Binomial Distribution Calculator
Find binomial probabilities: exactly k successes, at most k, at least k, and more, from the number of trials n and the success probability p, with the mean, standard deviation and a full table.
Results
| Item | Value |
|---|
Steps
| Step | Working |
|---|
Probability distribution
| k | P(X = k) | P(X ≤ k) |
|---|
Quick Answer
The binomial formula is P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). For 10 coin flips, the chance of exactly 6 heads is 0.205078, about 20.5%. The chance of 6 or fewer heads is 0.828125.
Embed this calculator
Adjust the size if needed, then copy the code below into your page or blog. The calculator will load in an iframe and stay updated automatically.
Table of Contents
- Binomial Distribution Calculator
- Quick Answer
- Related Calculators
- How This Binomial Distribution Calculator Works
- How to Use the Binomial Distribution Calculator
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Binomial vs Normal Distribution
- Coin Flips: 10 Trials, p = 0.5
- When to Use the Binomial Distribution
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Binomial Distribution Calculator Works
Enter how many times you repeat a trial (n), the chance of success each time (p), and the number of successes you care about (k). The calculator works out every probability from 0 to n successes, then shows the chance of exactly k, at most k, fewer than k, at least k and more than k. It also gives the mean, standard deviation and a table of the distribution.
Quick explanation: a binomial experiment has a fixed number of independent yes-or-no trials with the same chance of success.
- n: number of trials.
- p: chance of success on one trial.
- k: number of successes.
How to Use the Binomial Distribution Calculator
- Enter the number of trials, n.
- Enter the probability of success on each trial, p, as a decimal.
- Enter the number of successes, k.
- Press Calculate.
- Read the exact and cumulative probabilities, and scan the table.
Core Formulas
- Exactly k: P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)
- Combinations: C(n, k) = n! ÷ (k!(n − k)!)
- At most k: P(X ≤ k) = sum of P(X = i) for i = 0 to k
- Mean: np, and standard deviation: √(np(1 − p))
Key Terms
- Trial: one repetition of the experiment, such as one coin flip.
- Success: the outcome you are counting, such as heads.
- Probability mass function: the chance of each exact number of successes.
- Cumulative probability: the chance of k or fewer successes.
- Expected value: the long-run average number of successes, np.
Worked Example
You roll a die 5 times. What is the chance of exactly two sixes?
- n = 5, p = 1/6, k = 2
- C(5, 2) = 10, so P = 10 × (1/6)² × (5/6)³
- P = 0.160751, about 16.1%
Real-Life Examples
- Quality control: if 2% of parts are faulty, the chance of no faulty parts in a box of 50 is 0.3642.
- Free throws: a 75% shooter making at least 8 of 10 has a chance of 0.5256.
- Surveys: if 60% of people agree, the chance that exactly 12 of 20 agree is 0.1797.
- Multiple choice: guessing 10 questions with 4 options each, the chance of 5 or more right is 0.0781.
Binomial vs Normal Distribution
- Binomial: counts successes in a fixed number of trials. Values are whole numbers.
- Normal: a smooth bell curve for continuous values.
- Link: for large n, the binomial looks like a normal curve with mean np and standard deviation √(np(1 − p)).
Coin Flips: 10 Trials, p = 0.5
| Heads (k) | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.0010 | 0.0010 |
| 1 | 0.0098 | 0.0107 |
| 2 | 0.0439 | 0.0547 |
| 3 | 0.1172 | 0.1719 |
| 4 | 0.2051 | 0.3770 |
| 5 | 0.2461 | 0.6230 |
| 6 | 0.2051 | 0.8281 |
| 7 | 0.1172 | 0.9453 |
| 8 | 0.0439 | 0.9893 |
| 9 | 0.0098 | 0.9990 |
| 10 | 0.0010 | 1.0000 |
When to Use the Binomial Distribution
- Fixed trials: the number of trials is set in advance.
- Two outcomes: each trial is a success or a failure.
- Same chance: p stays the same for every trial.
- Independent: one trial does not affect another.
Common Mistakes to Avoid
- Entering p as a percent. Enter 0.25, not 25.
- Mixing up at least and at most. At least k includes k and everything above it.
- Using it when p changes. Drawing without replacement from a small group needs another model.
- Forgetting independence. Trials must not affect each other.
- Rounding probabilities too early. Small values add up.
Frequently Asked Questions
Use P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), or enter n, p and k here.
Add the probabilities for k through n, or compute 1 − P(X ≤ k − 1).
np. For 20 trials with p = 0.3, the mean is 6.
√(np(1 − p)).
A common rule is when np and n(1 − p) are both at least about 10.
Yes. Then the outcome is certain: always 0 successes or always n.
The number of ways to choose k successes from n trials. Our Combination Calculator explains it.
Only when p = 0.5. Otherwise it leans to one side.
Binomial has a fixed number of trials. Poisson counts events in a time or space with no fixed limit.
Very small probabilities are shown like 1.234e-8 so they are not rounded to zero.
Up to 10,000.
It is 1 minus the chance of zero successes: 1 − (1 − p)^n. For 10 tries with p = 0.1, that is about 0.65, or 65%.
A binomial test uses these probabilities to judge whether an observed count is unusual.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Binomial distribution | Formula, mean, variance and properties. | https://en.wikipedia.org/wiki/Binomial_distribution |
| NIST/SEMATECH e-Handbook, Binomial Distribution | Definition and uses. | https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm |