Odds and Probability

Probability measures successes against all trials; odds measure successes against failures. A probability of 0.75 is odds of 0.75/0.25 = 3, quoted as "3 to 1 on"; odds of 1:4 against, entered here as 0.25, correspond to P = 0.25/1.25 = 0.2. Enter probabilities as decimals from 0 to 1 (the % unit works too) and odds as the number of units in favour per one unit against.

Bookmakers quote the mirror image — "4/1" means four units against one in favour, so P = 1/5 = 0.2 — and they deliberately publish odds implying probabilities that sum to more than 1 across a field, the overround that pays the shop. Odds also make Bayes' theorem elegant: posterior odds equal prior odds times the likelihood ratio, which is why epidemiologists and machine-learning models work in log-odds. The usual trap is reading "3 to 1" as a 1-in-3 chance; it is 1 in 4.