Probability Calculator (Binomial)
Compute binomial probability for X=k.
Binomial probability explained
This probability calculator answers a specific and very common question: if you repeat an experiment n times and each attempt succeeds with probability p, what is the chance of exactly k successes? That is the binomial distribution, and it covers coin flips, quality-control sampling, free throws, click rates and any other repeated yes-or-no trial.
How to use the Probability Calculator
- Enter the number of Trials (n).
- Enter the number of Successes (k) you are asking about.
- Enter the Probability of success (p) for one trial, as a decimal.
- Read the probability and the number of combinations.
Formula and a worked example
P(X = k) = C(n, k) × pk × (1 − p)n − k
C(n, k) = n! ÷ (k! × (n − k)!)
C(n, k) counts how many different orderings produce k successes — that is why ten heads in twenty flips is far more likely than twenty heads in twenty flips, even though every individual sequence is equally rare.
Worked example
Flip a fair coin 10 times. Exactly 5 heads: C(10,5) = 252 orderings, each with probability 0.5¹⁰, giving 24.6%. Exactly 10 heads has just one ordering, so 0.098% — 252 times less likely.
Frequently asked questions
What is the difference between exactly k and at least k?
This gives exactly k. For at least k, add the probabilities of k, k+1, … up to n.
When does the binomial distribution apply?
When trials are independent, each has only two outcomes, and p stays the same throughout. Drawing cards without replacement breaks the last condition.
What does C(n, k) mean?
The number of ways to choose k items from n when order does not matter. C(10, 5) = 252.
Why is exactly half not the most likely outcome by a wide margin?
Because probability spreads across neighbouring values. In 10 flips, 5 heads is the single most likely result at 24.6%, but 4 or 6 heads each come in around 20.5%.
Related calculators
Other tools people use alongside the probability calculator.