Binomial Probability: formula
The model fails if trials are not independent, such as drawing cards without replacement, or if the probability changes between trials. For large n, a normal approximation is often used. Rare events over many trials are better modeled by a Poisson distribution.
Rounding hides small probabilities.
Binomial Probability: worked example
| Number of trials (n) | 10 |
|---|---|
| Successes wanted (k) | 3 |
| Chance of success each trial | 50.0% |
| Exactly k successes | 11.7% |
| k or fewer successes | 17.2% |
| Expected successes (n × p) | 5 |
The binomial model uses n independent trials, a required success count k and a single-trial success probability entered as a percentage. Each trial must have only two outcomes and the same probability, and trials must be independent, like coin flips or defective parts in a large batch. The tool gives the chance of exactly k successes and of k or fewer, and the expected number of successes.
Binomial Probability: changing Number of trials (n)
| Number of trials (n) | Exactly k successes | k or fewer successes | Expected successes (n × p) |
|---|---|---|---|
| 8 | 21.9% | 36.3% | 4 |
| 9 | 16.4% | 25.4% | 4.5 |
| 10 (example) | 11.7% | 17.2% | 5 |
| 11 | 8.1% | 11.3% | 5.5 |
| 12 | 5.4% | 7.3% | 6 |
Binomial Probability: compare input ranges
| Input | Tested from | To | Exactly k successes: low input | Exactly k successes: high input | Swing |
|---|---|---|---|---|---|
| Successes wanted (k) | 2 | 4 | 4.4% | 20.5% | 16.1% (137%) |
| Number of trials (n) | 9 | 11 | 16.4% | 8.1% | 8.3% (71%) |
| Chance of success each trial | 48.0% | 52.0% | 13.6% | 9.9% | 3.7% (32%) |
Binomial Probability: individual input comparisons
| Input | Value | Exactly k successes | k or fewer successes | Expected successes (n × p) |
|---|---|---|---|---|
| Number of trials (n) | 8 | 21.9% | 36.3% | 4 |
| 9 | 16.4% | 25.4% | 4.5 | |
| 10 | 11.7% | 17.2% | 5 | |
| 11 | 8.1% | 11.3% | 5.5 | |
| 12 | 5.4% | 7.3% | 6 | |
| Successes wanted (k) | 1 | 0.977% | 1.1% | 5 |
| 2 | 4.4% | 5.5% | 5 | |
| 3 | 11.7% | 17.2% | 5 | |
| 4 | 20.5% | 37.7% | 5 | |
| 5 | 24.6% | 62.3% | 5 | |
| Chance of success each trial | 46.0% | 15.6% | 24.5% | 4.6 |
| 48.0% | 13.6% | 20.7% | 4.8 | |
| 50.0% | 11.7% | 17.2% | 5 | |
| 52.0% | 9.9% | 14.1% | 5.2 | |
| 54.0% | 8.2% | 11.4% | 5.4 |
Binomial Probability: combined input changes
| Number of trials (n) / Successes wanted (k) | 1 | 3 | 5 |
|---|---|---|---|
| 8 | 3.1% | 21.9% | 21.9% |
| 9 | 1.8% | 16.4% | 24.6% |
| 10 | 0.977% | 11.7% | 24.6% |
| 11 | 0.537% | 8.1% | 22.6% |
| 12 | 0.293% | 5.4% | 19.3% |
Binomial Probability: which assumptions move the result?
Binomial Probability: values and definitions
| Quantity and definition | Value used |
|---|---|
| Number of trials (n) — How many independent attempts are made, up to 1,000. | 10 |
| Successes wanted (k) — The exact count of successes you are asking about. | 3 |
| Chance of success each trial — The probability that any single trial succeeds, from 0 to 100 percent. | 50.0% |
| Exactly k successes | 11.7% |
| k or fewer successes | 17.2% |
| Expected successes (n × p) | 5 |
Binomial Probability: half and double Successes wanted (k)
| Successes wanted (k) | Exactly k successes |
|---|---|
| 1.5 | 4.4% |
| 3 | 11.7% |
| 6 | 20.5% |
Common error: Reading the expected value as a promise. Five heads in ten flips is the average, yet the single most likely result still has less than a 25 percent chance of occurring.
Statistics & Probability calculators: Normal Distribution Probability Calculator, P-Value from Z-Score Calculator, Expected Value Calculator, Bayes' Theorem Calculator, Mean & Median Calculator, Standard Deviation Calculator.
Ten coin flips will land on exactly five heads only about a quarter of the time, and on exactly three about 12 percent. People expect the most likely outcome to be near-certain, and it never is. An exact-count probability covers one outcome.
A cumulative probability adds all outcomes up to and including that count; it is the lower-tail probability.
Check that the trials are independent and the chance is the same each time. Drawing cards without replacement breaks both rules.
See simple chances on the probability calculator, counting on the permutation and combination page, and the normal approximation on the normal probability calculator. The expected value and sample size pages apply the results, and the z-score page standardizes counts.
How do I find the chance of at least k successes?
Subtract the chance of k minus 1 or fewer successes from 100%, since at least k successes is the complement of fewer than k successes.
When can I use the binomial distribution?
When there is a fixed number of independent trials, each with two outcomes and the same chance of success.
What is the expected number of successes?
The number of trials times the chance of success, so 10 flips of a fair coin expect 5 heads.