Binomial Probability Calculator
n independent trials, success probability p: the single point P(X = k) and both tails, computed in log space so big n never underflows — with the np(1−p) shape note fired live when the normal approximation would be tempting.
Binomial Probability Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Independent trials with constant p — no hypergeometric (without replacement) mode
- Direct summation capped at n = 100,000 (approximations named as approximations)
In short: Ten fair coins, exactly five heads: P(X = 5) = C(10,5)(0.5)^10 = 252/1,024 = 0.246094 — the single most quoted binomial point. Both tails agree here: P(X ≤ 5) = 0.623047 and P(X ≥ 5) = 0.623047, because p = 0.5 makes the distribution exactly symmetric. The moments come free: mean np = 5, variance np(1−p) = 2.5. Roll a die twelve times and ask for three sixes: P = 0.197396, P(≤ 3) = 0.874822. And the boundaries behave: p = 0 returns 1 at k = 0 and 0 elsewhere — a certainty is not an approximation.
Formula
P(X = k) = C(n,k) p^k (1−p)^(n−k) · C(n,k) = n!/(k!(n−k)!)
The page computes C(n,k) in log space via the Lanczos log-gamma, then exponentiates — no factorials overflow, no probability underflows to a lying zero before k = 100,000.
Worked Example
- Give the trial count n, the success count k, and the per-trial probability p.
- Read the exact point P(X = k), then both tails beside it.
- Check the shape note when p is extreme — it tells you when the normal approximation convention would apply.
- Boundaries p = 0 and p = 1 are answered exactly, not approximately.
Defaults: P(X = 5) = 0.246094, P(≤ 5) = P(≥ 5) = 0.623047, mean 5, variance 2.5. The dice drive (n = 12, p = 1/6, k = 3) gives 0.197396 and ≤ 3 → 0.874822. np < 5 fires the shape note live.
Strengths & Limits Of This Model
Where this engine is strong
- Log-space arithmetic: no overflow, no underflow, p = 0/1 answered exactly
- Both tails plus the np ≥ 5 shape convention printed against your numbers
Where it stops
- No continuity-corrected normal comparison
- No inverse question (‘smallest k with tail below 5%’) — quoted as a boundary
Practical Use Cases
Quality
P(exactly 3 defectives in a lot)
Medicine
P(this many responders if p is the rate)
Games
dice and card counts, verified exactly
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Binomial Probability Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What exactly is “binomial” about it?
Two outcomes per trial (success/failure), the same p every trial, and independence between trials. Break any of those — a changing p, or trials that influence each other — and the formula answers a question nobody asked.
Why does P(X = 5) peak at p = 0.5?
C(10,5) = 252 is the largest coefficient, and p = 0.5 weights every ordering equally. At p = 0.5 the whole distribution is symmetric, which is why both tails print the same 0.623047 — the page shows the symmetry instead of asserting it.
How large can n be?
The arithmetic is direct log-space summation, honest to n = 100,000; beyond that the sum itself gets long and the normal or Poisson approximation is the right tool — the page states the cap rather than slowing to a crawl or faking a normal tail as exact.
When may I swap in the normal curve?
The working convention is np ≥ 5 AND n(1−p) ≥ 5 — a rule of thumb, not a theorem, and this page says so while printing whether your numbers clear it. Below the line the binomial is lumpy and one-sided; the normal smooths exactly the features you care about.
What happens at p = 0 or p = 1?
Exact answers, no approximation: at p = 0 the count is certain — P(X = 0) = 1 and every other k gets 0; at p = 1 the count is certain at k = n. A degenerate distribution is a distribution, and the page answers it exactly.
Why is P(X ≤ 5) not just 1 − P(X ≥ 5)?
Because they SHARE the middle: P(X ≤ 5) + P(X ≥ 5) = 1 + P(X = 5). At p = 0.5, n = 10 both come to 0.623047 and indeed 0.623047 × 2 = 1.246094 = 1 + 0.246094. The double-counted point is the classic binomial tail mistake.
My trials are not independent — now what?
Then n and p do not describe your process: drawing cards without replacement, or herders copying each other, need hypergeometric or model-based thinking. This page computes the independent-trials answer and names its assumptions; feeding it dependent trials produces a precise answer to the wrong question.
Where does the Poisson page take over?
When n is huge and p tiny with np = λ moderate — rare events — the binomial collapses to the Poisson with λ = np. The Poisson page owns that regime; the two agree to several decimals long before the swap is forced, which makes a good cross-check.