Troy stops at the Wild Rose Casino. He will play roulette, betting on 16 until he wins 3 times. We are interested in questions related to the number of times he plays. For this type of problem, we will use the negative binomial distribution. Recall that we formulate this distribution by the number of losses, X, before the 3rd win. In other words, if we want the probability that the 3rd win occurs on the 18th spin, we will calculate P (X = 15). (a) Give the PMF for X. Recall that the probability of a 16 is 1 38 for each spin. Make sure to give the values that X can take on. (b) Calculate the probability that the 3rd win occurs on the 18th spin, i.e., P (X = 15). Make this calculation using the dnbinom function in R. (c) Calculate the probability that the 3rd win occurs on 18 or fewer spins, i.e., P (X 15). Make this calculation using the pnbinom function in R. (d) Give E(X) and SD(X). Note: Use the formulas on the Families of Discrete Distribu tions sheet. (e) In roulette, a bet on a single number pays 35 to 1