108. 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 To do so, let s also consider ai  P(Allan wins 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. Hint: Condition 0

a2  P1Allan is the winner 0 he starts with $2 2 P1D 0 B2 ¨ C2 c  P1C 0 B1 2  P1C 0 B2 2, 1D 0 B1 ¨ C2 on the result of the rst coin toss, realizing that if it is aH, then from that pointAllan starts with $3.

c. Using the logic described in (b), develop a system of equations relating ai (i  1, 2, 3, 4) to ai1 and ai1. Then solve these equations. Hint:

Write each equation so that ai  ai1 is on the left hand side. Then use the result of the rst equation to express each other ai  ai1 as a function of a1, and add together all four of these expressions (i  2, 3, 4, 5).

d. Generalize the result to the situation in which Allan s initial fortune is $a and Beth s is $b.

Note: The solution is a bit more complicated if p  P(Allan wins $1)  .5.