How to use the Binomial Probability Calculator
- 1. Set n and pEnter the number of trials and the chance of success on each.
- 2. Pick k and the typeChoose exactly k, at most k, at least k or a range.
- 3. Get the probabilityRead the result and the table of every outcome.
The binomial formula
Use the binomial distribution when you have a fixed number of independent trials, each with the same chance of success. The probability of exactly k successes is P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) = n! ÷ (k! (n − k)!).
Worked example: 10 fair coin flips, 3 heads. C(10, 3) = 120, and 0.5¹⁰ = 1/1024, so P(X = 3) = 120 ÷ 1024 = 0.1172. For at most 3 heads add the cases 0, 1, 2 and 3: (1 + 10 + 45 + 120) ÷ 1024 = 0.1719. The mean is np = 5 and the standard deviation is √(np(1 − p)) = 1.5811.
Where it is used
- Marketing: if 4% of visitors buy, how likely are 10 or more sales from 150 visitors?
- Quality control: chance of finding 2 defects in a batch of 50 when the defect rate is 1%.
- Exams: chance of passing by guessing on a multiple-choice paper.
- A/B testing: the basis for comparing conversion rates.
The calculator adds exact probabilities, using logarithms of factorials so it stays accurate even for large n. When n is large and p is not tiny, the normal approximation with mean np and SD √(np(1 − p)) is close; you can compare with the normal distribution calculator. If trials are not independent, or p changes from trial to trial, the binomial model does not apply.
Frequently asked questions
What is the difference between at most and fewer than?
At most k includes k itself (≤). Fewer than k stops at k − 1 (<).
Why is the table hidden for big n?
For more than 30 trials the table would be very long, so only the summary values are shown.
Is the Binomial Probability Calculator free?
Yes, the Binomial Probability Calculator is free. The first time you see your first result, you are asked for your name and WhatsApp number once. After that, this and every other Uncodemy tool opens straight away without the form.