Gambler’s Ruin. Allan and Beth currently have $2 and $3, respectively. A fair coin is tossed. If the result of the toss is H, Allan wins $1 from Beth, whereas if the coin toss results in T, then Beth wins $1 from Allan This process is then repeated, with a coin toss followed by the exchange of $1, until one of the two players goes broke (one of the two gamblers is ruined). We wish to determine a² = P (Allan is the winner Ι he starts with $2) To do so, let’s also consider probabilities a_i = P (Allan wins Ι he starts with $i) for i = 0; 1; 3; 4; and 5:

a. What are the values of a₀ and a₅?
b. Use the Law of Total Probability to obtain an equation relating a₂ to a₁ and a₃.

c. Using the logic described in (b), develop a system of equations relating a_i (i = 1, 2, 3, 4) to a_i–1 and a_i+1. Then solve these equations.

d. Generalize the result to the situation in which Allan’s initial fortune is $a and Beth’s is $b. [The solution is a bit more complicated if p = P(Allan wins $1) 6 ≠ .5.]