State the mean of X B(50, 0.2). [1 mark]

For X B(6, 0.5), find P(X = 6). [2 marks]

Northern IrelandMathsSyllabus dot point

How do we model the probabilities of a discrete random variable, and when does the binomial distribution apply?

Discrete random variables and their probability distributions, the binomial distribution and its conditions, calculating binomial probabilities, and the mean of a binomial distribution.

A CCEA A-Level Mathematics answer on discrete random variables and their probability distributions, the conditions for the binomial distribution, calculating binomial probabilities with the formula and cumulative tables, and the mean of a binomial distribution.

Generated by Claude Opus 4.813 min answerUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section

What this dot point is asking
The answer
Examples in context
Try this

What this dot point is asking

CCEA wants you to understand discrete random variables and their probability distributions, recognise the conditions for the binomial distribution, calculate binomial probabilities (with the formula and cumulative tables), and find the mean of a binomial distribution. This is the first formal probability model in the course and is extended to the normal distribution at A2.

The answer

Discrete random variables

The binomial distribution and its conditions

Calculating binomial probabilities

The mean of a binomial distribution

The mean (expected number of successes) of $X \sim B(n, p)$ is

E(X) = np.

So out of

20

trials each with success probability

0.3

, you expect

20 \times 0.3 = 6

successes on average.

Worked example: a cumulative binomial probability

Find a "more than" binomial probability

In a factory, $15\%$ of items are faulty. A sample of $12$ items is taken. Find the probability that more than $2$ are faulty.

step State the distribution

Let $X$ be the number of faulty items. With a fixed $n = 12$ , independent items and constant $p = 0.15$ , $X \sim B(12, 0.15)$ .

step Rewrite "more than 2" using the complement

P(X > 2) = 1 - P(X \le 2) = 1 - \big[P(X = 0) + P(X = 1) + P(X = 2)\big].

step Compute each term

$P(X = 0) = (0.85)^{12} = 0.1422.$

$P(X = 1) = \binom{12}{1}(0.15)(0.85)^{11} = 12 \times 0.15 \times 0.1673 = 0.3012.$

$P(X = 2) = \binom{12}{2}(0.15)^2(0.85)^{10} = 66 \times 0.0225 \times 0.1969 = 0.2924.$

So $P(X \le 2) = 0.1422 + 0.3012 + 0.2924 = 0.7358$ , and $P(X > 2) = 1 - 0.7358 = 0.264$ (to three significant figures).

Examples in context

Example 1. Multiple-choice guessing. A student guessing each of $20$ four-option questions has $X \sim B(20, 0.25)$ , expecting $20 \times 0.25 = 5$ correct. The binomial models any fixed run of independent yes-or-no trials with the same chance.

Example 2. Quality assurance. A supplier claiming a low defect rate can be checked by modelling defects in a sample as binomial; an unusually high count casts doubt on the claim. This idea becomes a formal hypothesis test in the A2 statistics content.

Try this

Q1. A random variable has $P(X = 1) = 0.2$ , $P(X = 2) = 0.5$ and $P(X = 3) = p$ . Find $p$ . [1 mark]

Cue. The total is $1$ , so $p = 1 - 0.7 = 0.3$ .

Q2. State the mean of $X \sim B(50, 0.2)$ . [1 mark]

Cue. $np = 50 \times 0.2 = 10$ .

Q3. For $X \sim B(6, 0.5)$ , find $P(X = 6)$ . [2 marks]

Cue. $(0.5)^6 = \frac{1}{64} \approx 0.0156$ .

Exam-style practice questions

Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

CCEA 20206 marksA biased coin lands heads with probability

0.4

. It is tossed

10

times. Let

X

be the number of heads. Find

P(X = 3)

and the mean of

X

Show worked answer →

Here $X \sim B(10, 0.4)$ , so $P(X = r) = \binom{10}{r}(0.4)^r(0.6)^{10 - r}$ .

For $r = 3$ :

$P(X = 3) = \binom{10}{3}(0.4)^3(0.6)^7 = 120 \times 0.064 \times 0.02799 = 0.215$ (to three significant figures).

The mean of a binomial is $np = 10 \times 0.4 = 4$ .

Markers reward stating $X \sim B(10, 0.4)$ , the binomial formula with the correct $r$ , the numerical probability, and the mean $np$ .

CCEA 20195 marksThe random variable

X

has the probability distribution

P(X = x) = kx

for

x = 1, 2, 3, 4

. Find the value of

k

and

P(X \ge 3)

Show worked answer →

The probabilities must sum to $1$ :

$k(1) + k(2) + k(3) + k(4) = 10k = 1,$ so $k = \frac{1}{10} = 0.1$ .

Then $P(X \ge 3) = P(X = 3) + P(X = 4) = 0.1(3) + 0.1(4) = 0.3 + 0.4 = 0.7$ .

Markers reward setting the total probability to $1$ , solving for $k$ , and adding the correct probabilities for $X \ge 3$ .

Related dot points

Sources & how we know this

CCEA GCE Mathematics specification — CCEA (2018)