Binomial Probability

Experiment Setup

P(X=5) = C(10,5)·0.5⁵·0.5⁵

n (trials)

p (success %)

k (successes)

Enter p as a percentage (e.g. 50 for 50%). k must be between 0 and n.

Examples

P(X≤5) for n=10, p=0.5

n (trials)

p (success %)

k (threshold)

Computes P(X=k), P(X≤k), P(X≥k), P(X<k), and P(X>k).

Examples

Result

Enter values above and press Calculate to see the probability, formula breakdown, and distribution stats.

P(X = k)

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All Cumulative Probabilities

Formula Breakdown

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Distribution Statistics

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μ = np

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σ² = np(1−p)

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σ = √(np(1−p))

Binomial Conditions (BINS)

P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ

B — Binary outcomes: Each trial is either a success or a failure. No middle ground.

I — Independent trials: The outcome of one trial does not affect the next.

N — Fixed Number of trials: You must know n before you start.

S — Same probability: p is constant across all n trials.

C(n,k) = n! / (k!(n−k)!) counts the ways to choose k successes from n trials. It is always a whole number.

Expected Value & Spread

μ = np · σ² = np(1−p) · σ = √(np(1−p))

μ = np gives the expected (average) number of successes over many repetitions of the experiment.

σ² = np(1−p) is the variance — it measures how spread out the distribution is. The spread is largest when p = 0.5.

σ = √(np(1−p)) is the standard deviation in the same units as X.

Example: 10 coin flips at p = 0.5 → μ = 5 heads expected, σ = √(10·0.5·0.5) = √2.5 ≈ 1.58.