Binomial Distribution Calculator - Free Online Calculator

Number of Trials (n)

11,000

Probability of Success (p)

0100

Number of Successes (k)

01,000

P(X = k)

17.62%

P(X ≤ k)

58.81%

P(X ≥ k)

58.81%

Expected Value (μ)

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.

Why Use a Binomial Distribution Calculator?

Computing binomial probabilities by hand requires calculating factorial-based combinations and raising probabilities to potentially large powers, calculations that quickly become impractical to do manually once n exceeds 10 or 15, and prone to numerical overflow errors even in spreadsheet software for larger n.

This calculator uses a numerically stable logarithmic computation internally, so it produces accurate results even for hundreds of trials, and reports both the exact and cumulative probabilities together, giving you the complete picture in one calculation.

Who Should Use This Calculator?

Students in statistics, probability, and data science courses learning to compute and interpret discrete probability distributions.

Quality control engineers calculating the probability of finding a specific number of defective units in a sample, to design acceptance sampling plans.

Marketing and sales analysts estimating the probability of a specific number of conversions or closed deals from a fixed batch of leads with a known conversion rate.

Clinical researchers and biostatisticians computing the probability of a specific number of treatment successes or side effects across a fixed group of patients.

Game designers and probability enthusiasts calculating the odds of specific outcomes across repeated independent trials (dice rolls, card draws with replacement, loot drop chances).

What Insights Does This Calculator Give You?

P(X = k), Exact Probability, the headline result: the probability of getting exactly the specified number of successes.

P(X ≤ k), Cumulative Probability, the probability of k or fewer successes, useful for "at most" questions like defect tolerance thresholds.

P(X ≥ k), Upper Cumulative Probability, the probability of k or more successes, useful for "at least" questions like minimum conversion targets.

Expected Value (μ) and Standard Deviation (σ), the theoretical average outcome and its natural variability, giving context for whether your specific k value is close to or far from what's typically expected.

How to use this Binomial Distribution calculator

Enter the number of trials (n), how many independent yes/no events you're analyzing.
Enter the probability of success (p), as a percentage, based on historical data or a known/assumed rate.
Enter the target number of successes (k), the specific outcome count you want to evaluate.
Read P(X = k), the probability of exactly that many successes.
Check the cumulative probabilities, P(X ≤ k) and P(X ≥ k) for "at most" or "at least" style questions.
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.

Show formula & methodology ↓Show less ↑

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

What is a binomial distribution?

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.

How do you calculate binomial probability P(X = k) by hand?

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%.

What conditions must be met for the binomial distribution to apply?

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.

What is the difference between P(X = k) and P(X ≤ k)?

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.

How do you calculate the mean and standard deviation of a binomial distribution?

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.

How does binomial distribution relate to real-world quality control?

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.

What happens to the binomial distribution's shape as n gets very large?

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.

Why does this calculator use logarithms internally for the combination formula?

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.

How is the binomial distribution different from the Poisson distribution?

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.

What's a practical example of using this calculator in marketing or sales analysis?

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.

Can probability of success (p) be based on historical data?

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.

How many successes should I expect on average, and how much does it vary?

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