The binomial distribution

The idea

$X \sim B(n, p)$ counts successes in $n$ trials when the number of trials is fixed, trials are independent, there are two outcomes and $p$ stays the same. $P(X = r) = \binom{n}{r}p^r(1 - p)^{n - r}$ .

Check the four conditions in context.
Identify $n$ and $p$ .
Use the formula, or your calculator's binomial PD function.

Watch out State the conditions in context, not just as a list.

See it worked

Worked example

$X \sim B(10, 0.3)$ . Find $P(X = 2)$ .

$P(X = 2) = \binom{10}{2}(0.3)^2(0.7)^8$ .
$= 45 \times 0.09 \times 0.057648$ .
$= 0.2335$ (4 d.p.).

Answer $0.2335$

Your turn

Question 1Calculator allowed3 marks

$X \sim B(8, 0.25)$ . Find $P(X = 3)$ to 4 decimal places.

Question 2Calculator allowed3 marks

$X \sim B(5, 0.2)$ . Find $P(X = 0)$ .

Question 3Non-calculator3 marks

A student guesses all $20$ answers on a multiple-choice test with $4$ options each. Explain why the number correct can be modelled by $B(20, 0.25)$ .

Write your answer, then show the working

There is a fixed number of trials: $20$ questions.
Each guess is right or wrong, with the same chance $\frac{1}{4}$ .
Guesses are independent of each other.

Answer Fixed number of questions, independent guesses, right or wrong, and $p = 0.25$ each time

In past papers

35 official questions in 27 papers since 2018. The latest:

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