Question

In women’s tennis, a player must win 2 out of 3 sets to win a match. If a player wins the first 2 sets, she wins the match and the third set is not played. Player V and Player M will compete in a match. Let V represent the event that Player V wins a set, and let M represent the event that Player M wins a set. List all possible sequences of events V and M by set played that will result in Player V winning the match. List all possible sequences of events V and M by set played that will result in Player M winning the match. Player V and Player M have competed against each other many times. Historical data show that each player is equally likely to win the first set. If Player V wins the first set, the probability that she will win the second set is 0.60. If Player V loses the first set, the probability that she will lose the second set is 0.70. If Player V wins exactly one of the first two sets, the probability that she will win the third set is 0.45. What is the probability that Player V will win a match against Player M? What is the probability that a match between Player V and Player M will consist of 3 sets given that Player V wins the match? What is the expected number of sets played when Player V competes in a match with Player M?

Answer 1

Answer:

1)All possible sequences of events V and M by set played that will result in Player V winning the match = VV; MVV; VMV

2) All possible sequences of events V and M by set played that will result in Player M winning the match:

MM; VMM; MVM

3) The probability player V wins the match = 0.4575

4) Probability that a match between Player V and Player M will consist of 3 sets given that Player V wins the match. = 0.3443

The expected number of sets played when Player V competes in a match with Player M = 2.35

Explanation:

Given:

Event player V wins a set = V

Event player M wins a set = M

1)All possible sequences of events V and M by set played that will result in Player V winning the match = VV; MVV; VMV

2) All possible sequences of events V and M by set played that will result in Player M winning the match:

MM; VMM; MVM

3) The probability player V wins the match:

P(V) = P(VV) + P(MVV) + P(VMV)

P(VV) means probability player V wins first and second set. In the first set she had an equal chance of winning, i.e 50%. We are told if she wins first set, probability of winning second set is 0.60.

Therefore, P(VV) = 0.50 * 0.60

= 0.30

P(MVV) means player V lost the first set, but won the remaining two sets.

P(MVV) = 0.50 *(1-0.70) * 0.45

= 0.0675

VMV means player V won the first set, lost the second, and won the third set.

P(VMV) = 0.50 * (1-0.60) * 0.45

= 0.09

Therefore

P(VV) + P(MVV) + P(VMV)

= 0.30 + 0.0675 + 0.09

= 0.4575

Probability V wins is 0.4575

4) Probability that a match between Player V and Player M will consist of 3 sets given that Player V wins the match.

Here it means player V must lose one of the first two sets, but will still win the match.

Calculating, we have:

$\frac{P(MVV) + P(VMV)}{P(V)}$

$= \frac{0.0675 + 0.09}{0.4575}$

= 0.344262 ≈ 0.3443

5) The expected number of sets played when Player V competes in a match with Player M.

Let's first find the probability 2 sets are played.

P(2) = P(VV) + P(MM)

We know P(VV) = 0.30

P(MM) = 0.50 * 0.70 = 0.35

Therefore, P(2) = 0.30 + 0.35

P(2)= 0.65

Let's now find probability of 3 sets happening.

P(3) = P(2) not happening

Therefore

P(3) = 1 - P(2)

P(3) = 1 - 0.65 = 0.35

Since probability of 2 sets is 0.65 and probability of 3 sets is 0.35, the expected number of sets will be :

(2 * 0.65) + (3 * 0.35)

= 1.30 + 1.05 = 2.35

The expected number of sets = 2.35