25. Independent trials, resulting in one of the outcomes 1, 2, 3 with respective probabilities p1, p2, p3,

3i

=1 pi = 1, are performed.

(a) LetN denote the number of trials needed until the initial outcome has occurred exactly 3 times. For instance, if the trial results are 3, 2, 1, 2, 3 ,2, 3 thenN = 7.

Find E[N].

(b) Find the expected number of trials needed until both outcome 1 and outcome 2 have occurred.