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RUDIKE
1 month ago
15

Player V and Player M have competed against each other many times. Historical data show that each player is equally likely to wi

n 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?
Mathematics
1 answer:
PIT_PIT [12.4K]1 month ago
5 0

Answer:

0.46 (46%)

Step-by-step explanation:

The following data is available:

- The probability of player V achieving victory in the first set:

p(1)=0.50

(as indicated, both players are equally able to win the first set)

- The chance of player V winning the 2nd set provided success in the 1st set:

p(2|1)=0.60

Thus, the probability of player V winning the first 2 sets is:

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

Conversely, the possibility of player V losing the 2nd set after winning the 1st set is 0.40 (=1-0.60), leading to

p(2^c|1)=0.40

Hence, the probability that player V wins the 1st set yet loses the 2nd set is

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

Also, it is noted that:

- The likelihood that player V loses the 1st set:

p(1^c)=0.50

- The probability of her losing the 2nd set in this situation amounts to 0.70, implying the chance of winning the 2nd set after losing the 1st is 0.30, hence:

p(2|1^c)=0.30

Thus, the probability that she loses the 1st set but wins the 2nd set is:

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

Combining (2) and (3) results in the probability that player V wins exactly 1 set from the first two is:

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

At this stage, the likelihood that she wins the 3rd set is

p(3)=0.45

Thus, the total probability that she wins the 3rd set after winning only 1 of the first 2 sets is:

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

Ultimately, the overall chance that player V triumphs against player M can be expressed as the sum of (1) and (5):

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

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