Statistics
Binomial Probability Calculator
Compute P(X = k), P(X ≤ k), or P(X ≥ k) for a binomial random variable with parameters n and p.
Compute exact, cumulative, or tail probabilities for a binomial experiment.
Binomial distribution
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ
The calculator also returns the distribution’s expected value (np) and variance (np(1 − p)).
How to use
- Provide the number of trials n and success probability p.
- Enter the number of successes k and choose exact, at-most, or at-least mode.
- Read the probability, expected value, and variance.
Example
Input: n = 10, p = 0.4, k = 4, mode = exact
Output: P(X = 4) ≈ 0.2508
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for Binomial Probability Calculator: binomial probability calculator, binomial distribution calculator, binomial cumulative calculator, probability of exactly k successes. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
Binomial Probability Calculator: Finds exact or cumulative probabilities for binomial experiments. It's built around binomial distribution, probability, discrete, so you can go from a raw question to a checked answer without switching tools.
The math behind it: The calculator also returns the distribution’s expected value (np) and variance (np(1 − p)). The core relationship is P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, shown above the calculator so you can see exactly how your inputs turn into the result.
To use it well: (1) Provide the number of trials n and success probability p. (2) Enter the number of successes k and choose exact, at-most, or at-least mode. (3) Read the probability, expected value, and variance. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering n = 10, p = 0.4, k = 4, mode = exact returns P(X = 4) ≈ 0.2508. Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
How does rounding work?
Probabilities are displayed with up to four decimal places, but calculations use double precision internally.
Is cumulative probability inclusive?
Yes. P(X ≤ k) includes the k term, and P(X ≥ k) includes k as well.
What formula does the Binomial Probability Calculator use?
The calculator also returns the distribution’s expected value (np) and variance (np(1 − p)).
How do I use the Binomial Probability Calculator?
Provide the number of trials n and success probability p. Enter the number of successes k and choose exact, at-most, or at-least mode. Read the probability, expected value, and variance.