Binomial Probability Calculator
Given n independent Bernoulli trials with success probability p, the binomial distribution tells you how often you’ll see exactly k successes. The calculator returns the exact probability P(X = k) plus the tails P(X < k), P(X ≤ k), P(X ≥ k) and P(X > k) in one shot, using log-space arithmetic so the results stay accurate across the full range up to n = 1,000.
How to calculate binomial probability
-
1
Enter n (number of trials)
Must be a non-negative integer. Typical values: 10 coin flips, 100 A/B test visitors, 1,000 manufacturing samples.
-
2
Enter p (success probability)
A value between 0 and 1. For a fair coin p = 0.5; for a 12% click-through rate p = 0.12.
-
3
Enter k (target number of successes)
An integer from 0 to n.
-
4
Read the probabilities
The five probabilities: exact P(X = k), less than P(X < k), at most P(X ≤ k), at least P(X ≥ k) and more than P(X > k).
The formula
P(X = k) = C(n, k) · p^k · (1-p)^(n-k)
Where C(n, k) is the binomial coefficient “n choose k”. The tool uses log-space arithmetic to avoid overflow when n is large.
Worked example: 10 coin flips, exactly 7 heads
- n = 10, p = 0.5, k = 7
- C(10, 7) = 120
- P(X = 7) = 120 · 0.5^7 · 0.5^3 = 120 / 1024 ≈ 0.1172
So about 11.7% of the time you’ll see exactly 7 heads in 10 flips.
When the binomial distribution applies
All four Bernoulli assumptions must hold:
- Fixed number of trials (n is decided in advance).
- Each trial is independent of the others.
- Only two outcomes per trial (success / failure).
- Constant success probability p across trials.
If any assumption breaks (dependent draws without replacement, variable p, more than two outcomes), reach for the hypergeometric, Poisson-binomial or multinomial distribution instead.
Mean, variance and normal approximation
- Mean: μ = np
- Variance: σ² = np(1-p)
- Standard deviation: σ = √(np(1-p))
When np ≥ 10 and n(1-p) ≥ 10, the binomial is well-approximated by Normal(μ, σ²) with a continuity correction. Keep this rule as a cross-check for large n, since the calculator always returns exact values.
Frequently Asked Questions
P(X = k) is the probability of exactly k successes; P(X ≤ k) is the cumulative probability of at most k. For 10 flips of a fair coin, P(X = 5) ≈ 0.246 but P(X ≤ 5) ≈ 0.623.
Yes. The calculator returns P(X ≥ k) = 1 - P(X ≤ k-1). For “more than k”, subtract one more: P(X > k) = P(X ≥ k+1).
Up to 1,000 in the guided steps, which is stable thanks to the log-space computation and covers typical coin-flip, survey and A/B test cases. Beyond that, use the normal approximation or the Poisson approximation (valid when p is small and n is large).
Then you need the Poisson-binomial distribution, not the plain binomial. This calculator assumes a single constant p across all n trials.
Related Tools
BMI Calculator
Calculate body mass index from height and weight. Shows WHO category, healthy-weight range and limitations of BMI.
Social Security Calculator
Estimate a US Social Security retirement benefit from your PIA and claiming age. Compare monthly and annual amounts at 62, FRA 67 and 70.
Age Calculator
Calculate exact age in years, months and days from a birth date, plus total days, hours and the next birthday countdown.
One Rep Max Calculator
Estimate your one-rep max (1RM) from weight lifted and reps completed using Epley, Brzycki and Lander formulas.
Amp-Hour Calculator
Work out battery runtime and stored energy from amp-hours, voltage and load current. Get runtime in hours and minutes plus energy in watt-hours and kilowatt-hours.
Scientific Calculator
Evaluate supported real-number expressions with trig, logs, powers and parentheses locally in your browser.