1. If a die is rolled four independent times, what is the probability of one four, two fives, and one six, given that at least one six is produced?

2. Let the p.d.f. f( x) be . positive on, and only , on, the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, so that f( x) = [(11 - x)/] f( x - I ), x = 1,2, 3, . . . , 10. Find /(x).

3. Let X and Y have a bivariate normal distribution with µ, = 5, µ2 = 10, ai = 1, = 25, and p = j. Compute Pr (7 <>Y <>19lx = 5).

4. Say that Jim has three cents and that Bill has seven cents. A coin is tossed ten independent times. For each head that appears, Bill pays Jim two cents, and for each tail that appears, Jim pays Bill one cent. What is the probability that neither person is in debt after the ten trials?