A clinic specializes in shoulder injuries. A patient is randomly selected from the population of all clinic clients. Let \(S\) be the number of doctor visits for shoulder problems in the past six months. Assume the values of \(S\) are \(s=1,2,3\), or 4 . Patients at the shoulder clinic are also asked about knee injuries. Let \(K=\) the number of doctor visits for knee injuries during the past six months. Assume the values of \(K\) are \(k=0,1\) or 2 . The joint probability distribution of the numbers of shoulder and knee injuries is shown in Table P.10. Use the information in the joint probability distribution to answer the following questions. Show brief calculations for each

a. What is the probability that a randomly chosen patient will have two doctor visits for shoulder problems during the past six months?

b. What is the probability that a randomly chosen patient will have two doctor visits for shoulder problems during the past six months given that they have had one doctor visit for a knee injury in the past six months?

c. What is the probability that a randomly chosen patient will have had three doctor visits for shoulder problems and two doctor visits for knee problems in the past six months?

d. Are the number of doctor visits for knee and shoulder injuries statistically independent? Explain.

e. What is the expected value of the number of doctor visits for shoulder injuries from this population?

f. What is the variance of the number of doctor visits for shoulder injuries from this population?

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TABLE P.10 Table for Exercise P.14 Knee = K 0 1 2 f(s) 1 0.15 0.09 0.06 Shoulder S 2 0.06 3 0.02 0.10 0.2 4 0.02 0.08 0.10 f(k) 0.33