How it works
Sum the probabilities for zero through k successes. With n = 4, p = 0.5 and k = 1, the lower-tail probability is (1 + 4)/16 = 0.3125. The complement is the probability of more than k, not of at least k. Enter 0.5 for 50%, and use one fixed success probability for independent trials.
Inputs and units
| Input | Example value | What to enter |
|---|---|---|
| Number of trials n | 10 | Enter a numeric value. Use a whole number. Use a non-negative value. Allowed range: 0–170. |
| Maximum successes k | 3 | Enter a numeric value. Use a whole number. Use a non-negative value. Allowed range: 0–170. |
| Success probability (0–1) | 0.5 | Enter a numeric value. Use a non-negative value. Allowed range: 0–1. |
Worked example
Show example inputs and results
This example is generated by the same calculation engine using the starting values in the form. Your own inputs produce your own result.
- Number of trials n
- 10
- Maximum successes k
- 3
- Success probability (0–1)
- 0.5
Complete example results
- Probability of at most k (0–1)
- 0.171875
- Probability of more than k (0–1)
- 0.828125
Reading the result
Check the probability scale, sample size and assumptions stated by the method. A decimal probability of 0.5 represents 50%. Mean and spread inputs must describe the same dataset; the calculation does not validate those estimates.
Questions and answers
Can I use the example values for my own calculation?
The example values illustrate the calculation. Replace every input with your own information, especially dates, rates and the selected operation.
Does k = n give certainty?
Yes. A binomial count cannot exceed the number of trials. Very small complementary tails can lose precision when subtracted from one; the displayed rounding is not an exact tail guarantee.
Why are some results rounded?
Results display up to 8 decimal places. Calculations use the unrounded values; very small values use scientific notation. JavaScript floating-point arithmetic can introduce tiny rounding differences.
Are my inputs sent to a server?
The calculator processes inputs in your browser. Its code does not send your values to a server.
Method sources: itl.nist.gov