1. Write a program to compute the probability wx of Exercise 24 for given values of x, p, and T . Study the probability that the gambler will ruin the bank in a game that is only slightly unfavorable, say p = .49, if the bank has significantly more money than the gambler.

2. We considered the two examples of the Drunkard's Walk corresponding to the cases n = 4 and n = 5 blocks (see Example 11.13). Verify that in these two examples the expected time to absorption, starting at x, is equal to x(n - x).

See if you can prove that this is true in general. Hint : Show that if f (x) is the expected time to absorption then f (0) = f (n) = 0 and

f (x) = (1/2)f (x - 1) + (1/2)f (x + 1) + 1

for 0 x n. Show that if f1(x) and f2(x) are two solutions, then their difference g(x) is a solution of the equation

g(x) = (1/2)g(x - 1) + (1/2)g(x + 1).

Also, g(0) = g(n) = 0. Show that it is not possible for g(x) to have a strict maximum or a strict minimum at the point i, where 1 ≤ i ≤ n - 1. Use this to show that g(i) = 0 for all i. This shows that there is at most one solution. Then verify that the function f (x) = x(n - x) is a solution.