.

4. Determine the probability that Venus wins at least one of the three sets.

5. Determine the probability that Serena wins at least two of the three sets.

Answer 1

Given ,

Probability of Serena defeating Venus in a set of tennis = 0.7

S : Event of Serena defeating Venus in a set of tennis

P(S) = 0.7

V : Event of Venus defeating in a set of tennis

P(V) = 1 - P(S) = 1 - 0.7 = 0.3

1. Probability that Serena wins the match by winning the first two sets.

S: Event of Serena winning first Set

S : Event of Serena winning the second set

Event of Serena wins the match by winning the first two sets : S and S

Probability that Serena wins the match by winning the first two sets = P (S and S)

P (S and S) = P(S) x P(S) = 0.7 x 0.7 = 0.49

2. Probability that the match requires three sets to be played (meaning that each player wins one of the first two sets).

The match requires three sets to be played( (meaning that each player wins one of the first two sets).will happen

SV (Serena Wins first set, Venus second set ) or VS (Venus wins first set and serena wins second)

Probability that the match requires three sets to be played ( meaning that each player wins one of the first two sets) = P(SV or VS )

P(SV or VS) = P(SV) + P(VS)

P(SV) = P(S)xP(V)= 0.7 x 0.3= 0.21

P(VS) = P(V)xP(S)xP(S) = 0.3 x 0.7 = 0.21

P(SV or VS) = P(SV) + P(VS) = 0.21 + 0.21 = 0.42

Probability that the match requires three sets to be played = 0.42

3. Probability that Serena wins the best-of-three-set match

Event of Serena wins the best-of-three-set match can happen : SS or SVS or VSS

Probability that Serena wins the best-of-three-set match = P(SS or SVS or VSS)

P(SS or SVS or VSS) = P(SS)+P(SVS)+P(VSS)

P(SS) = P(S)xP(S) = 0.7x0.7 = 0.49

P(SVS) = P(S)xP(V)xP(S) = 0.7x0.3x0.7 = 0.147

P(VSS) = P(V)xP(S)xP(S) = 0.3 x 0.7 x 0.7 = 0.147

P(SS or SVS or VSS) = P(SS)+P(SVS)+P(VSS) = 0.49 + 0.147 + 0.147 = 0.784

suppose that they play a total of three sets, regardless of who wins the sets

4. Probability that Venus wins at least one of the three sets

Probability that Venus wins at least one of the three sets = 1- Probability of Venus losing all the sets

Probability of Venus losing all the sets

Event of Venus losing all the sets i.e Serena Winning all the sets i.e SSS

Probability of Venus losing all the sets = P(SSS)

P(SSS) = P(S) x P(S) x P(S) = 0.7 x 0.7 x 0.7 = 0.343

Probability of Venus losing all the sets = 0.343

Probability that Venus wins at least one of the three sets = 1- Probability of Venus losing all the sets = 1-0.343 = 0.657

Probability that Venus wins at least one of the three sets = 0.657

5. probability that Serena wins at least two of the three sets

Event of Serena wins at least two of the three sets can happen i.e winning 2 sets or three sets:

SSS(three sets) or

SSV or SVS or VSS (Two sets)

probability that Serena wins at least two of the three sets = P(SSS or SSV or SVS or VSS)

P(SSS or SSV or SVS or VSS) = P(SSS) + P(SSV) + P(SVS) + P(VSS)

P(SSS) = P(S)xP(S)xP(S) = 0.7 x 0.7 x 0.7 = 0.343

P(SSV) = P(S) x P(S) x P(V) = 0.7 x 0.7 x 0.3 = 0.147

P(SVS) = P(S) x P(V) x P(S) = 0.7 x 0.3 x 0.7 = 0.147

P(VSS) = P(V) x P(S) x P(S) = 0.3 x 0.7 x 0.7 = 0.147

P(SSS) + P(SSV) + P(SVS) + P(VSS) = 0.343+0.147+0.147+0.147= 0.784

P(SSS or SSV or SVS or VSS) = 0.784

probability that Serena wins at least two of the three sets = 0.784