The Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same two possible outcomes and the same probability of success. In GCSE Statistics it is used to calculate probabilities such as P(X = r), for example the number of heads in several coin tosses, using P(X=r) = nCr x p^r x (1-p)^(n-r).
Before you start
Make sure you're comfortable with these topics first:
Method
- Check the situation fits a binomial model: a fixed number of trials (n), each trial has only two outcomes (success/failure), the trials are independent, and the probability of success (p) stays constant.
- Identify n (number of trials), p (probability of success) and r (number of successes required).
- Work out the number of ways to arrange r successes among n trials using nCr (combinations).
- Multiply by p^r (probability of the successes) and (1-p)^(n-r) (probability of the failures).
- For 'at least' or 'at most' questions, list and add the probabilities of every outcome that satisfies the condition.
- Check the final probability lies between 0 and 1, and round sensibly, usually to 3 significant figures, unless an exact value is required.
Worked example
A biased coin lands on heads with probability 0.6. The coin is tossed 4 times. Find the probability of getting exactly 3 heads.
- Identify the binomial variables: n = 4, p = 0.6 (probability of a head), r = 3.
- Find the number of ways to get 3 heads out of 4 tosses: 4C3 = 4.
- Find the probability of 3 heads: p^3 = 0.6^3 = 0.216.
- Find the probability of 1 tail: (1-p)^(4-3) = 0.4^1 = 0.4.
- Multiply all three parts together: 4 x 0.216 x 0.4 = 0.3456.
- Final answer: P(exactly 3 heads) = 0.3456.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1A fair coin is tossed 3 times. Find the probability of getting exactly 2 heads.Show answer
Answer: 0.375 (3C2 x 0.5^2 x 0.5^1 = 3 x 0.25 x 0.5).
Q2A biased dice lands on a six with probability 0.2. The dice is rolled 5 times. Find the probability of getting no sixes.Show answer
Answer: 0.32768 (0.8^5).
Q3A quality inspector finds that 10% of bulbs from a factory are faulty. In a random sample of 6 bulbs, find the probability that exactly 1 is faulty.Show answer
Answer: 0.354 (3 sf) (6C1 x 0.1 x 0.9^5 = 6 x 0.1 x 0.59049).
Q4In a binomial distribution, n = 4 and p = 0.5. Find P(X = 4).Show answer
Answer: 0.0625 (0.5^4).
Q5A basketball player has a free-throw success probability of 0.7. She takes 5 free throws. Find the probability that she makes at least 4.Show answer
Answer: 0.528 (3 sf) (P(X=4) + P(X=5) = 0.36015 + 0.16807).
Q6A binomial variable X has n = 8 and p = 0.25. Find P(X = 2), giving your answer to 3 significant figures.Show answer
Answer: 0.311 (3 sf) (8C2 x 0.25^2 x 0.75^6 = 28 x 0.0625 x 0.177979...).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A biased coin lands on heads with probability 0.3. The coin is tossed 3 times. Find the probability of getting exactly 2 heads.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 3 available
A machine produces bolts, 5% of which are faulty. A random sample of 10 bolts is taken. Find the probability that exactly 2 bolts are faulty, giving your answer to 3 significant figures.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
A call centre finds that 80% of calls are resolved on the first attempt. A random sample of 6 calls is selected. (a) Find the probability that exactly 5 calls are resolved on the first attempt. (b) Find the probability that all 6 calls are resolved on the first attempt. (c) Hence find the probability that at least 5 calls are resolved on the first attempt.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 5 available
See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the the binomial distribution worksheet pack - 15 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 32 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
Next topics
Not quite what you needed?
Tell us what is missing on the binomial distribution, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.