Binomial Distribution Calculator

Binomial Distribution Probability Calculator

This is a discrete probability calculator built specifically for the binomial distribution — a core topic in probability theory that sits alongside, but is distinct from, general descriptive statistics. Where a statistics calculator (mean, median, standard deviation of a data set) summarizes data you already collected, this binomial probability calculator predicts the likelihood of a specific outcome before it happens, based on a fixed number of independent trials, each with the same probability of success. That distinction matters for Math, Engineering, and Data Science coursework: probability distributions like the binomial and normal distribution model uncertainty and randomness, while statistics summarizes observed data — and this calculator is squarely a probability tool.

The binomial distribution applies whenever an experiment has exactly two possible outcomes per trial — success or failure, heads or tails, defective or non-defective, pass or fail — repeated a fixed number of times (n), with each trial independent of the others and the probability of success (p) staying constant across all trials. This makes it one of the most widely used discrete probability distributions in coin-flip problems, quality-control sampling, A/B testing, survey analysis, genetics, and reliability engineering.

Binomial Probability Formula

The probability of getting exactly x successes in n independent trials, each with success probability p, is given by the binomial probability mass function (PMF):

P(X = x) = C(n, x) · p^x · (1 − p)^(n − x)

Here, C(n, x) = n! / (x! · (n − x)!) is the binomial coefficient — the number of different ways to choose x successes out of n trials — commonly read as 'n choose x'. The term p^x is the probability of getting x successes in a row, and (1 − p)^(n − x) is the probability of the remaining trials being failures. Multiplying all three together accounts for every possible arrangement of successes and failures that adds up to exactly x successes.

To find the probability of a range of outcomes rather than an exact count, this calculator uses the binomial cumulative distribution function (CDF): P(X ≤ x) = Σ P(X = i) for i = 0 to x. From this, P(X ≥ x) = 1 − P(X ≤ x − 1), and P(a ≤ X ≤ b) is the sum of the PMF from i = a to i = b.

Mean, Variance, and Standard Deviation of a Binomial Distribution

Beyond individual probabilities, every binomial distribution B(n, p) has three key summary values that describe its overall shape: the mean μ = n·p (the expected number of successes), the variance σ² = n·p·(1 − p) (how spread out the outcomes are), and the standard deviation σ = √(n·p·(1 − p)). These values are calculated automatically alongside your probability result, and they're especially useful for comparing how 'typical' a specific outcome is relative to the whole distribution.

How to Use This Calculator

Select what you want to find: the probability of exactly x successes, at most x successes, at least x successes, or a probability between two values. Enter the number of trials (n), the probability of success on a single trial (p, as a decimal between 0 and 1), and the number of successes (x). The calculator instantly returns the exact probability, its percentage form, and the mean, variance, and standard deviation of the full distribution, along with a bar chart showing every possible outcome from 0 to n successes.

• P(X = x) — exact probability, useful for questions like 'what's the chance of exactly 3 heads in 5 coin flips?'
• P(X ≤ x) — cumulative probability up to and including x, useful for 'what's the chance of at most 3 defective items in a batch?'
• P(X ≥ x) — probability of x or more, useful for 'what's the chance of at least 7 successful sign-ups out of 10 leads?'
• P(a ≤ X ≤ b) — probability within a range, useful for confidence-band or acceptance-sampling questions.

Worked Example

A fair coin is flipped 10 times (n = 10, p = 0.5). What is the probability of getting exactly 6 heads? Using the formula: P(X = 6) = C(10, 6) · (0.5)^6 · (0.5)^4 = 210 · 0.015625 · 0.0625 ≈ 0.2051, or about 20.51%. The mean number of heads across many repetitions of this experiment would be μ = 10 × 0.5 = 5, with a standard deviation of σ = √(10 × 0.5 × 0.5) ≈ 1.58. Enter n = 10, p = 0.5, and x = 6 into this calculator to see the same result instantly, along with every other possible outcome charted from 0 to 10 heads.

When to Use the Binomial vs. the Normal Distribution

The binomial distribution is discrete (it only counts whole-number successes), while the normal distribution is continuous. However, when n is large and p is not too close to 0 or 1, the binomial distribution starts to closely resemble a normal distribution — this is the basis of the normal approximation to the binomial. If you're working with a discrete count of successes out of fixed trials, use this binomial calculator; if you're working with a continuous measurement like height, weight, or a test score, use the normal distribution calculator instead.