Statistics / Probability
Binomial Distribution Calculator: Probability, CDF and Quantiles
Calculate P(X = x), P(X ≤ x) and inverse cumulative counts for a binomial distribution. Set the number of trials and the success probability, with editable starting examples.
Binomial Distribution Calculator: Probability, CDF and Quantiles: A binomial distribution counts successes in a fixed number of independent trials with the same success probability. Ten fair coin tosses provide a familiar example: with n = 10 and p = 0.5, the probability of exactly five heads is 0.24609375. This page starts with that example, but every parameter is editable. PD means the probability of exactly x successes. CD means the probability of at most x successes, including x itself. For at least x successes, subtract CD(x − 1) from 1. Inverse mode returns the smallest whole-number count whose cumulative probability meets or exceeds the probability you enter; this definition handles the jumps in a discrete distribution. Use the model for repeated independent checks, such as counting successful components under a fixed failure model. If the probability changes across trials or outcomes depend on each other, the binomial assumptions do not hold. The [permutation and combination calculator](/permutation-combination-calculator) explains the combinations term, and the [scientific calculator](/scientific-calculator) provides other distributions and statistical summaries. Runs 100% locally in your browser with zero server file uploads.
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More functions, constants and conversions
Use semicolons between arguments: ncr(52;5), dms(30;15;30), randint(1;6). Rnd rounds to the selected significant digits. E is Euler's number until you store a value in E; C is the speed of light until you store C.
- GCD
- 6
- LCM
- 144
- Integer quotient and remainder
- 2; 12
- 48 = 2 × 18 + 12
- 18 = 1 × 12 + 6
- 12 = 2 × 6 + 0
Constants: c, h, hbar (ħ), qe (elementary charge), me, mp, na, kb, gasr, grav (G), g (standard gravity), eps0, mu0, amu and sigma. Measured constants carry uncertainty; the displayed values are not exact. NIST / CODATA constants
Type or tap. Multiplication can be implied, as in 2π or 3(4+1); ^ is a power, ! a factorial, and % divides by 100. Enter calculates; Escape clears.
Normal PD is a density, not a point probability. CD includes the upper endpoint. Discrete inverse returns the smallest integer with CDF ≥ p; use 0 < p < 1. Tested central probabilities have about 12 decimal places of absolute accuracy; extreme tails may lose precision.
Binomial model
NIST Engineering Statistics Handbook, Binomial Distribution (https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm), documents the mass function and its assumptions. Trials must be independent with a fixed success probability.
Cumulative probabilities
NIST DLMF, Incomplete Beta Functions (https://dlmf.nist.gov/8.17), describes the special function used for the cumulative binomial calculation. Logarithmic gamma evaluation supports large counts; central-range tests target about 1e-12 absolute accuracy.
How to use it
- Enter the number of trials n and success probability p.
- Choose PD for exactly x successes, CD for at most x, or inverse for a cumulative probability.
- Read the probability or count and copy it, or switch to the normal and Poisson models.
Privacy & limitations
Trial counts and probabilities are calculated in your browser and are not uploaded.
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Frequently asked questions
What values can n, p and x take?
n is a whole number from 0 to ten million. p lies between 0 and 1, including both endpoints. For PD, x is a whole-number success count; impossible counts have probability 0. CD includes all integer counts up to the entered threshold.
Is this a normal approximation?
No. Point probabilities use the binomial formula through logarithmic gamma arithmetic; cumulative probabilities use the incomplete beta function. These avoid explicitly forming huge factorials, although the final results are decimal approximations.
Why might inverse return a count with probability above my target?
A discrete CDF jumps at each integer. Often no count matches the target exactly, so inverse chooses the first count reaching or exceeding it. Enter a probability strictly between 0 and 1.
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