Probability to Odds and Logit Calculator

StatisticsLast updated: August 22, 2026

Probability, odds, and the logit express the same information on three different scales. Probability is squeezed between 0 and 1; odds = p / (1 − p) remove that ceiling and open the range from 0 to infinity; the logit = ln(odds) gives a symmetric scale running from minus infinity to plus infinity. The dependent variable of logistic regression is exactly this logit scale, which is why model coefficients are interpreted as changes in log-odds rather than in probability.

This calculator converts a value entered on any one of these scales into the other two, and optionally lets you enter a logistic regression coefficient (β). When β is supplied, the odds ratio e^β, the new probability after the coefficient is applied, the change in percentage points, and the marginal effect p(1 − p)·β are computed together. The results also include a conversion table for common probability values and a logistic curve drawn on the logit axis.

Pick the scale your value is measured on; all of the other scales are then calculated.
Both a decimal point and a decimal comma are accepted (e.g. 0.25).
The effect of a one-unit increase on the log-odds. Leave it empty to see the conversions only.

Probability, Odds, and Logit Conversion Formulas

Odds (in favor):  odds = p / (1 − p)
Odds (against):   odds' = (1 − p) / p = 1 / odds
Probability:      p = odds / (1 + odds)
Logit:            L = ln(odds) = ln[ p / (1 − p) ]
Inverse logit:    p = e^L / (1 + e^L) = 1 / (1 + e^(−L))
Odds ratio:       OR = e^β
Coefficient:      L_new = L + β   →   p_new = e^(L+β) / (1 + e^(L+β))
Marginal effect:  ∂p/∂x ≈ p · (1 − p) · β
Logistic model:   ln[ p / (1 − p) ] = β₀ + β₁x₁ + … + βₖxₖ

The logit transformation opens the [0, 1] probability range onto the whole real line; the linear model of logistic regression is built on that scale. Odds and logit are undefined at p = 0 and p = 1, so those endpoints are not accepted.

How to Calculate

  1. Choose the type of value you have: probability, percentage, odds in favor, odds against, or logit.
  2. Type the value; both a decimal point and a decimal comma are accepted (e.g. 0.25).
  3. If you are interpreting logistic regression output, enter the coefficient (β) as well; leave it empty to see the conversions only.
  4. Read the probability, odds, betting odds, and logit equivalents together in the results card.
  5. If you entered β, compare the odds ratio (e^β) with the change in the probability in percentage points; the odds ratio is constant, but the change in percentage points depends on the baseline probability.
  6. Use the conversion table to see neighboring probability values, and the logistic curve to see how the logit maps onto probability.

Worked Examples

From probability to odds and logit

A probability of p = 0.25 corresponds to 0.3333 on the odds scale (betting odds of 1 : 3.00) and to -1.0986 on the logit scale. A coefficient of β = 0.7 gives an odds ratio of e^0.7 = 2.0138; the new logit is -0.3986, the new odds are 0.6713, and the new probability is 0.4016 — so the probability rises from 25.00% to 40.16%, an increase of 15.16 percentage points.

Probability (p): 0.250000 · Complement probability (1 − p): 0.750000 · Odds (in favor): 0.333333

From betting odds to probability

An event with odds in favor of 4 (that is, 4 : 1) has probability p = 4 / 5 = 0.80; the odds against are 0.25 and the logit is ln 4 = 1.3863. The decimal odds are 1 / 0.8 = 1.25, which in a fair bet means a total payout of 1.25 units on a stake of 1 unit.

Probability (p): 0.800000 · Complement probability (1 − p): 0.200000 · Odds (in favor): 4.000000

Interpreting a logistic regression coefficient

With a baseline risk of 10% (odds 0.1111, logit -2.1972), a one-unit increase in a variable with coefficient β = 1.1 gives an odds ratio of e^1.1 = 3.0042. The new logit is -1.0972 and the new probability is 25.03%: the risk rises by 15.03 percentage points. The odds triple, but the probability does not — and that distinction is the heart of interpreting an odds ratio.

Probability (p): 0.100000 · Complement probability (1 − p): 0.900000 · Odds (in favor): 0.111111

Frequently Asked Questions

What exactly is the difference between odds and probability?
Probability is the ratio of favorable outcomes to all outcomes and lies between 0 and 1. Odds are the ratio of favorable outcomes to unfavorable ones and range from 0 to infinity. For example, the probability of rolling a 6 on a six-sided die is 1/6 ≈ 0.167, while the odds are 1/5 = 0.20; in everyday language that is what "5 to 1" means.
Why is the logit transformation necessary?
A linear model that tries to predict a probability directly can produce values below 0 or above 1; moreover, the error variance depends on the probability, so the constant-variance assumption breaks down. The logit opens the [0, 1] range onto the whole real line and lets the linear model be built on an unbounded scale. When the prediction is converted back with the inverse logit, the result is always a valid probability.
How large does e^β have to be to count as an effect?
An odds ratio of 1 means no effect. Roughly speaking, values around 1.5 are read as small, 2 to 3 as moderate, and 4 or above as strong; values below 1 point to a protective (risk-reducing) effect. The real criterion, though, is the confidence interval: if it includes 1, the effect is not statistically significant no matter how large it looks.
Are the odds ratio and the relative risk the same thing?
No. The relative risk is a ratio of two probabilities, the odds ratio a ratio of two odds. When the event is rare (roughly p < 0.10) the two converge, but as the event becomes more common the odds ratio drifts away from the relative risk and makes the effect look larger than it is. The relative risk cannot be computed in case-control studies, which is why the odds ratio is used there; in cohort studies the relative risk should be preferred.
Why do I get an error when I enter p = 0 or p = 1?
At p = 1 the odds are 1/0, which is undefined, and at p = 0 the logit is ln(0), which goes to minus infinity. In real data this situation usually signals complete separation, and the logistic regression coefficients fail to converge. The usual remedies are Firth's penalized likelihood or a small correction to the event counts (adding 0.5, for instance).
How do I perform these conversions in Excel, SPSS, and R?
In Excel it is enough to write =p/(1-p) for the odds, =LN(p/(1-p)) for the logit, and =EXP(L)/(1+EXP(L)) for the inverse logit. In R the functions qlogis(p) and plogis(L) do the same job; the coefficients from glm(y ~ x, family = binomial) are on the log-odds scale and are converted to odds ratios with exp(coef(model)). In the SPSS Binary Logistic output, the B column holds the coefficient and the Exp(B) column gives the odds ratio directly.
Why does the same odds ratio produce different changes in probability?
The logistic curve is not linear; it is steepest around p = 0.5. The same coefficient β produces a large change in percentage points when the baseline probability is near 0.5 and a very small one when it is close to the extremes (0 or 1). This is why policy and clinical decisions should report the absolute risk difference and the marginal effect alongside the odds ratio.