Question

2. Suppose you have 15 identical dollar bills to give to your friends Adam, Beth and Charlie. You give all 15 dollar bills away by randomly selecting a person for each dollar. Each person is equally likely to recieve a dollar for each dollar (a) What is the probability that Adam gets exactly 4 dollars? (b) If you know that each person recieved at least one dollar, what is the probability that Adam gets exactly 4 dollars? (c) What is the expected number of dollars that Adam gets? (d) If you know that each person recieved at least one dollar, what is the expected number of dollars that Adam gets?

Answer 1

2a, X = no. of dollars received by Adam y = no. of dollar s received by Beth Z no. of dollars received by Charlie Then (X, Y, Z)~Multinomial (15, 1/3, 1/3,1/3) Then marginal distribution of XBinomial(15, 1/3) Probability that Adam gets eractly 4 dollars 15-4 15 12 = 0.1948 b. Now, (X, Y, z)l(x 21,Y 2 1, Z21Multinomial(12,1/3, 1/3, 1/3) Since we know that each per son received at least one dollar, so first we have to give one dollar to each and then remaining 12 dollars are distributed among three Then X|(X1,Y 21 Z 1Binomial(12, 1/3) P(x -3x 21,Y 2 1,z21)//3) 0.2120 (c) Expected number of dollars that Adam gets 15 (1/3) 5 (d) Expected number of dollars that Adam gets given that each per son received at least one dollar 12 (1/3)4