Binomial Probability Calculator
The probability of exactly k successes in n independent trials.
Result
Probability
0.117188
Probability (%)
11.7188
How it works
The binomial distribution gives the probability of exactly k successes in n independent trials, when each trial succeeds with the same probability p. Four conditions define it: a fixed number of trials, independence, only two outcomes per trial, and a constant p. The formula P = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ has two parts: pᵏ(1−p)ⁿ⁻ᵏ is the probability of one specific sequence of k successes and n−k failures, and C(n,k) counts how many such sequences exist. For exactly 3 heads in 5 coin flips: C(5,3)·(0.5)⁵ = 10/32 ≈ 31.25%. Two handy companions: the expected number of successes is n·p, and the variance n·p·(1−p). And mind the wording — this calculator gives exactly k; "at least k" requires summing the probabilities from k up to n.
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Frequently asked questions
When is it binomial?
When trials are independent, fixed in number, and each has the same two outcomes with constant probability.
What is p?
The probability of success on a single trial — for a fair coin, 0.5.
How do I get "at least k successes" instead of "exactly k"?
Add up the exact probabilities from k to n (or compute 1 minus the sum from 0 to k−1). This calculator returns the "exactly k" building block you sum.
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