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The binomial distribution models the probability of getting exactly k successes in n independent trials, each with the same probability of success — the classic example being how many heads you'd expect from a fixed number of coin flips.
This calculator finds both the exact probability P(X = k) and the cumulative probability P(X ≤ k), with a full step-by-step solution.
How it works
Enter the number of trials (n), the probability of success on each trial, and the number of successes (k) you're interested in. The calculator applies the binomial probability formula, using combinations to count the number of ways k successes can occur among n trials, then sums the probabilities from 0 to k for the cumulative result.
- Enter number of trials (n).
- Enter success probability (%).
- Enter number of successes (k).
- Click Calculate to see your results.
Examples
5 heads in 10 fair coin flips
With 10 trials at a 50% success probability, the exact probability of getting exactly 5 successes is 24.61%, and the cumulative probability of getting 5 or fewer successes is 62.30%.
Who should use it
- Calculating the probability of a specific number of successes in quality control sampling.
- Estimating outcomes in probability and statistics coursework.
Industry applications
- Quality control and manufacturing
- Statistics and probability education
Advantages
- Calculates both exact and cumulative probabilities in one step.
- Includes a complete, formula-based step-by-step solution.
Limitations
- Only applicable to independent trials with a constant success probability — not appropriate for dependent or varying-probability scenarios.
Common mistakes to avoid
- Applying the binomial distribution to trials that aren't actually independent or don't share the same success probability.
- Confusing the exact probability P(X = k) with the cumulative probability P(X ≤ k) — they answer different questions.
Best practices
- Double check that your scenario truly has independent trials with a fixed success probability before applying the binomial distribution.
Tips
- To find the probability of "at least k" successes rather than "at most k," subtract the cumulative probability of k-1 from 100%.