Binomial Probability Calculator
The Binomial Probability Calculator takes the number of trials, the number of successes, and the per-trial success probability, and returns the exact probability of that many successes, the cumulative probability up to it, the mean, and the standard deviation.
What this calculator does
The Binomial Probability Calculator estimates your result instantly from the values you enter, using standard statistics formulas. The Binomial Probability Calculator takes the number of trials, the number of successes, and the per-trial success probability, and returns the exact probability of that many successes, the cumulative probability up to it, the mean, and the standard deviation. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.
Result
About Binomial Probability Calculator
This calculator works out binomial probabilities for a fixed number of independent trials that each succeed with the same probability. Give it the trial count, the number of successes you are interested in, and the success probability, and it returns both the exact and cumulative probabilities.
It also reports the distribution mean and standard deviation so you can see the expected number of successes and its spread.
Formula & Mathematical Calculation
P(X = x) = f(Number of trials (n), Number of successes (x), Probability of success (p))
Calculated according to standard stats formulas implemented in the calculator core.
Calculation Example
For n = 10 trials, x = 3 successes, and p = 0.5, the exact probability is P(X = 3) = C(10,3) × 0.5³ × 0.5⁷ = 120 ÷ 1024 = 0.117188.
The cumulative probability sums the terms for 0 through 3 successes: (1 + 10 + 45 + 120) ÷ 1024 = 176 ÷ 1024 = 0.171875. The mean is np = 10 × 0.5 = 5 and the standard deviation is √(10 × 0.5 × 0.5) = √2.5 = 1.5811.
Key Benefits
- Computes the exact probability of an outcome count using the binomial formula
- Adds the cumulative probability from 0 up to your success count
- Reports the distribution mean np and its standard deviation
- Handles any valid trial count with a success probability between 0 and 1
How to Use this Calculator
-
Enter the number of trials
Type the total number of independent trials n into the trials field. -
Enter the number of successes
Type the number of successes x you want the probability for, between 0 and n. -
Enter the success probability
Type the per-trial probability of success p as a decimal from 0 to 1. -
Calculate
Press Calculate and read the P(X = x), P(X <= x), Mean, and Standard Deviation boxes.
Assumptions & Limitations
What the calculation assumes
- Every value you enter is treated as an exact, known amount.
- Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
- Results use standard published formulas and account only for the values entered above.
- Rounding is applied for display; the underlying calculation uses your full precision.
- Results are returned in the same units you enter, unless the tool states the output unit.
What it does not cover
- Binomial Probability Calculator produces a calculated result for reference and learning, not a certified or official measurement.
- Real-world outcomes differ when tolerances, rounding rules or local standards apply.
- Definitions and standards vary by region and change over time; verify the convention that applies to your case.
- Figures are only as accurate as the inputs you supply — check them before acting on the result.
- For work where the result carries real consequences, confirm it against an authoritative source.
Frequently Asked Questions
Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.