Short answer
A binomial experiment has a fixed number of trials n, two outcomes per trial, the same success probability p each time, and independent trials. Ten coin flips qualify; drawing cards without replacement does not.
1. The four conditions
- 1Fixed number of trials n, decided in advance.
- 2Two outcomes per trial: success or failure.
- 3Constant probability p of success on every trial.
- 4Independent trials: one result does not change another.
2. Examples that fit
| Situation | n, p | Example result |
|---|---|---|
| Heads in 10 fair flips | 10, 0.5 | P(X = 6) = 0.205078 |
| Sixes in 12 rolls | 12, 1/6 | P(X = 3) = 0.197396 |
| Defects in 50 units at 2% | 50, 0.02 | P(X ≤ 2) = 0.921572 |
| Patients responding out of 30 at 70% | 30, 0.7 | P(X ≥ 25) = 0.076595 |
3. Examples that do not fit
- Cards drawn without replacement: each draw changes the next probability, so the count follows a hypergeometric distribution.
- Flip until the first head: the number of trials is not fixed. That is a geometric distribution.
- A die with more than two outcomes can be made binomial by defining success as, say, rolling a six.
4. Sampling note
Sampling from a large population without replacement is close to binomial when the sample is a small part of it, because the probability barely changes between draws.
Once your situation fits, enter n, p and k into the Binomial Probability Calculator. For the formula, see how to calculate binomial probability; for choosing items, the Permutation and Combination Calculator.
5. Frequently asked questions
What are the conditions for a binomial distribution?expand_more
A fixed number of trials, two outcomes, constant probability and independent trials.
Are card draws without replacement binomial?expand_more
No. The draws are dependent, so the hypergeometric distribution applies.
Is flipping until you get heads binomial?expand_more
No. The number of trials is not fixed, which makes it geometric.
Can a die roll be binomial?expand_more
Yes, if you define one outcome as success, such as rolling a six, and treat all others as failure.