Find the gambler's ruin probability and the expected duration for the following set of parameters given: P = 0.487, N = 12, and k = 6. (a) Gambler's ruin probability = (b) Expected duration = In addition, derive the following in the written part: (c) For p = 0.5, by providing the details of first-step analyses, derive the linear recurrence relationship for uk and ex (the gambler's ruin probability and the expected duration, respectively). (d) Then using an approach similar to solving linear differential equations with constant coefficients (i.e., solving a characteristic equation), solve the linear recurrence relationships in the previous part. That is, derive the general formulae for uk and ex (the gambler's ruin probability and the expected duration, respectively) when p = 0.5