Binomial Probability Calculator
Find the probability of exactly or at most a chosen number of independent successes.
Result for the values shown.
Calculations run in your browser.
What to enter
| Input | Meaning and units |
|---|---|
| Number of trials | integer0..10000 |
| Successes / event count | integer0..n |
| Success probability | decimal[0,1] |
Understanding your result
These outputs describe the probability model and event definitions you entered. A probability is a number from 0 to 1; a probability density is not itself the probability of one exact continuous value.
Common mistakes
- Entering 25 instead of 0.25 for a probability, or assuming events are independent when the model requires that assumption.
- Confusing an exact-count probability with a cumulative probability, or replacement with sampling without replacement.
Check your calculation
- Check a zero/certain-event boundary and the stated support of the distribution against the worked example.
Calculation checks, sources and review limits
What is the probability of exactly, or at most, this many independent successes?
Find the probability of exactly or at most a chosen number of independent successes.
Common uses
- Find the probability of exactly or at most a chosen number of independent successes.
How it works
PMF=C(n,k)*p^k*(1-p)^(n-k); CDF=sum j0..k PMF(j). Use loggamma/stable recurrence. Trials/draws/counts are limited to 10,000; negative-binomial r and k may each be at most 10,000. Log-factorial arithmetic avoids forming large factorials. Extremely small probabilities can underflow to zero; outputs are numerical estimates, not guarantees about real events.
Worked example
Enter Number of trials: 3; Successes / event count: 2; Success probability: 0.5. The result is Pmf: 0.375; Cdf: 0.875.
FAQ
What is the probability of exactly, or at most, this many independent successes?
PMF=C(n,k)*p^k*(1-p)^(n-k); CDF=sum j0..k PMF(j). Use loggamma/stable recurrence. Trials/draws/counts are limited to 10,000; negative-binomial r and k may each be at most 10,000. Log-factorial arithmetic avoids forming large factorials. Extremely small probabilities can underflow to zero; outputs are numerical estimates, not guarantees about real events.
What assumptions and limits apply?
Fixed n independent Bernoulli trials, same p; p=0/1 boundary logic explicit.