Risk Taking - European Odds

In this article, I will teach you how to interpret a set of European odds.

(Disclaimer: I am not promoting gambling! But if you really choose to gamble, do it responsibly!)

For learning purpose, I shall use a coin-flipping game. There are only one bookmaker and one punter in this example. The coin used is believed to be fair and hence the chance of showing a head is the same as the chance of showing a tail. Lastly, we also assume that the bookmaker is not out to make money or lose money after many rounds of coin-flipping game.

The most rational set of odds set by the bookmaker will be 2 for Head and 2 for Tail. It means that if punter bet Head and the coin indeed shows Head, the punter will win $1 for every $1 the punter bets. However, if the coin shows Tail, the punter will lose every dollar that the punter bets. Why is the odd set at 2? It is because the probability of showing Head for a fair coin is ½. Over a long run, the bookmaker has 50% chance of losing the bet and 50% chance of winning the bet. If the bookmaker loses the bet, the bookmaker will lose $1 for each $1 bet placed by the punter. If the bookmaker wins the bet, the bookmaker will win $1 for each $1 bet placed by the punter. Over the long run, the expected gain by the bookmaker = ½(-$1)+1/2(+$1) = $0 for every dollar betted. This is what the bookmaker wants – not making or losing money after many games.

In fact, if the odds for Head = 1/x and the odds for Tail = 1/y, it means that bookmaker BELIEVES that the chances of the coin getting Head is x and the chance for Tail is y.

So, in what situation should a rational person enter the game and place the bet?

A smart punter will enter a game if the punter does not agree with the probabilities that the bookmakers believe. If punter somehow manage to get the coin to show Tail all the time (without alerting the bookmaker), it means that the punter believes that the probability of Tail =1. As the foolish bookmaker still thinks that the coin is fair, he/she will still believe that probability of Tail =1/2 (despite using an unfair coin). The smart punter will definitely enter the game and bet on Tail. Over the long run, the expected return of the punter is 1(+$1)+0(-$1) = $1 for every dollar betted. The smart punter is better off by betting. However, the smart punter will stay out from betting Head as probability of Head perceived by bookmarker > probability of Head perceived by the smart punter. The punter will be worse off betting Head as the expected return will be 0(+$1)+1(-$1) = -$1 per dollar betted.

In the next article, I will discuss about bookmaker’s rule and the non-existence of arbitrage in a single betting game if the bookmaker is rational.