Binomial Cumulative Probability Calculator
Enter trials n, probability p and count k for the binomial distribution.
Result
How it is calculated
The tool adds the binomial probabilities for every count from 0 to k: P(X ≤ k) = Σ C(n, i) × pi × (1 − p)n−i, for i = 0 … k. n is the number of independent trials, p is the chance of success on one trial as a fraction between 0 and 1 (not a percentage), and k is the largest number of successes you accept. The terms are summed in log space, so a large n does not overflow; the tool accepts up to 1,000,000 trials. The second result is the single term P(X = k) and the third is the mean n × p. Inputs that cannot describe a real experiment show no result: a fractional n or k, n below 1, a negative k, k larger than n, or p outside 0 to 1. Probabilities are shown with four decimals.
Worked example
Take 20 independent trials with a 0.3 chance of success each, and ask for the probability of at most 8 successes. The tool adds the terms for i = 0 to 8 and the total is 0.8867. The single term for exactly 8 successes is C(20, 8) × 0.38 × 0.712 = 0.1144. The mean is 20 × 0.3 = 6.00. The chance of 9 or more successes is therefore 1 − 0.8867 = 0.1133.
Common mistakes
- Typing p as a percentage. A 30 % chance is entered as 0.3; 30 gives no result.
- Reading the main result as the chance of exactly k. That is the separate pmf figure; the main result is the chance of k or fewer.
- Needing at least k successes but not converting. Take 1 minus the cumulative probability at k − 1.
- Applying it when trials are not independent or the chance changes between trials, such as drawing without replacement from a small batch.
- Entering a k larger than n, or a decimal k or n, and expecting a rounded answer. These inputs are rejected instead of adjusted.
About this calculator
For a series of n independent trials that each succeed with probability p (like flipping a biased coin n times), the binomial cumulative probability P(X ≤ k) gives the chance of getting at most k successes in total. It's calculated by summing the individual binomial probabilities C(n, i) × p^i × (1−p)^(n−i) for every count i from 0 up to k, where C(n, i) is the number of ways to choose which i trials succeeded.
Cumulative binomial probabilities are the standard way to answer “at least” or “at most” questions in quality control (the probability that no more than k defective items appear in a batch), clinical trials and medical statistics (the chance that at most k patients respond to a treatment), and everyday probability problems like the odds of winning at least a certain number of games or coin flips out of a fixed number of attempts, where the distribution's mean is simply n × p.
Enter the number of trials n, the success probability p, and a count k, and this calculator returns the cumulative probability P(X ≤ k), the exact probability at k, and the mean.
Was this helpful?
You can pick up to 3 reactions.
Popular calculators
Loan Calculator
Monthly payment, total interest and full amortization schedule.
Percentage Calculator
Percent change, percent of, and ratio calculations.
Discount Calculator
Calculate final price, savings, and discount amount.
BMI Calculator
Body mass index from height and weight (metric or imperial).
Age Calculator
Compute your exact age in years, months and days from a birth date.
Compound Interest Calculator
Calculate the final amount and interest earned with compounding.
All Calculators
No calculators found
Comments (0)