Binomial Distribution Calculator

Calculate exact and cumulative binomial probabilities, plus distribution table and chart.

The binomial distribution gives the probability of getting exactly k successes in n independent yes/no trials, each with the same success probability p. Use it for fixed-count, two-outcome situations — coin flips, pass/fail, click/no-click. This calculator returns P(X = k), the cumulative P(X ≤ k) and P(X ≥ k), plus the mean np and standard deviation √(np(1−p)). It assumes a fixed n, constant p, and independent trials. Results are validated to full precision.

Parameters

When the binomial applies

Four conditions must hold (the “BINS” checklist): a Binary outcome (success/failure), a fixed number of Independent trials, a constant probability p on each (No change), and you are counting Successes. Coin flips and quality-control pass/fail fit; drawing cards without replacement does not, because p changes.

Mean and spread

Over n trials you expect np successes, give or take √(np(1−p)). For n = 100, p = 0.5, that is 50 ± 5.

Worked example

A multiple-choice quiz has 10 questions with 4 options each (p = 0.25 by guessing). The chance of getting exactly 3 right is about 0.25, and of getting 5 or more right only about 0.08. Enter n, p, and k above for the exact and cumulative probabilities.

Common mistakes

Related distributions

When n is large and both np and n(1−p) exceed about 10, the binomial is well approximated by the normal distribution. For rare events over a continuous interval, use the Poisson distribution instead. Details are on the methodology page. For a printable reference, see the cumulative binomial table (n = 1 to 20).

Guide: Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.

Classic examples:

  • How many heads in 10 coin flips?
  • How many patients recover from a treatment out of 50?
  • How many defective items in a batch of 100 with a 2% defect rate?

P(X = k): The probability of getting exactly k successes in n trials. For example, P(X=6) when flipping 10 fair coins is the chance of getting exactly 6 heads.

P(X ≤ k): Cumulative probability — the chance of getting k or fewer successes. This is the CDF at k.

P(X ≥ k): The chance of getting k or more successes. This is 1 − P(X ≤ k−1). Useful for asking "what is the probability of at least k successes?"

  • Fixed n: The number of trials is determined in advance
  • Binary outcomes: Each trial results in exactly success or failure
  • Constant p: The probability of success is the same on every trial
  • Independence: The outcome of one trial does not affect others

If sampling without replacement from a small population, use the hypergeometric distribution instead.

When n is large and p is not too close to 0 or 1, the binomial distribution is well approximated by the normal distribution with μ = np and σ = √(np(1−p)).

A common rule: the normal approximation is adequate when np ≥ 5 and n(1−p) ≥ 5. Below these thresholds, use the exact binomial probability from this calculator.

  • A/B testing: If your baseline conversion is 5% (p=0.05) and you show 200 visitors the new design (n=200), P(X≥15) gives the probability of seeing 15 or more conversions by chance alone
  • Medical: If a drug works 70% of the time (p=0.7), P(X≥8 | n=10) tells you the probability at least 8 of 10 patients respond
  • Quality control: If a production line has a 1% defect rate (p=0.01), P(X≥3 | n=100) gives the probability of 3 or more defects in a batch

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