What is a binomial distribution?

A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes (success or failure) and the probability of success is the same on every trial. For example, flipping a fair coin 10 times and counting heads follows a binomial distribution with n = 10 and p = 0.5. The distribution is fully described by two parameters: n (number of trials) and p (probability of success on a single trial).

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

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) = n! / (k! × (n − k)!) is the binomial coefficient. For example, the probability of getting exactly 3 heads in 5 coin flips (p = 0.5) is C(5,3) × 0.5³ × 0.5² = 10 × 0.125 × 0.25 = 0.3125. The binomial coefficient counts the number of ways to arrange k successes in n trials.

What is the mean of a binomial distribution?

The mean (expected value) of a binomial distribution is μ = np, where n is the number of trials and p is the probability of success. For example, if you flip a coin 20 times, the expected number of heads is 20 × 0.5 = 10. The variance is σ² = np(1 − p) and the standard deviation is σ = √(np(1 − p)). For the coin example, variance = 20 × 0.5 × 0.5 = 5 and SD ≈ 2.24.

When can you use the normal approximation to the binomial distribution?

The normal approximation to the binomial is reliable when both np ≥ 10 and n(1 − p) ≥ 10. Under these conditions, the binomial distribution is roughly bell-shaped and can be approximated by a normal distribution with mean μ = np and standard deviation σ = √(np(1 − p)). For small n or extreme p values (near 0 or 1), use the exact binomial formula instead. A continuity correction (adding or subtracting 0.5) improves the approximation.

What is a Bernoulli trial?

A Bernoulli trial is a single experiment with exactly two possible outcomes — success (probability p) and failure (probability 1 − p). The binomial distribution is the sum of n independent Bernoulli trials. Examples of Bernoulli trials include: a coin flip (heads = success), a free throw (made = success), a quality inspection (defective = failure), or a medical treatment (effective = success). The key requirement is that each trial is independent and has the same probability p.

How do you calculate P(X ≤ k)?

P(X ≤ k) is the cumulative distribution function (CDF) and equals the sum of P(X = 0) + P(X = 1) + ... + P(X = k). For example, to find the probability of getting 3 or fewer heads in 5 flips, calculate P(0) + P(1) + P(2) + P(3) = 0.0313 + 0.1563 + 0.3125 + 0.3125 = 0.8125. The calculator shows P(X ≤ k), P(X ≥ k), P(X k) all at once, along with a probability table for n ≤ 20.

What are real-world examples of the binomial distribution?

The binomial distribution models any situation with repeated yes/no trials: quality control testing (how many defects in a batch of 100 units?), clinical trial outcomes (how many patients respond to a drug?), genetics (how many offspring inherit a recessive trait when each has a 25% chance?), and marketing (how many users click a button with a 5% click-through rate?). Any time you count "how many successes in n attempts," binomial distribution is your model.

When should I use binomial distribution vs. normal distribution?

Use the binomial distribution when trials are discrete and countable (exactly n attempts, each success or failure). Use the normal distribution for continuous measurements or when the binomial conditions are met and both np ≥ 10 and n(1−p) ≥ 10 — in that case, the binomial distribution is approximately normal and easier to work with. For very rare events with large n, the Poisson distribution may be a better approximation.

What is the expected value in a binomial experiment?

The expected value (mean) is μ = np. It tells you the average number of successes you would see if you ran the experiment many times. For example, if you roll a die 60 times and count sixes (p = 1/6), the expected count is 60 × 1/6 = 10 sixes. The expected value does not mean you will see exactly 10 sixes in any given run — it is the long-run average across many repetitions.

Binomial Distribution Calculator

Calculates P(X=k), P(X≤k), and P(X≥k) for any binomial experiment — shows mean, variance, and probability table.

How to Use the Binomial Distribution Calculator

This binomial distribution calculator computes exact probabilities for any binomial experiment — enter n (the total number of trials), p (the probability of success on a single trial, between 0 and 1), and k(the specific number of successes you are interested in). The calculator instantly computes P(X = k), P(X ≤ k), P(X ≥ k), and the distribution's mean, variance, and standard deviation. For n ≤ 20, a complete probability table shows every possible outcome from 0 to n with its individual probability and cumulative probability.

The selected value of k is highlighted in the table so you can see exactly where your result falls in the full distribution. For questions about continuous probability distributions, see our normal distribution calculator.

The Binomial Probability Formula

The probability of observing exactly k successes in n independent trials, each with success probability p, is:

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)

Where C(n, k) = n! / (k! × (n − k)!) is the binomial coefficient, representing the number of ways to choose k successes from n trials. The term p^k accounts for the k successes, and (1 − p)^(n−k) accounts for the remaining failures.

Worked Example

A quality control inspector samples 10 items from a production line where the defect rate is 5% (p = 0.05). What is the probability that exactly 2 items are defective (k = 2)?

P(X = 2) = C(10, 2) × 0.05² × 0.95⁸ = 45 × 0.0025 × 0.6634 ≈ 0.0746

There is approximately a 7.46% chance of finding exactly 2 defective items. The cumulative probability P(X ≤ 2) ≈ 0.9885, meaning there is only a 1.15% chance of finding 3 or more defectives in a sample of 10.

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Distribution Properties: Mean, Variance, and Standard Deviation

The binomial distribution has simple closed-form expressions for its summary statistics, making it easy to characterize without computing every probability.

Mean (μ = np): The expected number of successes. For 20 free throws with p = 0.75, μ = 20 × 0.75 = 15 made shots.
Variance (σ² = np(1 − p)): Measures the spread of the distribution. For the free throw example, σ² = 20 × 0.75 × 0.25 = 3.75.
Standard deviation (σ = √np(1 − p)): The typical deviation from the mean in the original units. For the free throw example, σ = √3.75 ≈ 1.94 made shots.

Variance is maximized when p = 0.5 (equal chance of success and failure) and approaches 0 as p approaches 0 or 1 (outcomes become more certain).

Cumulative Probabilities Explained

In many real-world scenarios, you need more than just the probability of an exact outcome — you need to know the probability of a range of outcomes.

P(X ≤ k): The probability of k or fewer successes. Used when you want to know if results are unexpectedly low.
P(X ≥ k): The probability of k or more successes. Used when you want to know if results are unexpectedly high.
P(X < k): Strictly less than k (equals P(X ≤ k − 1)).
P(X > k): Strictly greater than k (equals 1 − P(X ≤ k)).

For example, if a drug has a 30% success rate (p = 0.3) and is given to 10 patients (n = 10), the probability of seeing 5 or more successes is P(X ≥ 5) ≈ 0.1503. This tells a researcher that 5+ successes would be a relatively unusual but possible result.

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When to Use the Binomial Distribution

The binomial distribution applies when all four conditions are met:

Fixed number of trials (n): You run the experiment a predetermined number of times.
Two outcomes: Each trial results in exactly one of two outcomes — success or failure.
Constant probability (p): The probability of success is the same on every trial.
Independence: The outcome of one trial does not affect any other trial.

Common applications include: predicting the number of defective items in a batch, modeling the number of customers who make a purchase, analyzing test results where each question is answered correctly or incorrectly, and estimating the probability of winning a certain number of games in a sports season.

When n is large and both np ≥ 10 and n(1 − p) ≥ 10, the binomial is approximately normal. Use our confidence interval calculator when working with sample proportions from large populations.

Sources & References

Binomial Distribution — NIST/SEMATECH e-Handbook of Statistical Methods
Binomial Probability Formula — Khan Academy

Binomial Distribution Calculator