# Exam P Practice Problem 23 – Medical Screening

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Problem 23A
A researcher in a medical study screens prospective participants for having high serum cholesterol. Let $Y$ represent the number of prospective participants screened until the fifth person wtih high serum cholesterol is found. It is known from past experience that the expected number of prospective participants screened until finding the fifth person with high serum cholesterol is 25. The researcher wants to know the likelihood of achieving his objective by screening just 12 to 15 prospective participants.

What is the probability of finding 5 participants with high serum cholesterol by screening 12 to 15 prospective participants (inclusive for both endpoints)?

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Problem 23B
One ball is drawn successively with replacement from an urn containing one red ball and three white balls. Find the probability that at most 6 draws are necessary to obtain 3 red balls.

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Answers

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### 2 responses

1. for A. I believe Y has negative binomial with p=1/6 and r=5, then this is what I did, I used 12 chose 5, times (1/6)^5 times (5/6)^7, and the same with 13, 14 and 15, just changes the numbers above p and q, and I added the results. but my answer is totally different from the answer key. could you please display the actual solution. Thank you so much.

2. Bebe

There are two versions of the negative binomial distributions. One is to count the # of failures to get the rth success. The other is to count the # of trials to get the rth success. This problem calls for the second version. Hope this gives you a good start.

Dan Ma