=+Example 7.6. The gambler has initial capital a and plays at unit stakes until his capital increases to c (0) ≤a ≤

c) or he is ruined. Here Fo = a and W„ = 1, and so F =a + S, The policy is bounded by

c, and F, is c or 0 according as the gambler succeeds or fails. If p = ; and if s is the probability of success, then a = Fo = E[F, ] = sc. Thus s = a /c. This gives a new deriva-

tion of (7.7) for the case p = ;. The argument assumes however that play is certain to terminate. If p ≤ ;, Theorem 7.2 only gives s ≤ a/c, which is weaker than (7.7).