Binomial Distribution Calculator

Formula

P(X=k) = C(n,k) p^k (1-p)^(n-k)

Worked example

With Number of trials: 10 trials, Number of successes: 3 successes, Success probability per trial: 0.2, probability of exactly k successes is 0.201327.

Method and inputs

For each trial there must be two outcomes labeled success and failure, with the same success probability p. The probability of exactly k successes is the number of possible success arrangements times p to the k power times one minus p to the remaining-trial power.

Worked example

For ten independent trials with success probability 0.2, the chance of exactly three successes is C(10,3) × 0.2 cubed × 0.8 to the seventh, about 0.2013. The expected count is 10 × 0.2 = 2.

Exactly three successes in the example has probability about 20.13 percent. The event of at least three successes would add probabilities for three, four and every higher count; it must be larger. This distinction prevents a frequent misreading of an exact-count calculator.

How to read the result

The probability describes one exact count, not three or more and not at most three. The expectation is a long-run average across many repeated sets of ten trials; it need not equal the most likely count in any one set.

If a trial succeeds with certainty, p equals one and only k = n can occur. If p equals zero, only k = 0 can occur. Those boundary cases are useful checks on the formula, while ordinary cases require the independence assumption.

Assumptions and checks

Trials must be independent and share one fixed probability. Sampling without replacement from a small population violates those assumptions. The success count is a whole number; if it exceeds the number of trials, its exact probability is zero.

Do not add probabilities for overlapping events without checking overlap. A near-zero result for an extreme count is possible even with valid inputs. The combination count is calculated multiplicatively to avoid the huge factorials that overflow quickly.

Common questions

What does the binomial distribution calculator calculate?

It returns the probability of the exact requested success count under a binomial model and the expected number of successes.

When should I recalculate or check another method?

Recalculate if trial count or success probability changes. If the probability changes between trials, or draws depend on one another, use a different probability model rather than labeling the result binomial.

Sources and references

• NIST Binomial Distribution