Probability Calculator

Probability Calculator: run probability calculations online. Formula, assumptions, and interpretation guide.

This Probability Calculator uses a binomial model: it counts successes across a fixed number of independent trials, with the same success chance on every trial. Enter the number of trials n, the number of successes k, and the per-trial probability p. The calculator reports the probability of exactly k successes, the probability of at most k successes, and the number of ways to choose k successes from n trials. It is not a general event-probability, conditional-probability, or Bayesian calculator.

Use the binomial distribution calculator for a matching model, or compare results with the mean calculator and confidence interval calculator when you need a different statistical summary.

Enter n as a whole number from 0 to 20. It is the fixed number of trials in the model.
Enter k as a whole number from 0 to n. This is the success count you want to evaluate.
Enter p from 0 to 1 as a decimal probability for success on each trial; for example, 25% is 0.25.
Read P(X = k) for exactly k successes and P(X ≤ k) for k or fewer successes.
Read C(n,k) as the number of distinct trial positions that can contain the k successes.

Binomial probability formulas

Exactly k successes: P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ.

At most k successes: P(X ≤ k) = Σᵢ₌₀ᵏ C(n,i) × pⁱ × (1−p)ⁿ⁻ⁱ.

Ways to place k successes: C(n,k) = n! / (k! × (n−k)!).

The same p is used for every trial. The model assumes trials are independent and each trial has only two outcomes: success or failure.

Interpreting the binomial results

Exactly k is different from k or fewer

P(X = k) counts one exact success total. P(X ≤ k) adds the probabilities for zero, one, and every other success count up to k. These values answer different questions; the cumulative result cannot be substituted for the exact result.

C(n,k) is a count of arrangements, not a probability. It says how many trial sequences have k successes, while the p terms weight those sequences according to the model's per-trial success and failure chances.

Statistics tips and best practices

Define what counts as success before choosing p, and use the same definition for every trial.
Use p as a probability between 0 and 1, not a whole-number percentage such as 25.
Independence matters: one trial's outcome must not change the chance on a later trial.
Use a different model when trials have changing probabilities, more than two outcomes, or dependence.
For a question about at least k successes, use the complement of P(X ≤ k−1); this page reports the at-most result directly.

With n = 2, k = 1, and p = 0.5, P(X = 1) is 50%, P(X ≤ 1) is 75%, and C(2,1) is 2.
When p = 0, the model assigns probability 1 to zero successes; when p = 1, it assigns probability 1 to n successes.
The input limit n ≤ 20 keeps this page focused on small, transparent exact calculations.

Common mistakes to avoid

Using the model for dependent trials, such as repeated draws without replacement.
Changing the meaning of success or the probability p from one trial to another.
Reading P(X ≤ k) as the probability of exactly k successes.
Treating C(n,k) as a percentage rather than a count of arrangements.
Using this page for Poisson, Bayesian, conditional, or arbitrary combined-event calculations.

This is a mathematical model based on the inputs and assumptions shown. It does not establish that trials are independent or that p is known accurately; check those assumptions before using the result in a decision.