Binomial distribution calculator with cumulative probabilities
Multiply the number of ways to choose k successes, C(n, k), by p^k and by (1 − p)^(n − k). For 10 fair coin flips, exactly 3 heads has probability 120 × 0.5¹⁰ ≈ 0.1172. For “at most”, “at least” or “between”, add the probabilities of every count in the range.
Inputs used by this calculator
Binomial Distribution Calculator uses the fields below. The starting values reproduce the page example; replace them with values from the same units and time period before using the result.
| Input | Starting example | Accepted value | How it is used |
|---|---|---|---|
| Number of trials (n) | 10 | 1 to 1000 | Supplies the number of trials (n) quantity used by the published equation or result breakdown. |
| Probability of success on each trial (p) | 0.5 | 0 to 1 | Supplies the probability of success on each trial (p) quantity used by the published equation or result breakdown. |
| Probability of | Exactly k successes | Exactly k successes · At most k (≤ k) · At least k (≥ k) · 3 more choices | Chooses which Binomial Distribution Calculator method or unknown the engine evaluates. |
| Number of successes (k) | 3 | 0 to 1000 | Supplies the number of successes (k) quantity used by the published equation or result breakdown. |
| Upper number of successes (k₂) | 6 | 0 to 1000 | Supplies the upper number of successes (k₂) quantity used by the published equation or result breakdown. |
When the binomial distribution applies
Use it when you count successes in a fixed number of trials and four conditions hold: the number of trials n is fixed, each trial has two outcomes, the trials are independent, and the probability of success p is the same every time. Coin flips, free throws, defective parts in a batch and survey responses are common examples.
- A fixed number of trials, n.
- Two outcomes per trial: success or failure.
- Independent trials.
- The same probability p on every trial.
The probability of exactly k successes
C(n, k) counts the different orders in which k successes can appear among n trials. Each particular order has probability p^k (1 − p)^(n − k). Multiplying the two gives the probability mass function.
P(X = k) = C(n, k) p^k (1 − p)^(n − k)C(n, k) = n! ÷ (k! (n − k)!)At most, at least and between
Cumulative questions add several exact probabilities. “At most 3” is P(0) + P(1) + P(2) + P(3). “At least 3” is 1 − P(at most 2). Watch the wording: “fewer than 3” stops at 2, and “more than 3” starts at 4. The calculator shows all three common forms for your k so you can check which one the problem asks for.
Mean, variance and the most likely count
The expected number of successes is np and the variance is np(1 − p). The standard deviation is the square root of the variance. The most likely count (the mode) is the whole number part of (n + 1)p; when (n + 1)p is a whole number, that value and the one below it tie.
mean = npvariance = np(1 − p)SD = √(np(1 − p))Worked example
A basketball player makes 70% of free throws. In 8 attempts, the chance of making exactly 6 is C(8, 6) × 0.7⁶ × 0.3² = 28 × 0.117649 × 0.09 ≈ 0.2965. The expected number made is 8 × 0.7 = 5.6.
Large n and the normal approximation
Textbooks often approximate the binomial with a normal distribution when np and n(1 − p) are both at least 5 or 10, using a continuity correction. This calculator computes exact binomial terms with logarithms for up to 1,000 trials, so no approximation is needed; use the approximation only when a course asks for it.
10 fair coin flips (n = 10, p = 0.5)
| Heads (k) | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|---|---|---|
| 0 | 0.000977 | 0.000977 | 1.000000 |
| 1 | 0.009766 | 0.010742 | 0.999023 |
| 2 | 0.043945 | 0.054688 | 0.989258 |
| 3 | 0.117188 | 0.171875 | 0.945313 |
| 4 | 0.205078 | 0.376953 | 0.828125 |
| 5 | 0.246094 | 0.623047 | 0.623047 |
| 6 | 0.205078 | 0.828125 | 0.376953 |
| 7 | 0.117188 | 0.945313 | 0.171875 |
| 8 | 0.043945 | 0.989258 | 0.054688 |
| 9 | 0.009766 | 0.999023 | 0.010742 |
| 10 | 0.000977 | 1.000000 | 0.000977 |
Frequently asked questions
What is the binomial probability formula?
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where n is the number of trials, k the number of successes and p the probability of success.
How do I find P(X ≤ k)?
Add P(X = 0) through P(X = k). For 10 fair coin flips, P(X ≤ 3) = 0.171875.
How do I find P(X ≥ k)?
Use 1 − P(X ≤ k − 1), or add P(X = k) through P(X = n).
What is the mean of a binomial distribution?
np. With 20 trials and p = 0.3, the expected number of successes is 6.
What is the difference between binomial and normal distributions?
The binomial counts whole numbers of successes in n trials; the normal is continuous. For large n the binomial looks approximately normal.
When should I not use the binomial distribution?
When trials are not independent, p changes from trial to trial, or you sample without replacement from a small group.
What does binompdf versus binomcdf mean?
On graphing calculators, binompdf gives P(X = k) and binomcdf gives P(X ≤ k). Choose “Exactly” or “At most” here for the same results.
What to keep in mind
The binomial model needs a fixed number of independent trials, each with the same probability of success. Sampling without replacement from a small population calls for the hypergeometric distribution instead.
Results are rounded for display in US format, with money in dollars and cents; CSV exports keep full calculation precision. This is a calculation summary, not an official certificate.