Abstract

The mathematical study of probability originated in the seventeenth century, when mathematicians were invited to tackle problems arising in games of chance. In such games gamblers want to know which betting odds on unpredictable events are advantageous. This amounts to a concern with probability, because probability and fair betting odds appear linked by the principle that odds of m to n for a bet on a repeatable event E are fair if and only if the probability of E is n/(m + n). For example, suppose E is the occurrence of double‐six on a throw of two dice and that its probability is 1/36 = 1/(35 + 1). Because of the intuitive connection between the probability of an event and its long‐run frequency in repeated trials, we would expect E to occur on average 1 time in (35 + 1) throws. Since to set odds on E at 35 to 1 is to agree that for a stake B, the one who bets on E gains 35B/1 if E occurs and loses B to his opponent if it does not, those must be the fair odds for a bet on E. At them, we can expect 1 win of 35B to be balanced in the long run by 35 losses of the stake B, so that neither party to the bet enjoys net gain.