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Category Statistics & Probability

Number of Trials (n) Integer from 0 to 10,000

Probability of Success (p) Value from 0 to 1

Calculation Mode

Number of Successes (k) Integer from 0 to n

Presets:

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About

Miscalculating binomial probabilities leads to flawed A/B tests, incorrect quality control decisions, and unreliable risk models. The binomial distribution models the number of successes X in n independent Bernoulli trials, each with success probability p. This calculator computes exact point probabilities P(X = k), cumulative probabilities P(X ≤ k) and P(X ≥ k), and range probabilities using log-gamma arithmetic to avoid overflow for trials up to n = 10,000. It also derives the mean μ = np, variance σ² = np(1 − p), and skewness. This tool approximates using IEEE 754 double-precision floating point. For n > 1,000 with extreme p, probabilities below 10⁻¹⁵ may round to zero.

Formulas

The probability mass function (PMF) of the binomial distribution gives the probability of exactly k successes in n trials:

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

where n = number of independent trials, k = number of successes (0 ≤ k ≤ n), p = probability of success on a single trial, and (1 − p) = probability of failure, often denoted q.

The cumulative distribution function (CDF) sums probabilities up to k:

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

To avoid factorial overflow for large n, computation uses log-space arithmetic via the Lanczos approximation of the log-gamma function:

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

Descriptive statistics derived from the parameters:

Mean: μ = np Variance: σ² = npq Skewness: γ = 1 − 2p√npq

where q = 1 − p.

Reference Data

Scenario	n	p	Mean (μ)	Variance (σ²)	Std Dev (σ)	Skewness	Application
Coin Flips (Fair)	10	0.50	5.00	2.50	1.58	0.00	Probability teaching
Quality Control	100	0.02	2.00	1.96	1.40	0.69	Defect rate monitoring
Drug Efficacy Trial	50	0.80	40.00	8.00	2.83	−0.21	Clinical trials
Survey Response	200	0.30	60.00	42.00	6.48	0.06	Market research
Network Packet Loss	1000	0.01	10.00	9.90	3.15	0.31	Network reliability
Free Throw Shooting	20	0.75	15.00	3.75	1.94	−0.26	Sports analytics
Email Open Rate	500	0.22	110.00	85.80	9.26	0.06	Email marketing
Genetic Inheritance	4	0.25	1.00	0.75	0.87	0.58	Mendelian genetics
A/B Test Conversion	1000	0.05	50.00	47.50	6.89	0.13	Web optimization
Insurance Claims	10000	0.001	10.00	9.99	3.16	0.32	Actuarial science
Dice Rolling (6)	12	0.167	2.00	1.67	1.29	0.52	Gaming probability
Election Poll Sample	400	0.52	208.00	99.84	9.99	−0.004	Political forecasting
Rare Disease Screening	5000	0.0005	2.50	2.50	1.58	0.63	Epidemiology
Server Uptime (daily)	365	0.999	364.63	0.37	0.60	−1.63	SLA compliance
Lottery (match 1 number)	6	0.102	0.61	0.55	0.74	1.07	Gambling odds

Frequently Asked Questions

The normal approximation is generally reliable when both np ≥ 5 and n(1 − p) ≥ 5. Under these conditions, the binomial distribution is approximately symmetric and bell-shaped. For stricter accuracy, some sources require both products to exceed 10. This calculator uses exact log-gamma computation, so the normal approximation is unnecessary regardless of n.

IEEE 754 double-precision floating point has a minimum positive subnormal near 5 × 10⁻³²⁴. When n is large (e.g., 10,000) and k is far from the mean μ = np, the log-probability ln P falls below −745, which exponentiates to zero. These probabilities are genuinely negligible for practical purposes. The calculator displays them as < 10⁻¹⁵ when they underflow.

When p = 0, the only possible outcome is k = 0 with probability 1. When p = 1, the only possible outcome is k = n with probability 1. The calculator detects these degenerate cases before invoking log-space arithmetic (since ln(0) is undefined) and returns deterministic results directly.

The PMF P(X = k) answers questions like "What is the probability of exactly 3 defects?" The CDF P(X ≤ k) answers "What is the probability of at most 3 defects?" For quality control and hypothesis testing, CDF-based thresholds are standard. For instance, an acceptance sampling plan rejects a batch if the number of defects exceeds a critical value c, computed from P(X ≤ c) ≥ 0.95.

Yes. For a one-sided test of H₀: p = p₀ vs. H₁: p > p₀, compute P(X ≥ k) under p₀ using the "P(X ≥ k)" mode. If this p-value falls below your significance level α (typically 0.05), reject H₀. For two-sided tests, compute both tails and sum them.

Skewness γ = (1 − 2p) ÷ √npq. When p < 0.5, skewness is positive (right-tailed), meaning the distribution has a long right tail and the mean exceeds the mode. When p > 0.5, the distribution is left-skewed. At p = 0.5, skewness is zero (symmetric). Higher n reduces skewness magnitude, making the distribution more symmetric regardless of p.