AP Stats Binomial Distribution Calculator

Find exact binomial probabilities for a fixed number of independent trials with the same success probability. Enter the number of trials, the probability of success, and the number of successes you care about.

What Is the Binomial Distribution?

The binomial distribution gives the probability of getting a fixed number of successes in a fixed number of independent trials, when each trial has the same probability of success. Flipping a fair coin 10 times and counting heads is a binomial setting: n = 10 trials, p = 0.5. The probability of exactly k successes is C(n, k) times p to the power k times (1 - p) to the power (n - k).

How to Use This Calculator

1. Enter n, the total number of trials — for example, 10 coin flips or 20 free throws.
2. Enter p, the probability of success on a single trial, as a decimal between 0 and 1 (for a fair coin, use 0.5; for a 70% free-throw shooter, use 0.7).
3. Enter x, the exact number of successes you want the probability for.
4. Click Calculate. You will get P(X = x), P(X ≤ x), and P(X ≥ x), plus the full distribution when n is 30 or less.

How to Calculate

For a binomial random variable X with n trials and success probability p, the probability of exactly x successes is:

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

where C(n, x) = n! / (x! × (n − x)!) counts the ways to choose which x trials are successes.

Worked example: n = 10, p = 0.5, x = 5

Suppose you flip a fair coin 10 times. What is the probability of exactly 5 heads?

C(10, 5) = 252. So P(X = 5) = 252 × 0.5^5 × 0.5^5 = 252 × 0.0009765625 = 0.2461 (about 24.6%).

Cumulative probabilities add up the individual values: P(X ≤ 5) = 0.6230 and P(X ≥ 5) = 0.6230 in this symmetric case — note that P(X ≥ 5) includes P(X = 5), so P(X ≤ 5) + P(X ≥ 5) is more than 1. Try it in the calculator above to confirm.

Tips and Common Mistakes

• Check BINS first: Binary outcomes, Independent trials, fixed Number of trials, and the Same probability of success on each trial. If any condition fails, the binomial formula does not apply.
• Do not confuse p and 1 − p: decide clearly what counts as a “success” before you enter p. If success is rolling a 6, p = 1/6, not 5/6.
• “At least” vs “at most”: “at least x” means P(X ≥ x) and “at most x” means P(X ≤ x). Both include x itself.
• “More than” and “fewer than” are different: P(X > x) = 1 − P(X ≤ x), and P(X < x) = P(X ≤ x − 1). Do not just use the calculator’s ≥ or ≤ result for these.
• Enter p as a decimal, not a percent: 50% is 0.5 here. Entering 50 will be rejected as out of range.
• Watch for independence: sampling without replacement from a small population is not binomial — that is hypergeometric. Sampling with replacement, or from a very large population, is approximately binomial.

Frequently Asked Questions

What is the difference between binompdf and binomcdf?

binompdf gives an exact probability, P(X = x). binomcdf gives a cumulative probability, P(X ≤ x). This calculator gives you both: the exact value and the cumulative values P(X ≤ x) and P(X ≥ x).

What are the binomial conditions (BINS)?

Binary: each trial has two outcomes, success or failure. Independent: trials do not affect each other. Number: n is fixed in advance. Same probability: p is the same on every trial.

How do I find “at least” a number of successes?

Use P(X ≥ x), shown by this calculator. For example, “at least 7 heads in 10 flips” is P(X ≥ 7), which sums P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10).

How do I find “at most” a number of successes?

Use P(X ≤ x). For example, “at most 3 successes in 20 trials” is P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).

What are the mean and standard deviation of a binomial distribution?

The mean is μ = np and the standard deviation is σ = √(np(1 − p)). For n = 10 and p = 0.5, the mean is 5 and the standard deviation is about 1.58.

Can I use this for a geometric distribution?

No — a geometric distribution counts trials until the first success, with no fixed n. Use a binomial calculator only when the number of trials is fixed in advance.

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