Probability Calculator

Binomial and Poisson probabilities, combinations and permutations, and probabilities of combined events (union, intersection, conditional and independence checks).

The Probability Calculator computes exact, cumulative and tail probabilities for binomial and Poisson distributions, nCr and nPr counts, and the union, intersection and conditional probabilities of two events.

Free, runs in your browser. Nothing you enter is uploaded.

How to use it

Choose a mode.
Enter the parameters.
Read the probabilities.

What you enter

Parameters for the chosen mode

What you get

Exact and cumulative probabilities
Mean, variance, SD
nCr / nPr

Formula

P(X = k) = C(n,k) pᵏ (1−p)ⁿ⁻ᵏ   |   P(X = k) = λᵏ e^(−λ) / k!   |   P(A∪B) = P(A) + P(B) − P(A∩B)

Binomial counts successes in n independent trials with probability p; Poisson counts rare events with mean λ. Logarithms of the gamma function keep the calculation exact for large n.

Worked example

Input: Binomial n = 10, k = 3, p = 0.5

Result: P(X = 3) = 0.1172, P(X ≤ 3) = 0.1719

Three heads in ten fair coin tosses happens about 12 % of the time.

Questions

What is the difference between P(X ≤ k) and P(X < k)?

P(X ≤ k) includes k; P(X < k) = P(X ≤ k − 1). The calculator shows both, plus P(X ≥ k) and P(X > k).

Combination or permutation?

Combinations when order does not matter (choosing a committee); permutations when it does (arranging a sequence).

Limitations

Binomial assumes independent trials with constant p.

References

Ross SM (2014). A First Course in Probability, 9th ed. Pearson.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.