Binomial Probability Calculator
Find the probability of exactly, at most, or at least k successes across n independent trials with a fixed success probability.
P(X = k) = C(n,k) × pk × (1−p)n−k
Example: flipping a fair coin (p = 0.5) 10 times (n = 10) — probability of exactly 6 heads (k = 6).
Probability
📊 Distribution Stats
The Binomial Probability Formula, Explained
A binomial distribution applies whenever you repeat the same yes/no trial a fixed number of times, with the same success probability every time — coin flips, pass/fail quality checks, whether each of n customers converts. The formula for exactly k successes in n trials is:
C(n, k) — read "n choose k" — counts how many different orderings of successes and failures produce exactly k successes (e.g. HHT, HTH and THH are 3 different sequences that all give 2 heads in 3 flips, so C(3,2) = 3). pk is the probability of getting k successes in a row, and (1−p)n−k is the probability of the remaining n−k trials all failing — multiplied together and then by C(n,k) to account for every possible ordering.
Exactly, At Most, or At Least — Which One Do You Need?
- P(X = k) — exactly k: "What's the probability of exactly 3 heads in 10 flips?" Use this for a precise single count.
- P(X ≤ k) — at most k: "What's the probability of 3 or fewer defective items in a batch?" Sums P(X=0) through P(X=k).
- P(X ≥ k) — at least k: "What's the probability of at least 3 successful conversions?" Calculated as 1 − P(X ≤ k−1).
A common special case: "at least one success" is almost always easiest as 1 − P(X = 0) — the probability that every single trial fails — rather than summing every other possible count.
Mean, Variance, and Standard Deviation
Beyond a single probability, a binomial distribution has predictable shape statistics that this calculator also computes:
| Statistic | Formula |
|---|---|
| Mean (expected successes) | n × p |
| Variance | n × p × (1 − p) |
| Standard Deviation | √(n × p × (1 − p)) |
For 100 fair coin flips (n=100, p=0.5): mean = 50, standard deviation = 5 — so getting 45-55 heads is common, while 70+ heads would be a genuinely rare, notable result.
Frequently Asked Questions
When should I use this instead of a basic probability calculator?
When you have a fixed number of independent trials, each with the same success probability, and want the probability of a specific count of successes — like 6 heads in 10 coin flips. A basic calculator handles single/two-event questions; binomial handles repeated-trial questions.
What are the 4 conditions for a binomial distribution?
Fixed number of trials (n), each trial has only two outcomes, the success probability (p) stays constant across trials, and trials are independent. Sampling without replacement from a small population can violate this and technically needs a hypergeometric distribution instead.
What's the difference between P(X = k), P(X ≤ k) and P(X ≥ k)?
Exactly k, at most k (cumulative sum up to k), and at least k (1 minus the cumulative probability below k). 'At least one' is usually easiest as 1 − P(X = 0).
How are mean and standard deviation calculated?
Mean = n × p. Variance = n × p × (1−p). Standard deviation is the square root of variance. For 100 coin flips: mean 50, std dev 5.
Why does 'n choose k' appear in the formula?
Because multiple different orderings of successes/failures give the same count — C(n,k) counts how many such equally-likely sequences exist, multiplied by the probability of any one sequence.