User Rating 0.0 ★★★★★

Total Usage 0 times

Category Statistics & Probability

Number of trials (n) Integer from 1 to 10,000

Number of successes (k) Integer from 0 to n

Probability of success (p) Value from 0 to 1

Presets:

P(X = k) —

P(X ≤ k) —

P(X ≥ k) —

P(X < k) —

P(X > k) —

Mean (μ)—

Variance (σ²)—

Std Dev (σ)—

Skewness—

Kurtosis (excess)—

Full Distribution Table

k	P(X = k)	P(X ≤ k)

Is this tool helpful?

Your feedback helps us improve.

★ ★ ★ ★ ★

About

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability p. Miscalculating cumulative probabilities leads to flawed hypothesis tests, incorrect confidence intervals, and bad quality-control decisions. This calculator computes the exact probability mass function P(X = k) and all cumulative probabilities using log-gamma arithmetic to prevent floating-point overflow for large n. It handles up to 10,000 trials without approximation.

Results include the mean μ = np, variance σ² = np(1 − p), and a full distribution chart rendered on canvas. The tool assumes independence between trials and a constant probability per trial. For n > 1,000 the chart uses a Normal approximation for rendering speed while exact numeric results remain log-gamma based. Pro tip: verify your expected value against sample data before trusting any model. Real processes often violate the independence assumption.

Formulas

The probability of observing exactly k successes in n independent trials, each with success probability p:

P(X = k) = n!k! (n − k)! ⋅ p^k ⋅ (1 − p)^{n − k}

The cumulative distribution function (CDF) gives the probability of at most k successes:

P(X ≤ k) = k∑i=0 n!i! (n − i)! ⋅ pⁱ ⋅ (1 − p)^{n − i}

To avoid factorial overflow, the calculator uses the log-gamma function internally:

ln P(X = k) = lnΓ(n + 1) − lnΓ(k + 1) − lnΓ(n − k + 1) + k ⋅ ln(p) + (n − k) ⋅ ln(1 − p)

Descriptive statistics of the distribution:

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

Where n = number of trials (positive integer), k = number of successes (0 ≤ k ≤ n), p = probability of success on a single trial (0 ≤ p ≤ 1), Γ = gamma function (continuous extension of factorial), μ = expected value (mean), σ = standard deviation.

Reference Data

Scenario	n	p	Mean (μ)	Std Dev (σ)	P(X = 0)	Skewness
Coin flip (fair, 10 flips)	10	0.5	5.0	1.581	0.000977	0.000
Quality control (1% defect)	100	0.01	1.0	0.995	0.366	0.985
Drug efficacy trial	50	0.7	35.0	3.240	≈0	−0.390
Free throw shooting	20	0.85	17.0	1.597	≈0	−0.439
Survey response rate	200	0.3	60.0	6.481	≈0	0.062
Dice roll (getting a 6)	12	0.1667	2.0	1.291	0.112	0.516
Email open rate	500	0.22	110.0	9.261	≈0	0.060
Genetic trait (recessive)	4	0.25	1.0	0.866	0.3164	0.577
Server uptime (99.9%)	365	0.001	0.365	0.604	0.694	1.630
A/B test conversion	1000	0.05	50.0	6.892	≈0	0.130
Insurance claims	10000	0.002	20.0	4.470	≈0	0.223
Rare disease screening	5000	0.0001	0.5	0.707	0.6065	1.413
Election polling sample	1500	0.48	720.0	19.35	≈0	0.005
Manufacturing yield	250	0.95	237.5	3.446	≈0	−0.261
Lottery (6 from 49 approx)	49	0.1224	6.0	2.294	0.0012	0.346

Frequently Asked Questions

Instead of computing n! directly (which overflows JavaScript's Number at roughly n > 170), the calculator works in log-space. It computes ln(P) using the log-gamma function: ln Γ(n+1) − ln Γ(k+1) − ln Γ(n−k+1) + k·ln(p) + (n−k)·ln(1−p), then exponentiates the result. This keeps intermediate values within floating-point range for n up to 10,000 and beyond.

The standard rule of thumb requires both np ≥ 5 and n(1−p) ≥ 5. Under these conditions, the binomial distribution is well-approximated by a Normal distribution with mean μ = np and variance σ² = np(1−p). For more precise work, apply a continuity correction: P(X ≤ k) ≈ Φ((k + 0.5 − μ)/σ). This calculator uses exact log-gamma computation for numeric results regardless of n, but the chart rendering uses Normal density for n > 1,000 to maintain rendering speed.

When p = 0, every trial fails. P(X = 0) = 1 and P(X = k) = 0 for all k > 0. When p = 1, every trial succeeds. P(X = n) = 1 and P(X = k) = 0 for all k < n. The calculator handles these edge cases explicitly before entering the log-gamma pathway, since ln(0) is undefined.

Skewness equals (1 − 2p) / √(np(1−p)). When p = 0.5, skewness is zero and the distribution is perfectly symmetric. When p < 0.5, the distribution is right-skewed (tail extends toward higher k). When p > 0.5, it is left-skewed. Extreme skewness (p near 0 or 1 with small n) means the Normal approximation is unreliable.

No. The binomial distribution is a discrete distribution defined only for integer values of k from 0 to n. If you enter a non-integer, the calculator rounds to the nearest integer and displays a warning. For continuous analogs, consider the Beta distribution or Normal approximation.

A Bernoulli distribution is a special case of the binomial with n = 1. It models a single trial with outcomes 0 (failure) or 1 (success). The binomial distribution is the sum of n independent Bernoulli random variables, each with the same probability p. Setting n = 1 in this calculator gives you Bernoulli probabilities directly.

They overlap at k. P(X ≤ k) + P(X ≥ k) = 1 + P(X = k). The complementary pairs are P(X ≤ k) + P(X > k) = 1 and P(X < k) + P(X ≥ k) = 1. The calculator displays all five probabilities to prevent this common confusion.