Key takeaways
- P(X = k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ gives the chance of exactly k successes in n trials.
- Write p as a decimal between 0 and 1, not a percentage.
- The trials must be independent and p must stay the same every time.
- For at least k successes, add up the answers for k, k + 1, … up to n.
In this guide
What is binomial probability?
Binomial probability is the chance of getting an exact number of successes when you repeat the same yes-or-no trial several times. Coin flips, guessing on multiple-choice questions and free throws in basketball all fit.
A problem is binomial when it has four features: a fixed number of trials, two outcomes per trial, the same chance of success every time, and trials that do not affect each other.
Why does binomial probability matter?
Our intuition about repeated chances is often wrong. The binomial formula gives exact answers to questions like 'how likely is it that 3 of 20 items are faulty?' or 'what are my odds of passing by guessing?'
In 10 fair coin flips, exactly 5 heads is the single most likely result, yet it happens only about 24.6% of the time. The other 75% of the time you get some other number of heads.
The formula
| Symbol | What it means | Example |
|---|---|---|
| P(X = k) | Chance of exactly k successes | 0.2637 |
| n | Number of trials | 5 questions |
| k | Successes you want (0 to n) | 2 correct |
| p | Chance of success in one trial (0 to 1) | 0.25 |
| 1 − p | Chance of failure in one trial | 0.75 |
| nCk | Ways to place k successes among n trials | 5C2 = 10 |
The pᵏ part is the chance of k successes, the (1 − p)ⁿ⁻ᵏ part is the chance of the n − k failures, and nCk counts all the different orders they could happen in. Get nCk from the combinations calculator.
How to calculate binomial probability step by step
- 1Identify n, k and p. Check the four binomial features first.
- 2Find nCk. For 5 trials and 2 successes,
5C2 = 10. - 3Raise p to the power k.
0.25² = 0.0625. - 4Raise (1 − p) to the power n − k.
0.75³ = 0.421875. - 5Multiply all three.
10 × 0.0625 × 0.421875 ≈ 0.2637.
Try your own n, k and p in the calculator below. It shows the answer as a decimal and a percentage.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Confusing exactly with at least. This formula is for exactly k. For at least 2, add the answers for k = 2, 3, … up to n.
- Forgetting nCk. Without it you only get the chance of one particular order of results.
- Using it when p changes. Drawing cards without putting them back changes p each time, so it is not binomial.
- Swapping k and n − k. The power on p is the number of successes, and the power on (1 − p) is the number of failures.
Tips and tricks
- Quick check: the probabilities for k = 0, 1, … n always add up to exactly 1.
- Expected number of successes is
n × p. For 5 guesses at 0.25, you expect about 1.25 correct. - Just one trial? The basic probability calculator is all you need. For big nCk by hand, the factorial calculator helps.
Where you'll use it in real life
- Quality control: the chance of finding 2 faulty items in a sample of 20.
- Medicine: how many patients out of 10 might respond to a treatment.
- Sport: the chance a player makes exactly 7 of 10 free throws.
- Exams: the odds of passing a multiple-choice test by pure guessing.
Quick summary
Binomial probability gives the chance of exactly k successes in n independent trials with the same success chance p: nCk × pᵏ × (1 − p)ⁿ⁻ᵏ. Always enter p as a decimal. Use the calculator above to get the answer with every step shown.
Worked example
A quiz has 5 multiple-choice questions, each with 4 options. You guess every answer at random, so each guess has a 0.25 chance of being right. What is the probability that you get exactly 2 correct?
- 01P(X = k) = nCk × p^k × (1-p)^(n-k)
- 025C2 = 10
- 03P = 10 × 0.25^2 × 0.75^3
- 04P = 10 × 0.0625 × 0.4219
- 05P = 0.2637 (26.37%)
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the probability of exactly 5 heads in 10 coin flips?
10C5 × 0.5⁵ × 0.5⁵ = 252 ÷ 1024 ≈ 0.246, or about 24.6%.
How do I find the probability of at least k successes?
Add the binomial probabilities for k, k + 1, … up to n. Or find 1 minus the probability of fewer than k successes, which is often quicker.
What makes a problem binomial?
A fixed number of independent trials, two outcomes per trial (success or failure), and the same probability of success on every trial.
What is the chance of getting none right when guessing 5 questions with 4 options?
P(0) = 0.75⁵ ≈ 0.237, or about 23.7%. So there is about a 76.3% chance of getting at least one right.
