4. Generalize (7.2) and (7.1) in the following way:

(a) If a gambler is asked to bet m for a chance to win n, what is the probability of winning that will make this bet fair? Confirm that m ¼ 1 gives (7.2).

(b) If the gambler knows that the probability of a win is p, what ratio of amount bet to amount won, m n from part (a), will make this a fair bet? Confirm that m ¼ 1 gives

(7.1).

(c) Show that if p is irrational in part (b), a fair bet requires an irrational value for m

n . Conclude that since bets and payo¤s must be rational numbers, a bet with an irrational probability of winning can never be fair.