Expected values and variances in bookmaker payouts : a theoretical approach towards setting limits on odds

Abstract

This study summarizes the key methods of displaying probabilities as odds and provides simple mathematical derivation of a number of key statements in setting odds. Firstly it estimates the expected bookmaker profit as a function of wagers placed and bookmaker margin. Moreover it shows that odds set by bookmakers should have implied probabilities that add up to at least one, otherwise arbitrage is present. Bookmakers can increase profitability by offering more multiples, also known as accumulators, and can lower variation in payouts by maintaining the ratio of wagers and implied probability per outcome.