Everyday Jo practices her tennis serve by continually servin

Everyday Jo practices her tennis serve by continually serving until she has had a total of 50 successful serves. If each of her serves is, independent of previous ones, successful with probability 0.4, approximately what is the probability she will need more than 100 serves to accomplish her goal?

Hint: Imagine even if Jo is successful that she continues to serve until she has served exactly 100 times. What must be true about her first 100 serves if she is to reach her goal?

Use DeMoire\'s-LaPlace\'s Theorem?

Solution

Sum of P(50 serves) to P(100 serves).

Each serve has 0.6 failure.

Sum of (50 serves to 100 serves) means that at 50 serves, # failures is 0, at 100 serves, # of failures is 50.

So P(x <= 50) where x is # of failures is the probability you want to find, so that you can do 1 - P(x <= 50) to find the probability that she will need more than 100 serves to accomplish her goal. <= is Less than or equal to.

I used Minitab to compute this for me, I find that P(x <= 50) = 0.0270992.

By the way this is a binomial probability. If you want to do it by hand you have to find P(x = 0) all the way to P(x = 50).

So P(More than 100 serves) = 1 - 0.0270992 = 0.9729008