Kim has a strong first serve; whenever it is good (that is, in) she wins the point 70% of the time. Whenever her second serve is good, she wins the point 40% of the time. Fifty dash five percent of her first serves and 70% of her second serves are good. (a) What is the probability that Kim wins the point when she serves? (b) If Kim wins a service point, what is the probability that her 1st serve was good?
Accepted Solution
A:
Answer:a) There is a 63.35% probability that Kim wins the point when she serves.b) If Kim wins a service point, there is a 55.80% probability that her 1st serve was good.Step-by-step explanation:We have these following probabilitiesA 50.5% probability that her first serve is good.A 70% probability that her second serve is good.If her first serve is good, she has a 70% probability of winning the point.If her second serve is good, a 40% probability of winning the point.(a) What is the probability that Kim wins the point when she serves?This is the sum of 70% of 50.5% and 40% of 70%. So[tex]P = 0.7*(0.505) + 0.4*(0.7) = 0.6335[/tex]There is a 63.35% probability that Kim wins the point when she serves.(b) If Kim wins a service point, what is the probability that her 1st serve was good?This can be formulated as the following problem:What is the probability of B happening, knowing that A has happened.It can be calculated by the following formula[tex]P = \frac{P(B).P(A/B)}{P(A)}[/tex]Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.SoWhat is the probability of Kim's first serve being good, given that she won the point?P(B) is the probability of Kim's first serve being good. So [tex]P(B) = 0.505[/tex].P(A/B) is the probability of Kim's winning the point when her first serve is good. So [tex]P(A/B) = 0.70[/tex].P(A) is the probability of Kim's winning the point. From a), that is [tex]P(A) = 0.6335[/tex][tex]P = \frac{P(B).P(A/B)}{P(A)} = P = \frac{0.505*0.70}{0.6335} = 0.5580[/tex]If Kim wins a service point, there is a 55.80% probability that her 1st serve was good.