Binomial Probability Calculator
Solve common binomial-distribution probability questions with exact and cumulative results plus mean and standard deviation.
Exact and cumulative binomial probabilities answer different questions
P(X = k) is the probability of exactly k successes. P(X ≤ k) includes every success count from zero through k, while P(X ≥ k) includes k through n. Lynara calculates all three from the same n, k and p so you can match the wording of the problem instead of re-entering the model.
The calculator evaluates probabilities in log space before summing tails, which avoids the huge intermediate combinations that make naive factorial formulas overflow. The numerical result is still only as appropriate as the binomial assumptions: fixed n, two outcome categories, independent trials under the model and a constant success probability.
Privacy: calculator inputs are processed in your browser; Lynara does not need to upload these values to calculate the result.
Formula
P(X = k) = C(n,k) × p^k × (1 − p)^(n − k)
Example
For n = 10, p = 30% and k = 3, P(X = 3) is about 0.2668.
References for this method
Useful for
- Binomial probability homework
- Exactly / at most / at least probability
- Checking binomial mean and standard deviation
Questions
What conditions make a binomial model appropriate?
Use it when there is a fixed number of trials, each trial has two outcome categories, trials are independent under the model and the success probability stays constant.
What is the difference between exact, at most and at least?
Exact is P(X = k), at most is P(X ≤ k), and at least is P(X ≥ k).