Recall the assumptions that "A random experiment has exactly three possible outcomes, referred to as outcomes 1, 2, and 3, with probabilities p1 > 0, p2 > 0, and p3 > 0, where p1 + p2 + p3 = 1." We consider a sequence of independent trials, at each of which the specified random experiment is performed.) For i = 1, 2, let Ni be the number of trials needed for outcome i to occur, and put N := N1 N2.

(a) Show that N is independent of 1_{N1

before 7, what is the (conditional) expected number of rolls? (Ans: 3.272727)

(d) Under the assumptions above, find the expected number of times outcome 1 occurs before the first occurrence of outcome 2. (Ans: p1/p2)

(e) When rolling a pair of dice, find the expected number of times the total 4 (resp., 5; 6; 8; 9; 10) appears before the first total of 7. (Ans: 3/6 (resp., 4/6; 5/6; 5/6; 4/6; 3/6))