Question

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?

Answer 1

Answer:

0.46 (46%)

Step-by-step explanation:

We have the following data:

- Probability that player V wins the first set:

$p(1)=0.50$

(because the text says the two players are equally likely to win the first set)

- Probability that player V wins the 2nd set if he has won the 1st set:

$p(2|1)=0.60$

So, the probability that player V wins the first 2 sets is:

$p(12)=p(1)\cdot p(2|1)=(0.50)(0.60)=0.30$ (1)

Instead, the probability that player V loses the 2nd set if he has won the 1st set is 0.40 (=1-0.60), so

$p(2^c|1)=0.40$

So, the probabiity that player V winse the 1st set but loses the 2nd set is

$p(12^c)=p(1)\cdot p(2^c|1)=(0.50)(0.40)=0.20$ (2)

Also, we have:

- Probability that player V loses the 1st set:

$p(1^c)=0.50$

- Probability that she will lose the 2nd set in this case is 0.70, it means that the probability that she will win the 2nd set if she lost the 1st set is 0.30, so:

$p(2|1^c)=0.30$

So, the probability that she will lose the 1st set and win the 2nd set is:

$p(1^c2)=p(1^c)\cdot p(2|1^c)=(0.50)(0.30)=0.15$ (3)

Combined together (2) and (3), this means that the probability that player V wins exactly 1 set out of the first two sets is:

$p(1/2)=p(12^c)+p(1^c2)=0.20+0.15=0.35$ (4)

At this point, the probability that she will win the 3rd set is

$p(3)=0.45$

This means that the overall probability that she will win the 3rd set if she won exacty 1 of the first 2 sets is:

$p(1/2,3) = p(1/2)\cdot p(3)=(0.35)(0.45)=0.16$ (5)

So, the overall probability that player V will win a match against player M is the sum of (1) and (5):

$p(W)=p(12)+p(1/2,3)=0.30+0.16=0.46$