Binomial Probability P(X = k)
Enter trials n, successes k and success probability p to get P(X=k) = C(n,k)·p^k·(1−p)^(n−k) for a binomial random variable.
P(X=k)
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Binomial probability P(X = k)
The binomial distribution counts how many successes you get in n independent Bernoulli trials, where each trial succeeds with probability p: P(X = k) = C(n, k) · p^k · (1−p)^(n−k), with C(n, k) = n! / (k!(n−k)!) being the binomial coefficient. The mean works out to μ = n·p and the variance to σ² = n·p·(1−p). Take 10 tosses of a fair coin and ask for exactly 7 heads: C(10, 7) · 0.5⁷ · 0.5³ = 120/1024 ≈ 0.117, about 11.7%. Once n is large enough that np ≥ 5 and n(1−p) ≥ 5, a Normal(np, np(1−p)) gives a good approximation, which follows from the Central Limit Theorem.
Applications
A/B testing leans on it to count conversions, quality control to count defective items in a batch, and Mendelian genetics to predict offspring proportions. You'll also see it behind opinion polling (the margin of error for a proportion), digital communication (bit error rate, or BER), clinical trials measuring response to treatment, and reliability engineering, where it gives the chance of k failures among n components.
FAQ
When does the binomial model not apply? Two cases break it. One is when the trials aren't independent, such as sampling without replacement from a small population, where the hypergeometric is the right model instead. The other is when p shifts from trial to trial.
Binomial or Poisson? Reach for Poisson when n is very large and p very small, yet np stays moderate. That's the rare-events regime, and Poisson tracks the binomial closely there.
What's P(X ≤ k) vs P(X = k)? P(X = k) is the chance of getting exactly k successes. P(X ≤ k) is the cumulative version, the sum of P(X = i) from i = 0 up to k, which answers "at most k successes."
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Exact Binomial Test
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.