Probability
Probability is the ratio of the number of favorable outcomes to the total number of outcomes:
Probability always ranges from 0 to 1 (or equivalently 0% to 100%). A probability of 0 means the event cannot happen; a probability of 1 means it is certain.
Example: if 30 out of 100 patients respond to a treatment, the probability of response is 30/100 = 0.30, or 30%.
Odds
Odds is the ratio of the number of favorable outcomes to the number of unfavorable outcomes:
Odds range from 0 to infinity (not 0 to 1). Odds of 1 (or 1:1) mean the event is equally likely to occur or not occur. Odds greater than 1 favor the event occurring; odds less than 1 favor the event not occurring.
Using the same example: the odds of response = 30/70 = 0.429 (or "30 to 70", simplified "3 to 7").
Conversion Table
| Probability | Odds | Verbal |
|---|---|---|
| 0.10 (10%) | 0.10 / 0.90 = 0.111 (1:9) | 1 to 9 against |
| 0.25 (25%) | 0.25 / 0.75 = 0.333 (1:3) | 1 to 3 against |
| 0.50 (50%) | 0.50 / 0.50 = 1.0 (1:1) | Even odds |
| 0.75 (75%) | 0.75 / 0.25 = 3.0 (3:1) | 3 to 1 in favor |
| 0.90 (90%) | 0.90 / 0.10 = 9.0 (9:1) | 9 to 1 in favor |
Converting Between Probability and Odds
Probability to Odds
Example: p = 0.30 → Odds = 0.30 / 0.70 = 0.429
Odds to Probability
Example: Odds = 0.429 → p = 0.429 / 1.429 = 0.30
Why Odds Are Used in Research
Odds ratios (ORs) appear throughout medical and epidemiological literature, particularly in case-control studies and logistic regression. There are two main reasons:
- Logistic regression models the log-odds of an outcome as a linear function of predictors. The exponentiated regression coefficient directly gives the odds ratio.
- In case-control studies, you cannot directly measure the probability of disease (because you start by selecting cases and controls), but you can estimate the odds ratio from the study design. The odds ratio approximates the relative risk when the disease is rare.
- Odds ratios have a convenient symmetry: the OR for exposure given disease equals the OR for disease given exposure.
Common Misinterpretation
An odds ratio is not the same as a relative risk (risk ratio). When the event is common (probability > 10%), odds ratios overestimate the relative risk. Reporting an OR as if it were a risk ratio inflates the perceived effect size. Always check whether the paper reports odds ratios or relative risks, and which is appropriate for the study design.