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Read the ratio as two weights
For odds for an event of a to b, the favourable weight is a and the unfavourable weight is b. Normalize by their sum. The weights need not be literal observed frequencies.
a and b are nonnegative weights and a + b must be positive.
Convert a probability back into odds
For a probability p strictly between zero and one, odds for the event are p to 1 − p, or p/(1 − p) to 1. Odds against reverse those two components. At p = 0 or p = 1, one of the components is zero; a finite ratio-to-one representation may then be unavailable.
| Probability | Odds for | Odds against |
|---|---|---|
| 20% | 1 to 4 | 4 to 1 |
| 50% | 1 to 1 | 1 to 1 |
| 72% | 18 to 7 | 7 to 18 |
Avoid dividing by the unfavourable weight alone
For odds for of 2 to 9, dividing 2 by 9 gives an odds ratio of approximately 0.2222 to 1. It does not give the event probability. Dividing by 2 + 9 gives approximately 0.1818, the event’s share of the total weight.
Keep for and against attached to the event
A quotation can be numerically identical while referring to the opposite direction. “9 to 1 for default” and “9 to 1 against default” imply 90% and 10% default probabilities respectively. Omitting the direction changes the interpretation.