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Walk-through

How to Use This Calculator

4 steps

01

Enter trials, probability, and successes

Fill in the number of trials (n) — coin flips, shots attempted, items inspected — the success probability per trial (p, as a decimal between 0 and 1), and the number of successes (k) you're asking about. The calculator recalculates instantly as you type.

02

Choose exactly, at most, or at least

Pick the comparison that matches your question: 'Exactly k' gives P(X = k), 'At most k' gives the cumulative P(X ≤ k), and 'At least k' gives the upper-tail P(X ≥ k). Switching the dropdown recomputes the same n and p instantly.

03

Try a preset scenario

Use the sample-scenario chips — coin flips, free throws, or a manufacturing defect rate — to see the calculator populated with a realistic n, p, and k. Edit any field afterward to make it match your own numbers.

04

Read the probability, mean, and SD

The result card shows the probability plus the distribution's mean (np) and standard deviation. Open the Cumulative tab for the full P(X≤k)/P(X≥k) table, or the Distribution tab for a bar chart of every possible outcome with your k highlighted.

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Reference

Formula & Methodology

3 formulas

Binomial probability mass function

P(X = k) = C(n,k) · p^k · (1 − p)^(n − k) where C(n,k) = n! / (k!(n − k)!)

C(n,k), read 'n choose k', counts the number of ways to arrange k successes among n trials. p^k is the probability all k successes occur in one particular arrangement, and (1−p)^(n−k) is the probability the remaining trials are all failures. Multiplying the three together gives the probability of exactly k successes in any order.

Mean and standard deviation

μ = np; σ² = np(1 − p); σ = √(np(1 − p))

The mean is the expected number of successes over many repetitions of n trials — for n=10 coin flips (p=0.5) that's 5 heads on average. The standard deviation measures how much the actual count typically varies from that average; it's largest when p is near 0.5 and shrinks as p approaches 0 or 1.

Cumulative probability

P(X ≤ k) = Σ P(X = i) for i = 0..k; P(X ≥ k) = 1 − P(X ≤ k − 1)

'At most k' sums the exact-probability formula over every outcome from 0 successes up to k. 'At least k' is the complement of everything below k, computed as 1 minus the cumulative probability up to k−1 — faster than summing the (potentially long) upper tail directly.

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Glossary

Key Terms Explained

7 terms

Binomial distribution The probability distribution of the number of successes in a fixed number of independent trials, each with the same success probability. It applies whenever an experiment has a fixed trial count, two possible outcomes per trial, a constant success probability, and independence between trials.

Trial One repetition of the underlying experiment — a coin flip, a free throw, a manufactured item inspected for defects. The binomial model requires every trial to be independent of the others and to share the same success probability p.

Success probability (p) The probability that a single trial results in the outcome you're counting, entered as a decimal between 0 and 1 (0.5 for a 50% chance, 0.02 for a 2% defect rate). Failure probability is always 1 − p.

n choose k (combination) Written C(n,k) or 'n choose k', the number of distinct ways to pick k items from a set of n without regard to order. In the binomial formula it counts every possible arrangement of k successes among n trials, since a specific arrangement's probability alone would undercount all the equally-likely orderings.

Bernoulli trial A single trial with exactly two possible outcomes — success or failure — and a fixed success probability p. A binomial distribution is the sum of n independent, identical Bernoulli trials.

Probability mass function (PMF) A function that gives the probability of each possible discrete outcome. For the binomial distribution, the PMF gives P(X = k) for every integer k from 0 to n; plotted as a bar chart, it shows the full shape of the distribution.

Cumulative probability The running total probability up to (or from) a given point — P(X ≤ k) or P(X ≥ k) — rather than the probability of one exact outcome. Cumulative probabilities answer questions like 'what's the chance of at least 8 successes' instead of 'exactly 8'.

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Scenarios

Real-World Examples

3 worked examples

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Statistics student

Ten fair coin flips, exactly 4 heads

Trials (n) 10 Success probability (p) 0.5 Successes (k) 4 Comparison Exactly k

P(X = 4) = C(10,4) · 0.5⁴ · 0.5⁶ = 210 · 0.0009766 ≈ 0.2051 — about a 20.5% chance of landing exactly 4 heads in 10 flips. The distribution's mean is np = 5 heads, so 4 is just one below the expected count, which is why it's a fairly common outcome rather than a rare one.

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Basketball coach

A 75% free-throw shooter attempting at least 8 of 10

Trials (n) 10 Success probability (p) 0.75 Successes (k) 8 Comparison At least k

P(X ≥ 8) sums the exact probabilities for 8, 9, and 10 makes, coming out to roughly 53% — despite a 75% per-shot make rate, hitting at least 8 of 10 is close to a coin flip because variance across 10 shots is substantial. The mean is np = 7.5 makes, so 8+ is slightly above average.

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Quality control analyst

A batch of 50 units at a 2% defect rate

Trials (n) 50 Success probability (p) 0.02 Successes (k) 2 Comparison At most k

P(X ≤ 2) ≈ 92%, meaning there's roughly an 8% chance a batch of 50 has more than 2 defective units. Mean defects per batch = np = 1, with SD ≈ 0.99 — the distribution is right-skewed since p is small, so most batches see 0 or 1 defect and 3+ is the tail.

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Deep Dive

Understanding the Binomial Distribution Calculator

3 chapters · 4 min read

The binomial distribution answers a specific but extremely common question: if you repeat an independent yes/no experiment a fixed number of times, how likely is a given number of successes? This calculator computes exact, at-most, and at-least probabilities along with the distribution's mean and standard deviation, so you can go from 'how many trials and what odds' straight to a usable probability.

How the binomial formula works

The binomial PMF, P(X = k) = C(n,k)·p^k·(1−p)^(n−k), breaks into three pieces. C(n,k) — 'n choose k' — counts how many different orderings of k successes among n trials are possible; without it you'd only capture the probability of one specific ordering, not all the equally-likely ways to get k successes. p^k is the probability that a particular set of k trials all succeed, and (1−p)^(n−k) is the probability the remaining trials all fail. Multiplying the three terms together and, for cumulative questions, summing across every relevant k, gives the full probability picture.

Inputs and what they mean

Trials (n) is the fixed number of independent repetitions — it must be a whole number, and this calculator supports up to 1,000 for numerical stability. Success probability (p) is entered as a decimal between 0 and 1 (50% is 0.5, not 50), and represents the same fixed probability applied to every trial. Successes (k) is the count you're asking about, always between 0 and n. The comparison type — exactly, at most, or at least — determines whether you get one exact-probability answer or a cumulative sum; 'at least k' is usually the more useful real-world question (e.g., 'at least 8 of 10 free throws') since it captures every outcome that clears a bar rather than one precise count.

When it applies, and the normal approximation

The binomial model requires four conditions: a fixed number of trials, exactly two outcomes per trial, a constant success probability across trials, and independence between trials (one trial's outcome can't change another's odds). Sampling without replacement from a small population violates independence and calls for the hypergeometric distribution instead. When n is large and p isn't too close to 0 or 1 (a common rule of thumb: np ≥ 5 and n(1−p) ≥ 5), the binomial distribution is closely approximated by a normal distribution with the same mean and standard deviation — useful for quick mental estimates, though this calculator always computes the exact binomial value rather than the approximation.

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Questions

Frequently Asked Questions

6 questions

What's the binomial probability formula?+

P(X = k) = C(n,k) · p^k · (1 − p)^(n − k), where n is the number of trials, p is the success probability per trial, k is the number of successes, and C(n,k) is 'n choose k' — the number of ways to arrange k successes among n trials.

What does the mean of a binomial distribution represent?+

The mean, μ = np, is the expected number of successes if you repeated the n trials many times and averaged the results. It's not necessarily a possible single outcome (e.g., np = 7.5 for n=10, p=0.75) — it's a long-run average, and the PMF chart shows the actual probability of each whole-number outcome around it.

How is the standard deviation calculated?+

σ = √(np(1 − p)). It measures the typical spread of the actual success count around the mean. Standard deviation is largest when p = 0.5 (maximum uncertainty per trial) and shrinks toward 0 as p approaches either 0 or 1, since outcomes become more predictable.

What's the difference between 'exactly k' and 'at least k'?+

'Exactly k' — P(X = k) — is the probability of that one precise outcome and nothing else. 'At least k' — P(X ≥ k) — sums the probabilities of k, k+1, k+2, all the way to n, so it's always greater than or equal to P(X = k). Most real-world questions ('what's the chance of at least 8 successes') are cumulative, not exact.

When does the binomial distribution apply?+

When you have a fixed number of independent trials, each with exactly two possible outcomes (success/failure) and the same success probability throughout. Coin flips, pass/fail quality inspections, and free-throw attempts (assuming a constant shooting percentage) are classic examples. It doesn't apply if the success probability changes between trials or if trials aren't independent.

How is the binomial distribution different from the Poisson distribution?+

The binomial distribution models successes out of a fixed, known number of trials with a per-trial probability p. The Poisson distribution models the count of events in a fixed interval of time or space when events happen at a known average rate, with no natural 'number of trials' to divide by. When n is large, p is small, and np stays moderate, the binomial distribution converges to a Poisson distribution with mean λ = np — a useful approximation for rare-event counts.

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Binomial Distribution Calculator

Inputs

Cumulative probability table

Probability mass function

Methodology and assumptions

How to Use This Calculator

Enter trials, probability, and successes

Choose exactly, at most, or at least

Try a preset scenario

Read the probability, mean, and SD

Formula & Methodology

Key Terms Explained

Real-World Examples

Statistics student

Basketball coach

Quality control analyst

Understanding the Binomial Distribution Calculator

How the binomial formula works

Inputs and what they mean

When it applies, and the normal approximation

Frequently Asked Questions

Get a branded PDF of your results

Binomial Distribution Calculator

Inputs

Cumulative probability table

Probability mass function

Methodology and assumptions

How to Use This Calculator

Enter trials, probability, and successes

Choose exactly, at most, or at least

Try a preset scenario

Read the probability, mean, and SD

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Binomial Distribution Calculator

How the binomial formula works

Inputs and what they mean

When it applies, and the normal approximation

Frequently Asked Questions

Get a branded PDF of your results

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