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
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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.
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.
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.
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