binomial test calculator

What is a binomial test?

A binomial test is an exact hypothesis test used when your outcome has two categories (success/failure, yes/no, pass/fail) and you want to compare an observed success rate against a hypothesized probability.

Example: if a coin is supposed to be fair (p₀ = 0.5) and you flip it 20 times, a binomial test tells you whether your observed number of heads is unusually high or low under that assumption.

When should you use this calculator?

• You have a fixed number of independent trials (n).
• Each trial has only two possible outcomes.
• The probability of success is assumed constant under the null hypothesis.
• You want an exact p-value rather than a normal approximation.

Inputs explained

1) Number of trials (n)

This is the total number of observations. It must be a positive integer.

2) Observed successes (k)

The count of successes in your sample. It must be an integer between 0 and n.

3) Null probability (p₀)

This is the expected success probability if the null hypothesis is true. Common values are 0.5, 0.25, 0.1, and so on.

4) Alternative hypothesis

• Two-sided: tests whether the true probability differs from p₀.
• Greater: tests whether the true probability is greater than p₀.
• Less: tests whether the true probability is less than p₀.

How results are interpreted

The calculator reports an exact p-value. Compare that p-value with your significance level α:

• If p-value < α, reject the null hypothesis.
• If p-value ≥ α, fail to reject the null hypothesis.

“Fail to reject” does not prove the null is true; it means there is not enough evidence against it at the chosen alpha level.

Formula behind the test

Under the null, X ~ Binomial(n, p₀). The probability of exactly x successes is:

P(X = x) = C(n, x) p₀^x (1 - p₀)^(n - x)

For one-sided tests, p-values are tail probabilities like P(X ≤ k) or P(X ≥ k). For the two-sided exact test, this tool sums probabilities of outcomes that are as or more extreme than the observed outcome (based on probability mass under the null).

Practical example

Suppose a support team claims 80% first-contact resolution. You audit 50 tickets and find 33 resolved on first contact.

• n = 50
• k = 33
• p₀ = 0.80
• Alternative: p < 0.80

Enter those values to test whether performance is significantly below the target.

Common mistakes to avoid

• Using percentages instead of proportions (enter 0.8, not 80).
• Choosing the wrong tail direction for the research question.
• Interpreting p-value as the probability the null is true (it is not).
• Ignoring study design assumptions such as independence of trials.

Final notes

This binomial test calculator is ideal for quality checks, A/B outcomes, coin-flip style examples, and pass/fail process data. For large-sample settings, normal approximations can be useful, but exact binomial inference is often preferable when sample sizes are modest or proportions are near 0 or 1.