Binomial Probability Calculator
Calculate exact, cumulative, and complementary probabilities for a binomial distribution given the number of trials, probability of success, and target number of successes.
The highlighted bar is k.
Frequently asked questions
When does the binomial distribution apply?
The binomial model applies when: there are a fixed number of trials (n); each trial has exactly two outcomes (success or failure); trials are independent; and the probability of success (p) is the same on every trial. Classic examples include coin flips, quality control sampling, and clinical trials with a binary outcome. If the sample is drawn without replacement from a finite population, the hypergeometric distribution is more appropriate.
What is the difference between exact, cumulative, and complementary probability?
P(X = k) is the probability of getting exactly k successes. P(X ≤ k) is the probability of getting k or fewer successes — the left-tail cumulative probability. P(X ≥ k) is the probability of getting k or more — the right-tail or complementary probability. Note that P(X ≤ k) + P(X ≥ k) = 1 + P(X = k), so the two tails overlap at k.
How is the normal approximation related to the binomial?
For large n, the binomial distribution approaches a normal distribution with mean np and standard deviation √(np(1−p)). The approximation is reliable when np ≥ 5 and n(1−p) ≥ 5. For smaller n or extreme p values, the exact binomial calculation shown here is always preferable to the normal approximation.
What is the expected value and standard deviation of a binomial distribution?
The expected number of successes is μ = np. The standard deviation is σ = √(np(1−p)). Intuitively, more trials mean a higher expected count, and the spread is largest when p = 0.5 (maximum uncertainty per trial). The distribution is symmetric only when p = 0.5; otherwise it is skewed toward the more likely outcome.
Can this calculator handle large n?
Yes, up to n = 1000. The calculator uses log-space arithmetic to compute the probability mass function — multiplying very small probabilities in log space avoids numerical underflow that would occur when working directly with numbers like 0.5^1000.