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Binomial Distribution Calculator
Find the probability of a given number of successes in a fixed number of independent trials. Enter the number of trials, the chance of success on each and the number of successes to get the exact probability and all four cumulative ones, with the full distribution in a table.
- P(X ≤ k) at most k
- 0.171875
- P(X ≥ k) at least k
- 0.945313
- P(X < k) fewer than k
- 0.054688
- P(X > k) more than k
- 0.828125
- Mean (n × p)
- 5
- Standard deviation
- 1.5811
Full distribution
| k | P(X = k) | P(X ≤ k) |
|---|
Saved setups
Save a set of inputs you reuse — your usual rate, your loan, your room sizes — and load it back in one tap.
Your recent calculations
Results you calculate here are kept on this device so you can come back to them.
Formula
How to use it
- Enter the number of trials, n.
- Enter the probability of success on a single trial as a decimal — 0.25 for 25%.
- Enter the number of successes, k.
- Pick the line that matches your question: exactly k, at most k, at least k, fewer than k or more than k.
Worked examples
Exactly 3 heads in 10 fair coin flips
- P(X = k) exactly k
- 0.117188
- P(X ≤ k) at most k
- 0.171875
- P(X ≥ k) at least k
- 0.945313
- Mean (n × p)
- 5
- Standard deviation
- 1.5811
Two defective items in a batch of 20 when 10% are defective
- P(X = k) exactly k
- 0.28518
- P(X ≤ k) at most k
- 0.676927
- P(X ≥ k) at least k
- 0.608253
- Mean (n × p)
- 2
When the binomial distribution applies
It needs four things: a fixed number of trials, only two outcomes per trial, the same probability of success every time, and trials that do not affect each other. Coin flips, free throws at a steady success rate and defect checks on a large production run all fit.
A worked example
Exactly 3 heads in 10 fair flips: C(10, 3) = 120 ways, each with probability (1/2)¹⁰ = 1/1024, so 120/1024 = 0.1172. At most 3 heads adds the cases for 0, 1 and 2: (1 + 10 + 45 + 120)/1024 = 0.1719.
Results here are summed term by term rather than approximated with a normal curve, for up to 100,000 trials. On a TI calculator the same values come from binompdf(n, p, k) and binomcdf(n, p, k).
Questions people ask
What is the probability of exactly 3 heads in 10 coin flips?
11.72% (0.117188).
What is the probability of at least 3 heads in 10 flips?
94.53% (0.945313), which is 1 minus the chance of 0, 1 or 2 heads.
If 10% of items are defective, what is the chance of exactly 2 defects in 20?
28.52%. The chance of 2 or fewer is 67.69%.