1. Consider a gambler who on each bet either wins 1 with probability 18/38 or loses 1 with probability 20/38. (These are the probabilities if the bet is that a roulette wheel will land on a speci?ed color.) The gambler will quit either when he or she is winning a total of 5 or after 100 plays. What is the probability he or she plays exactly 15 times?
2. Show that
(a) E[XY|Y = y]= yE[X|Y = y]
(b) E[g(X, Y)|Y = y]= E[g(X, y)|Y = y]
(c) E[XY]= E[YE[X|Y]]