39. For the gambler's ruin model of Section 5, let Mf denote the mean number of games that must be played until the gambler either goes broke or reaches a fortune of N, given that he starts with /', / = 0, 1, . . . , N. Show that Mi satisfies M0 = MN = 0; M, = 1 + pMi+l + qMt_x, i = 1, ...,N - 1

(f ) Prove your guess in part

(e) either by showing that Equation (4.6) is satisfied or by using the results of Example 4.25.
(g) If ñ = 1, what is the expected time until there are only white balls in the urn if initially there are / white and TV - / black?