Suppose that an object can be at any one of n+1 equally spaced points x₀, x₁. . . x_n. When an object is at location x_i, it is equally likely to move to either x_i−1 or x_i+1 and cannot directly move to any other location. Consider the probabilities {Pi}ⁿ_i=0 that an object starting at location x_i will reach the left endpoint x₀ before reaching the right endpoint x_n. Clearly, P₀ = 1 and P_n = 0. Since the object can move to x_i only from x_i−1 or x_i+1 and does so with probability 1/2 for each of these locations, P_i = 1/2 P_i−1 + 1/2 P_i+1, for each i = 1, 2. . . n − 1.

a. Show that

b. Solve this system using n = 10, 50, and 100.

c. Change the probabilities to α and 1 − α for movement to the left and right, respectively, and derive the linear system similar to the one in part (a).

d. Repeat part (b) with α = 1/3 .

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1-2 0 0 1 1-2 12 0 0 12 1 1-2 1-2