12 A syndicate of college students spends weekends gambling in Las Vegas. They begin week 1 with W dollars.

At the beginning of each week, they may wager any amount of their money at the gambling tables. If they wager d dollars, then with probability p, their wealth increases by d dollars, and with probability 1  p, their wealth decreases by d dollars. Their goal is to maximize their expected wealth at the end of T weeks.

a Show that if p  1/2

, the students should bet all their money.

b Show that if p 1/2

, the students should bet no money.

(Hint: Define ft(w) as the maximum expected wealth at the end of week T, given that wealth is w dollars at the beginning of week t; by working backward, find an expression for ft(w).)