28. Answer exercise 4 in the case of a lottery rather than a bet. (Hint: A lottery is the same as a bet with di¤erent payo¤s.)

(a) If a gambler buys the lottery ticket for m, and either wins 0 or n, what is the probability of winning that will make this lottery fair?

(b) If the gambler knows that the probability of a win is p, what ratio of the cost of a ticket to amount won, m n from part (a), will make this a fair lottery?

(c) Show that if p is irrational in part (b), a fair lottery requires an irrational value for m n . Conclude that since ticket prices and payo¤s must be rational numbers, a lottery with an irrational probability of winning can never be fair.