The binomial distribution
When the same trial is repeated a fixed number of times and each one either succeeds or fails, the number of successes follows a binomial distribution. Spotting when it applies is half the marks.
Part 1 of 3: Learn it
In short
- X ~ B(n, p): n trials, each with probability p of success.
- P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ.
- For "at least" questions, use 1 − P(X ≤ r − 1).
Where this is in your specification
Spec points: DfE N1 (AQA 7357 N1, Edexcel 9MA0 Statistics topic 4, OCR A H240 2.04)
| Board | Topic: The binomial distribution |
|---|---|
| DfE content | N1 |
| AQA 7357 | N1 |
| Edexcel 9MA0 | Statistics topic 4 |
| OCR H240 | 2.04 |
When the model fits
- There is a fixed number of trials, n.
- Each trial has two outcomes: success or failure.
- The trials are independent of each other.
- The probability of success, p, is the same every time.
Single probabilities
The nCr counts the different orders in which r successes can happen among n trials. Calculators have a binomial probability function that does the whole thing.
Cumulative probabilities
| You want | Use |
|---|---|
| P(X ≤ r) | the cumulative function directly |
| P(X < r) | P(X ≤ r − 1) |
| P(X ≥ r) | 1 − P(X ≤ r − 1) |
| P(X > r) | 1 − P(X ≤ r) |
| P(a ≤ X ≤ b) | P(X ≤ b) − P(X ≤ a − 1) |
Because X only takes whole number values, the difference between < and ≤ matters. Writing the values out on a number line helps.
X ~ B(5, 0.2). Find P(X = 0).
Show the answer
0.8⁵ = 0.32768.
Part 2 of 3: See it worked
Worked examples
Example 1
X ~ B(10, 0.3). Find P(X = 3).
- 10C3 = 120
- 0.3³ = 0.027 and 0.7⁷ = 0.0823543
- 120 × 0.027 × 0.0823543 = 0.26683
Answer: 0.267 (3 s.f.).
Example 2
X ~ B(10, 0.3). Find P(X ≥ 4).
- P(X ≥ 4) = 1 − P(X ≤ 3)
- P(X ≤ 3) = 0.6496 (calculator)
- 1 − 0.6496 = 0.3504
Answer: 0.350 (3 s.f.).
Common mistakes
- Using 1 − P(X ≤ 4) for P(X ≥ 4). It should be 1 − P(X ≤ 3).
- Forgetting the nCr part when working out a single probability by hand.
- Using a binomial model when the trials are not independent, such as picking without replacement from a small group.
- Stating the conditions without linking them to the situation in the question.
Write P(X < 6) in terms of a cumulative probability.
Show the answer
P(X ≤ 5).
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
X ~ B(5, 0.2). Find P(X = 0).
0.8⁵ = 0.32768.
Write P(X < 6) in terms of a cumulative probability.
P(X ≤ 5).
Give two conditions needed for a binomial model.
Any two of: a fixed number of trials, two outcomes per trial, independent trials, a constant probability of success.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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