1.) In your job as a croupier at the River Rock Casino, you catch a cheater with a loaded (unfair) die. It is weighted to throw “6” half the time while all other numbers are equally likely. The following chart describes the probability distribution of this unfair die. Outcome X=1 X=2 X=3 X=4 X=5 X=6 Pr[X = x] (1/2)/5 = 1/10 = 0.10 (1/2)/5 = 1/10 = 0.10 (1/2)/5 = 1/10 = 0.10 (1/2)/5 = 1/10 = 0.10 (1/2)/5 = 1/10 = 0.10 1/2 = 0.50 What is the mean, variance, and standard deviation of a single toss of this unfair die? 2.) The weather at Seymour Mountain can be classified as one of two kinds of days: powder days (P) when there is fresh snow and other days (O) when there is not. The length of time spent waiting in lift lines is either Long (L) or Short (S). On powder days, the probability of a short wait is Pr(S|P) = 1/3. On other days, the probability of a short wait is Pr(S|O) = 1/2. Powder days occur with probability 1/3, a short wait is 1 minute, and a long wait is 15 minutes. What is the expected waiting time on any given day?