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Binomial Distribution Calculator

Statistics

Calculate binomial probability P(X = k) and cumulative P(X ≤ k) in seconds. Enter number of trials, probability of success, and number of successes to get results.

11,000
0100
01,000

P(X = k)

17.62%
P(X ≤ k)
58.81%
P(X ≥ k)
58.81%
Expected Value (μ)
10
Standard Deviation (σ)
2.236

This calculator computes your P(X = k), P(X ≤ k), P(X ≥ k), Expected Value (μ), Standard Deviation (σ) from the values you enter.

Inputs
Number of Trials (n)Probability of Success (p)Number of Successes (k)
Outputs
P(X = k)P(X ≤ k)P(X ≥ k)Expected Value (μ)Standard Deviation (σ)

What is a Binomial Distribution?

The Binomial Distribution Calculator computes the probability of getting exactly k successes across n independent trials, each with the same probability of success p. Enter the number of trials, the probability of success, and the target number of successes, and the calculator returns P(X = k), the cumulative P(X ≤ k), P(X ≥ k), and the distribution's mean and standard deviation.

The binomial distribution is one of the most widely applicable discrete probability distributions, modeling any scenario with a fixed number of independent yes/no trials — coin flips, quality control inspections, A/B test conversions, or medical treatment outcomes across a fixed patient group.

For events measured as a rate over time or space rather than a fixed number of trials, see the Poisson Distribution Calculator instead.

How to use this Binomial Distribution calculator

  1. Enter the number of trials (n) — how many independent yes/no events you're analyzing.

  2. Enter the probability of success (p) — as a percentage, based on historical data or a known/assumed rate.

  3. Enter the target number of successes (k) — the specific outcome count you want to evaluate.

  4. Read P(X = k) — the probability of exactly that many successes.

  5. Check the cumulative probabilities — P(X ≤ k) and P(X ≥ k) for "at most" or "at least" style questions.

  6. Compare to the mean and standard deviation — to understand whether your target k is close to the typical expected outcome or represents an unusual result.

Formula & Methodology

Binomial probability mass function:
P(X = k) = C(n, k) × pᵏ × (1−p)ⁿ⁻ᵏ

Combination formula:
C(n, k) = n! / (k! × (n−k)!)

Mean and standard deviation:
μ = n × p
σ = √(n × p × (1−p))

Variable definitions:
- n — number of independent trials
- p — probability of success on each trial
- k — target number of successes

Worked example:

A sales rep closes 25% of qualified leads on average (p = 0.25). Out of the next 20 leads (n = 20), what's the probability of closing exactly 6 deals (k = 6)?

Step 1 — C(20, 6) = 38,760

Step 2 — P(X = 6) = 38,760 × 0.25⁶ × 0.75¹⁴ ≈ 0.1686 (16.86%)

Step 3 — Mean: μ = 20 × 0.25 = 5 expected closes

Step 4 — Standard deviation: σ = √(20 × 0.25 × 0.75) = √3.75 ≈ 1.94

Closing exactly 6 deals (slightly above the mean of 5) has about a 16.9% chance — a plausible, unsurprising outcome given the natural variability of ±1.94 around the mean.

Assumption: This calculator assumes all trials are independent with a constant probability of success — violated if trials influence each other or if sampling is done without replacement from a small population.

Frequently Asked Questions

A binomial distribution models the probability of getting a specific number of successes (k) across a fixed number of independent trials (n), where each trial has the same probability of success (p) — like counting heads in a series of coin flips, or defective units in a batch of manufactured products. It applies whenever you have a fixed number of yes/no trials with a constant success probability.
P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, where C(n,k) is the number of combinations of k successes among n trials (n! / (k!(n−k)!)). For example, the probability of exactly 3 heads in 5 coin flips (p = 0.5): C(5,3) = 10, so P(X=3) = 10 × 0.5³ × 0.5² = 10 × 0.125 × 0.25 = 0.3125, or 31.25%.
Four conditions must hold: (1) a fixed number of trials (n), (2) each trial has only two possible outcomes (success/failure), (3) the probability of success (p) is the same for every trial, and (4) trials are independent of each other. Common violations include sampling without replacement from a small population (which changes p slightly each draw) or trials that influence each other.
P(X = k) is the probability of exactly k successes. P(X ≤ k) is the cumulative probability of k or fewer successes, calculated by summing P(X = 0) through P(X = k). For example, P(X ≤ 2) in a 10-trial scenario adds up the individual probabilities of 0, 1, and 2 successes — useful when you care about 'at most' rather than 'exactly' a given outcome.
The mean (expected value) is μ = n × p, and the standard deviation is σ = √(n × p × (1−p)). For example, flipping a fair coin 20 times (n=20, p=0.5): mean = 20 × 0.5 = 10 expected heads, and standard deviation = √(20 × 0.5 × 0.5) = √5 ≈ 2.24.
Quality control teams use the binomial distribution to calculate the probability of finding a certain number of defective units in a random sample from a production batch, given the batch's known or assumed defect rate (p). This informs acceptance sampling plans — deciding whether to accept or reject an entire batch based on the number of defects found in a smaller sample.
As n grows large (with p staying moderate, not too close to 0 or 1), the binomial distribution's shape increasingly approximates a normal distribution — this is a direct consequence of the Central Limit Theorem. A common rule of thumb is that the normal approximation becomes reasonably accurate when both n×p and n×(1−p) are at least 5-10, at which point the [Normal Distribution Calculator](/normal-distribution-calculator/) can be used as a computational shortcut.
For large values of n and k, computing factorials directly (like 100!) produces numbers so large they exceed standard floating-point precision, causing calculation errors or overflow. This calculator computes the natural logarithm of the combination and probability terms instead, using a numerically stable gamma function approximation, then exponentiates only at the final step — avoiding overflow while preserving accuracy even for n in the hundreds.
The binomial distribution requires a fixed, known number of trials (n) with a constant success probability (p) per trial. The [Poisson Distribution Calculator](/poisson-distribution-calculator/) instead models the count of events occurring in a fixed interval of time or space, based only on an average rate (λ), without needing a discrete number of 'trials.' The Poisson distribution is actually the limiting case of the binomial distribution as n becomes very large and p becomes very small while n×p stays constant.
A sales team with a known 20% close rate on qualified leads could use this calculator to find the probability of closing exactly 8 deals out of the next 30 leads (n=30, p=0.20, k=8), helping set realistic short-term sales targets and understand the natural variability around their average close rate rather than expecting every batch of leads to convert at exactly the long-run average.
Yes — p is most reliably estimated from historical data on similar trials (e.g., a website's historical conversion rate, or a factory's historical defect rate), rather than assumed. Using a p value estimated from too small a historical sample introduces its own uncertainty into the binomial calculation, so it's good practice to base p on a reasonably large historical dataset when possible.
The expected value (mean) of a binomial distribution is n×p — this is your best single-number prediction. But real outcomes vary around this average by roughly one standard deviation (σ = √(n×p×(1−p))) in either direction most of the time, so don't be surprised if any individual batch of trials produces a count noticeably different from the mean; that's normal random variation, not evidence something has changed.
Also known as
binomial probability calculatorbinomial distributionbernoulli trial calculatorP(X=k) calculatorbinomial pmf calculator