2. A hat check girl returns three hats at random to the three customers who had previously checked them. Smith, Jones an

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2 A Hat Check Girl Returns Three Hats At Random To The Three Customers Who Had Previously Checked Them Smith Jones An 1 (64.86 KiB) Viewed 11 times

2. A hat check girl returns three hats at random to the three customers who had previously checked them. Smith, Jones and Brown, in that order, each receives one of the hats. (a) Discuss an appropriate probability model for this experiment. (b) Find the p.f. of M, the number of matches. (c) If there are n hats to be returned to n men, what is Pr(M = n) ? Pr(M = n 1) ? Discuss these results.
3. Suppose W has a p. f. given by W -1 0 1 Pr(W = W) 3C 3C 6C (a) Determine C. (b) What is the p.f. of X = 2W + 1? 4. A tetrahedron is tossed into the air and the bottom face on which it comes to rest is noted. Each of the four faces-numbered 1, 2, 3 and 4-has an equal probability of being on the bottom when the tetra- hedron comes to rest. Suppose the tetrahedron is tossed twice. (a) What is an appropriate sample space for this experiment? What is the probability associated with each outcome? Discuss carefully any assumptions which you have made in assigning these probabilities. (b) Define a r.v. X to be the sum of the outcomes on the two tosses. Find the p.f. of X.
6. A salesman has four different stores as indicated in the map below: 1.5 mi. 3 mi. X Home 5 mi 1.5 mi. B He decides to visit the store which telephones him first and then return home. The probability that the first call is from A is 1/6, from B is 1/3, from C is 1/3, from D is 1/6. Let X represent the distance to the first store and Y the total distance traveled to the store and back. (a) Find the p.f. of X. (b) Find the p. f. of Y.
11. A medical clinic is studying a new treatment. Suppose that each of five patients is given the new treatment and after one week the doctor diag- noses his condition as improved or not improved. (a) Discuss an appropriate probability model for this experiment if "improved" and "not improved" are equally likely. (b) If, after receiving the new treatment, the patients are twice as likely to improve, how does this affect the probability model devel- oped in (a)? (c) Define a random variable X, the number of patients who improve. Find the p. f. of X using the assumptions in (1) part (a), (2) part (b).