Odds

The odds on an event is the probability that the event occurs divided by the probability that the event does not occur. The word comes from gambling and represents the ratio of stakes by two parties who want to make a fair bet on whether or not the event happens.

For example, one in six male passengers survived the Titanic disaster, five in six died. The odds on surviving, for a man, were therefore (1/6) / (5/6) = 1/5, one says "the odds were one to five against". Choosing a male passenger at random and betting on their survival, it would be reasonable to place 1 euro on a bet that person survived against 5 euros that they didn't survive. The gambler who bets on survival places 1 Euro on the table, the gambler who bets on non survival places 5 euros on the table, the winner takes all.

In medical statistics and epidemiology, the term "odds ratio" is often used and stands for the ratio between the odds on the same event in two different circumstances which we want to compare.

For instance, the odds on a male passenger surviving the Titanic disaster were 1 to 5 against, the odds on a female passenger surviving the Titanic disaster were 2 to 1 in favour of survival. The odds ratio is (1/5) / (2/1) = 1/10.

Odds, and odds ratio, are important in the theory of the statistical analysis of 2×2 contingency tables. The log odds and log odds ratio turn out to be canonical parameters when the statistical model is seen as an exponential family.