Odds

Odds are a way of expressing a probability as a ratio between the chances of success and the chances of failure. Where probability compares the favourable cases to the total of all possible cases, odds compare the favourable cases directly to the unfavourable ones. The two notations carry exactly the same information.

The intuition is that of a balance. Odds of "1 to 3" mean: for one favourable case, three unfavourable cases. You place on one side what succeeds, on the other what fails, and read the ratio between the two pans.

Moving from one notation to the other is mechanical. A probability of 1 in 4 — that is, one favourable case out of four — leaves three unfavourable cases, which gives odds of "1 to 3". Conversely, odds of "1 to 5" bring together one favourable case and five unfavourable ones, six cases in all, hence a probability of 1/6. The general rule: odds of "a to b" correspond to a probability of a divided by the sum a + b.

The classic trap is to confuse the odds with the probability and to overestimate one's chances. Hearing "1 to 3" can suggest "one chance in three", whereas it is one chance in four, that is 25% and not 33%. Likewise, in a bet, the odds posted by an operator include its margin and therefore do not reflect the real probability of the event.

On this site, knowing how to convert odds into a probability helps you assess a drawing situation accurately rather than by guesswork. It is a useful habit of a responsible reading of chance: bringing ratios of chances back to clear numbers so as not to be mistaken about what is probable.