Binomial Distribution Calculator

Calculate Binomial Probabilities

Binomial Distribution Formula

Formula

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \qquad \binom{n}{k} = \frac{n!}{k!\,(n-k)!} \\ P(X \le k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{n-i}, \qquad P(X \ge k) = 1 - P(X \le k-1) \\ \mu = np, \qquad \sigma^2 = np(1-p), \qquad \sigma = \sqrt{np(1-p)} \]

Where:

\(n\) — number of independent trials
\(p\) — probability of success on a single trial (between 0 and 1)
\(k\) — number of successes of interest (\(0 \le k \le n\))
\(\binom{n}{k}\) — binomial coefficient, the number of ways to choose \(k\) successes out of \(n\) trials
\(\mu\), \(\sigma^2\), \(\sigma\) — mean, variance, and standard deviation of the distribution

The first formula gives the probability mass function (pmf): the chance of exactly \(k\) successes in \(n\) trials. The second gives the cumulative distribution (cdf), the chance of at most \(k\) successes, obtained by summing the pmf from 0 to \(k\).

Binomial Distribution – Calculation Example

Example: a fair coin is flipped 10 times. What is the probability of getting exactly 5 heads? Here \(n = 10\), \(p = 0.5\), \(k = 5\).

\[ P(X = 5) = \binom{10}{5} (0.5)^5 (0.5)^{10-5} = 252 \cdot \frac{1}{1024} = 0.246094 \approx 24.61\% \]

\[ P(X \le 5) = \sum_{i=0}^{5} \binom{10}{i}(0.5)^{10} = \frac{638}{1024} = 0.623047 \approx 62.30\% \]

\[ \mu = 10 \cdot 0.5 = 5, \qquad \sigma^2 = 10 \cdot 0.5 \cdot 0.5 = 2.5, \qquad \sigma = \sqrt{2.5} \approx 1.5811 \]

Result: P(X = 5) = 0.246094 (24.61%), P(X ≤ 5) = 0.623047 (62.30%), P(X ≥ 5) = 0.623047 (62.30%), mean = 5, variance = 2.5, standard deviation ≈ 1.5811.

How to use the calculator

Enter the Number of trials (n) — a whole number from 1 to 10,000.
Enter the Probability of success (p) as a percentage between 0% and 100% (e.g. enter 50 for a fair coin).
Choose the mode under Successes of interest: “Exactly k successes” for a single value, or “Between k₁ and k₂ successes” for a range.
Enter the Number of successes (k). In range mode this is the lower bound k₁.
If you selected the range mode, also enter the Upper bound (k₂), which must be at least k₁.
Read the results: P(X = k), P(X ≤ k), P(X ≥ k), and in range mode P(k₁ ≤ X ≤ k₂), plus the mean, variance, and standard deviation. The interpretation block also lists complementary probabilities such as P(X < k) and P(X > k).

What is the binomial distribution?

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success \(p\). Each trial has exactly two outcomes — success or failure — which is why it is called a Bernoulli trial. Classic examples include the number of heads in a series of coin flips, the number of defective items in a sample, or the number of patients who respond to a treatment.

The mean of the distribution is \(\mu = np\): on average you expect \(n \cdot p\) successes. The spread is measured by the variance \(\sigma^2 = np(1-p)\) and the standard deviation \(\sigma\). When \(p = 0.5\) the distribution is symmetric around \(n/2\); when \(p\) is close to 0 or 1 it becomes skewed, piling up near 0 or near \(n\).

The calculator distinguishes between exact and cumulative probabilities. The pmf answers “what is the chance of exactly k successes”, while the cdf answers “what is the chance of at most k successes”. Cumulative values are often more useful in practice: for example, if k is a quality threshold, P(X ≤ k) tells you the probability of staying within the limit, and 1 − P(X ≤ k) tells you the probability of exceeding it. Note that P(X ≥ k) includes k itself; if you need strictly more than k, use P(X > k) = 1 − P(X ≤ k), shown in the interpretation block.

Assumptions, limitations and tips

Fixed number of trials. \(n\) must be set in advance. If the number of trials itself is random, a different model (e.g. the Poisson distribution) is more appropriate.
Independence. The outcome of one trial must not affect another. Sampling without replacement from a small population violates this; use the hypergeometric distribution instead.
Constant success probability. \(p\) must be the same for every trial. If it changes from trial to trial, the binomial model no longer applies.
Large n. The calculator uses a numerically stable log-gamma method, so it handles large \(n\) (up to 10,000) without overflow. For very small probabilities the result is shown in scientific notation.
Interpreting tiny probabilities. A result like “≈ 0%” does not mean impossible — it can simply mean the probability is smaller than about \(10^{-8}\). Check the decimal value shown alongside the percentage.
Rule of thumb for approximation. For large \(n\) with \(np\) and \(n(1-p)\) both greater than about 5, the binomial distribution is well approximated by a normal distribution with mean \(np\) and standard deviation \(\sqrt{np(1-p)}\) — handy for quick mental estimates.

Frequently Asked Questions

How do I calculate a binomial probability by hand without a calculator?

Use the formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). First compute the binomial coefficient, for example \( \binom{10}{5} = \frac{10!}{5! \cdot 5!} = 252 \). Then multiply by \( p^k \) and \( (1-p)^{n-k} \). For small \( n \) you can also list all outcomes, but factorial values grow extremely fast, so manual calculation becomes impractical beyond about \( n = 20 \).

What is the difference between "at least", "at most", and "exactly" k successes?

"Exactly" is a single probability: \( P(X = k) \). "At most" means \( k \) or fewer: \( P(X \le k) \). "At least" means \( k \) or more: \( P(X \ge k) \). A common mistake is confusing "at least \( k \)" with "more than \( k \)" — the former includes \( k \) itself, the latter is \( P(X > k) = 1 - P(X \le k) \). The interpretation block in the results shows both versions.

How do I convert odds or "1 in N" chances into the probability p?

The calculator expects \( p \) as a percentage. If you have odds of "a to b", convert with \( p = \frac{a}{a+b} \) and multiply by 100. A "1 in 6" chance (like a die roll) means \( p \approx 16.67\% \); "1 in 100" means enter 1. Decimal probabilities convert directly: 0.35 equals 35%.

How do I find the probability of a range, like between 3 and 7 successes?

Select the "Between k₁ and k₂ successes" mode, enter 3 as the number of successes and 7 as the upper bound. The calculator computes \( P(k_1 \le X \le k_2) \) by summing the individual probabilities. By hand, the same result comes from \( P(X \le k_2) - P(X \le k_1 - 1) \). Both bounds are included, so the range 3–7 covers 3, 4, 5, 6, and 7 successes.

When should I use the binomial distribution instead of Poisson, normal, or hypergeometric?

Use binomial when the number of trials is fixed and outcomes are independent with a constant \( p \). Use the hypergeometric distribution when sampling without replacement from a small population (e.g. drawing cards). Use Poisson when counting events over a fixed interval rather than a fixed number of trials. The normal distribution is an approximation, accurate when \( np \) and \( n(1-p) \) are both above 5.

What does the mean np tell me, and can the actual result be far from it?

The mean \( \mu = np \) is the expected average number of successes over many repetitions — not a prediction for a single experiment. Individual outcomes typically fall within about \( \mu \pm 2\sigma \) (roughly 95% of the time), where \( \sigma = \sqrt{np(1-p)} \). For example, with \( n = 100 \), \( p = 0.5 \): expect about 50 successes, usually between 40 and 60.

Why is my result shown as "≈ 0%" — is the outcome really impossible?

No, it is not impossible. "≈ 0%" only means the probability is smaller than about \( 10^{-8} \). Check the decimal value displayed next to the percentage — very small probabilities appear in scientific notation, such as \( 2.5 \times 10^{-12} \). With large \( n \), values far from the mean \( np \) can be astronomically unlikely yet still mathematically possible.