In logistic regression, what is the difference between odds and probability?

## Two ways to express the same chance A probability `p` is a share of a whole: out of all accounts, what fraction churn? It is bounded, `0 <= p <= 1`. Odds re-express the same chance as a ratio of two complementary counts: how often the event happens versus how often it does not. ``` odds = p / (1 - p) p = odds / (1 + odds) ``` The two carry identical information — either can be recovered from the other — but they live on different scales. | Probability | Odds | Spoken | |---|---|---| | 0.05 | 0.0526 | about 1-to-19 against | | 0.20 | 0.25 | 1-to-4 | | 0.50 | 1 | even | | 0.80 | 4 | 4-to-1 on | | 0.99 | 99 | 99-to-1 on | Notice the asymmetry of the odds scale. Probability is symmetric around 0.5; odds are symmetric around 1, but multiplicatively — 0.25 and 4 are mirror images (each is the reciprocal of the other), while 0.2 and 0.8 are mirror images additively. ## Why odds have an unbounded range As `p` approaches 1, the denominator `1 - p` approaches 0 and the odds grow without limit. As `p` approaches 0, the odds approach 0 but never go negative. So odds occupy `(0, infinity)` — half of the real line. Taking a logarithm finishes the job: ``` logit(p) = log(p / (1 - p)) ``` The logit maps `(0, 1)` onto the entire real line: `logit(0.5) = 0`, `logit(0.8) = log 4 = 1.386`, `logit(0.2) = log 0.25 = -1.386`. It is symmetric in the sense that `logit(1 - p) = -logit(p)`. ## Why the model needs the unbounded scale A linear predictor `b0 + b1*x1 + b2*x2 + ...` can take any real value: push `x1` far enough and it will exceed 1 or drop below 0. If you set that expression equal to a probability, the model is making claims it cannot honour. Setting it equal to the **logit** instead is safe, because every real number corresponds to exactly one probability: ``` log(p / (1 - p)) = b0 + b1*x1 + ... p = 1 / (1 + exp(-(b0 + b1*x1 + ...))) ``` The second line is the logistic (inverse-logit) function — the familiar S-curve that flattens as it approaches 0 and 1 and is steepest at `p = 0.5`. This is what the "logit link" means: a function applied to the mean of the response so that the transformed mean is linear in the predictors. ## Consequences you should be able to state - **Additive on the log-odds scale, multiplicative on the odds scale.** Adding a coefficient to the linear predictor multiplies the odds by `exp(coefficient)`. That is the whole reason odds ratios are the natural currency of a logistic model. - **A zero linear predictor means even chance.** If the linear predictor evaluates to 0, the odds are 1 and the probability is 0.5. The intercept is the log-odds when every predictor is 0. - **The same change in log-odds is not the same change in probability.** Moving the logit from 0 to 0.5 moves probability from 0.500 to 0.622 (about 12 points). Moving it from 3 to 3.5 moves probability from 0.953 to 0.971 (under 2 points). The curve saturates. - **Odds are not percentages.** "Odds of 4" is not "a 4% chance" and not "a 400% chance"; it is a probability of 0.8. Confusing the two is the single most common slip in interview answers. ## When each scale is the right one to speak Inside the model, log-odds are the working scale — that is where the linearity lives. Odds and odds ratios are the reporting scale for statisticians and for the model's own summary table. Probabilities and differences in probabilities are the scale for everyone else: a stakeholder asking "how much more likely is churn?" is asking about probability, and answering in odds without converting is a reliable way to be misunderstood. A final vocabulary note: gamblers usually quote odds *against* ("5 to 1 against"), which is `(1 - p) / p`, the reciprocal of the statistical convention. Statistics uses odds *for* the event. If someone quotes odds and you cannot tell which convention they mean, ask before converting.