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:

All possible sequences of events V and M by set played that will result in player V winning are: VV, VMV, MVV

All possible sequences of events V and M by set played that will result in Player M winning the match are: MM, MVM, VMM

The probability that Player V will win a match against Player M is 0.4575

The probability that a match between Player V and Player M will consist of 3 sets given that Player V wins the match is 0.3443

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

Explanation:

All possible sequences of events V and M by set played that will result in player V winning are: VV, VMV, MVV

Because, in all of them V wins 2 sets.

At the same way, all possible sequences of events V and M by set played that will result in Player M winning the match are: MM, MVM, VMM

Now, the probability that Player V will win a match against Player M is calculated as:

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

Where:

P(VV) = 0.5*0.6 = 0.3

Because, there is equally likely to win the first set and if Player V wins the first set, the probability that she will win the second set is 0.60.

At the same way P(VMV) and P(MVV) are calculated as:

P(VMV) = 0.5*(1-0.6)*0.45 = 0.09

P(MVV) = 0.5*(1-0.7)*0.45 = 0.0675

So, the probability that Player V will win a match against Player M is:

P(V wins) = 0.3 + 0.09 + 0.0675 = 0.4575

On the other hand, the probability that a match between Player V and Player M will consist of 3 sets given that Player V wins the match is calculated as:

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

Because VMV and MVV are the events in which a match between Player V and Player M will consist of 3 sets and Payer V wins.

So, the probability that a match between Player V and Player M will consist of 3 sets given that Player V wins the match is:

$\frac{0.09+0.0675}{0.4575}=0.3443$

Finally, to find the expected number of sets played when Player V competes in a match with Player M, we need to find the probability that they play 2 sets and the probability that they play 3 sets as:

P(2 sets) = P(VV) + P(WW)

P(3 sets) = 1 - P(2 sets)

Where P(WW) is calculated as:

P(WW) = 0.5*0.7 = 0.35

Therefore: P(2 sets) = 0.3 + 0.35 = 0.65

                  P(3 sets) = 1 - 0.65 = 0.35

Then, the expected number of sets played when Player V competes in a match with Player M is:

E(sets) = 2*0.65 + 3*0.35 = 2.35

Because the probability that they play 2 sets is 0.65 and the probability that they play 3 sets is 0.35