Binomial Probability Calculator
Calculate the exact and cumulative probability of a given number of successes across a fixed number of independent trials.
Capped at 100
%
Results
P(X = 4)
20.508%
P(X ≤ 4)
37.695%
How to Use the Binomial Probability Calculator
Enter the number of independent trials (n), the number of successes you want the probability for (k), and the probability of success on each individual trial (p). The binomial distribution applies whenever you repeat the same trial a fixed number of times, each trial has only two outcomes, and the probability of success stays the same every time — like flipping a coin 10 times or testing 20 products for defects.
where C(n, k) = n! / (k! × (n − k)!)
Example
Flipping a fair coin (p = 50%) 10 times, the probability of getting exactly 4 heads is C(10,4) × 0.5⁴ × 0.5⁶ = 210 × 0.0625 × 0.015625 ≈ 20.5%. The cumulative probability of getting 4 or fewer heads out of 10 flips is about 37.7%.
Common Use Cases
- Finding the probability of a specific number of successes across repeated trials, like coin flips or dice rolls.
- Quality control — estimating the probability of finding a certain number of defective items in a sample batch.
- Statistics coursework covering the binomial distribution and its cumulative probability.
FAQs
- How is this different from the generic Probability Calculator? The generic Probability Calculator handles single events and combinations of two independent events (both happening, or at least one happening). This calculator is specifically for the binomial distribution — repeating the same trial many times and asking about the count of successes across all those repeated trials.
- Why is n capped at 100? Very large trial counts can push factorial-based combination math into floating-point overflow or precision loss. This calculator uses a numerically stable iterative method for C(n,k) rather than computing raw factorials, and the cap keeps results both fast and reliable.
- What's the difference between P(X = k) and P(X ≤ k)? P(X = k), the exact probability, is the chance of getting precisely k successes. P(X ≤ k), the cumulative probability, is the chance of getting k successes or fewer — it's the sum of the exact probabilities for every outcome from 0 up to k.