About the Binomial Probability
This calculator finds the probability of landing on exactly a chosen number of successes across a fixed number of independent trials, each carrying the same probability of success. It applies the binomial distribution, the standard model for repeated yes/no outcomes such as coin flips, pass/fail inspections, or survey responses. Enter the trial count, target number of successes, and per-trial success probability to get both the exact and cumulative probability.
How It Works
You enter the number of trials (n), the number of successes you want to check (k), and the probability of success on each trial (p, entered as a decimal between 0 and 1). The calculator builds the binomial coefficient C(n,k) using a running multiplicative product rather than full factorials, then multiplies it by p raised to the k successes and (1-p) raised to the remaining failures to get the exact probability. It also sums that same calculation for every outcome from 0 up to k to give the cumulative probability of getting k or fewer successes, and reports the distribution's mean (n times p) for reference.
Formula & Methodology
The calculator computes C(n,k) iteratively: starting from 1, it multiplies by (n-i)/(i+1) for i from 0 to k-1, which avoids overflow from computing huge factorials directly. That combinatorial count is then multiplied by p^k and (1-p)^(n-k) to weight each of the C(n,k) equally likely arrangements of successes and failures by its actual probability.
Examples
Fair coin, ten flips
With the default inputs (n=10 trials, p=0.5), the probability of getting exactly 4 successes is C(10,4) x 0.5^10 = 210/1024, about 20.51%. The probability of 4 or fewer successes comes out to about 37.70%, and the expected number of successes is 5.
Defect rate in a batch
For a process with a 10% defect rate (p=0.1) sampled 20 times (n=20), the probability of finding exactly 2 defective units is C(20,2) x 0.1^2 x 0.9^18, about 28.52%. The expected number of defects across the batch is 2.
Advantages
- Computes exact combinatorial probabilities instead of relying on a normal approximation, which stays accurate even for small samples where approximations break down.
- Returns both the exact and cumulative probability in one calculation, saving the need to sum multiple terms by hand.
- Surfaces the distribution's mean alongside the probabilities, giving quick context for whether an observed count is close to or far from what's typically expected.
Common Mistakes
- Entering the success probability as a whole percentage, like 50, instead of a decimal, 0.5, which the field expects.
- Assuming trials are binomial when the success probability actually changes between draws, such as sampling without replacement from a small population, which calls for the hypergeometric distribution instead.
- Reading P(X <= k) when the question actually asks for P(X = k), or vice versa, and drawing the wrong conclusion about how likely an exact count is.
Edge Cases to Watch For
- Inputs are rounded to whole numbers for n and k (fractional trial or success counts are not meaningful), so entering 4.6 successes is treated as 5.
- The calculator rejects k greater than n, negative counts, or a probability outside 0 to 1, since these values have no meaning in a binomial model.
- It only reports P(X = k) and P(X <= k); to find P(X >= k) or a range between two values you need to combine these outputs yourself, for example P(X >= k) = 1 - P(X <= k-1).
Common Use Cases
- Quality control staff estimating how likely a given number of defective units is in a fixed sample size.
- Students and instructors checking binomial distribution homework or exam problems by hand.
- Analysts evaluating the likelihood of a specific number of successes in survey responses, A/B test conversions, or repeated experiments.