Binomial Experiment Probability Calculator

Example: ten trials with success probability 0.30. Table shows probabilities for each possible number of successes.

Suppose a process has ten trials and success probability 0.30. You want the probability of observing at most two successes.

1. Set “Number of trials (n)” to 10.
2. Enter 0.30 as the probability and keep decimal mode.
3. Select “At most k successes (P(X ≤ k))” as calculation type.
4. Enter 2 in the k box and click “Calculate”.
5. The summary panel will show P(X ≤ 2) ≈ 0.382783.
6. This corresponds to about 38.28% chance of at most two successes.

Adjust n, p, and k to match your own binomial experiment.

Production quality control example

A factory inspects twenty items, each with defect probability 0.02.

Marketing email response example

A campaign sends fifty emails, each with response probability 0.08.

Clinical trial responder example

A study tracks thirty patients with individual success probability 0.40.

Service reliability over one week

A service operates seven days with daily success probability 0.95.

For a binomial experiment with n trials and success probability p:

Probability of exactly k successes:

P(X = k) = C(n, k) · p^k · (1 − p)^{n − k}

C(n, k) = n! / (k!(n − k)!) is the binomial coefficient.

Mean (expected value): μ = n · p

Variance: σ² = n · p · (1 − p)

Standard deviation: σ = √(n · p · (1 − p))

What is a binomial experiment?

A binomial experiment is a process consisting of a fixed number of independent trials, each having only two outcomes, commonly called success and failure, with the same probability of success.

When should I use this binomial calculator?

Use this calculator when your situation fits the binomial model: fixed number of trials, independent outcomes, constant success probability, and you are counting how many successes occur.

Should I enter probability as decimal or percentage?

Select decimal when you already know probability between zero and one. Select percentage when inputs are given like 30% or 8%, and the calculator will convert them automatically.

Why do probabilities not sum exactly to one?

Probabilities, percentages, and cumulative values are rounded for readability. Small rounding differences may cause totals to be slightly above or below one, which is normal in numerical reporting.

Are there limits on n or probability values?

Very large numbers of trials or extreme probabilities may cause numerical underflow or overflow. For stable results here, keep trials at or below two hundred and avoid probabilities extremely close to zero or one.