ABSTRACT
This needs some explanation. Suppose you are the player: if heads comes up on the first throw you get £2, if it comes up on the second throw you get £4, on the third £8, and so on. For each successive throw the payout doubles. The chance of heads on the first throw is 1⁄2; the chance that heads comes up first on the second throw is the chance of tails on the first × 1⁄2, that is, 1⁄4; the chance that heads comes up first on the third throw is 1⁄2 × 1⁄2 × 1⁄2, and so on. For each successive throw the chance halves, just as the payout doubles. The expected gain from the first toss is £2 × the probability that it is heads (1⁄2), that is, £1; from the second throw it is £4 × 1⁄4 = £1; from the third £8 × 1⁄8 = £1, and in general for the nth throw it is £2n × 1⁄2n = £1. Since there is no limit to the possible number of throws before heads comes up, the sum for the expected gains, 1 + 1 + 1 + . . . goes on for ever and the expectation is infinite. Yet would you pay any sum, however large, to participate?
