Probability Overview – A Level Maths Year 1

Probability is the mathematics of uncertainty. Statistics looks at data that have already happened; probability builds models of what could happen. The two meet in hypothesis testing, where a probability model is used to judge whether real data are surprising.

Year 1 has two halves. First, the language and rules for combining events — Venn diagrams, tree diagrams, mutually exclusive and independent events. Second, random variables: a number decided by chance, described by a probability distribution. The most important one at AS is the binomial distribution, which counts successes in a fixed number of independent trials.

What is new compared with GCSE

Set notation $A \cap B$, $A \cup B$, $A'$ · testing for independence
Discrete random variables and their distributions
The binomial distribution $\mathrm{B}(n, p)$ and its modelling assumptions

DfE content areas M · Probability and N · Statistical distributions. Your calculator will be expected to give binomial probabilities, both $\mathrm{P}(X = r)$ and the cumulative $\mathrm{P}(X \leqslant r)$. Learn exactly where these functions are on your model — and always write down the distribution and the probability you are finding, so method marks are available.

Word	Meaning	Example
Event	A set of possible outcomes.	"Roll an even number"
$A'$	Complement: $A$ does not happen.	"Roll an odd number"
$A \cap B$	Both $A$ and $B$.	Even and greater than $3$
$A \cup B$	$A$ or $B$ or both.	Even or greater than $3$
Mutually exclusive	Cannot happen together: $\mathrm{P}(A \cap B) = 0$.	Rolling a $2$ and rolling a $5$
Independent	One does not affect the other: $\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B)$.	Two separate coin tosses
Random variable	A numerical quantity determined by chance.	$X$ = number of heads
$X \sim \mathrm{B}(n, p)$	$X$ has a binomial distribution with $n$ trials and success probability $p$.	$\mathrm{B}(10, 0.5)$: heads in $10$ tosses

1 — The addition rule

$\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)$

2 — Testing independence

Compare $\mathrm{P}(A \cap B)$ with $\mathrm{P}(A) \times \mathrm{P}(B)$

3 — Tree diagrams

Multiply along branches; add the paths you want

4 — A probability distribution

$\sum \mathrm{P}(X = x) = 1$ — use this to find unknown constants

5 — The binomial distribution

$\mathrm{P}(X = r) = \dbinom{n}{r} p^r (1-p)^{n-r}$; $\mathrm{P}(X \geqslant r) = 1 - \mathrm{P}(X \leqslant r - 1)$

Worked Example — Several techniques in one question

At a café, each customer independently pays by card with probability $0.3$. Of the next $8$ customers, let $X$ be the number who pay by card. Find (a) $\mathrm{P}(X = 2)$, (b) $\mathrm{P}(X \geqslant 3)$, (c) the probability that the first two customers both pay by card.

①Fixed $n = 8$, two outcomes, independent, constant $p$: $X \sim \mathrm{B}(8, 0.3)$.

②(a) $\mathrm{P}(X = 2) = \dbinom82 (0.3)^2 (0.7)^6 = 28 \times 0.09 \times 0.117649 = 0.2965$.

③(b) $\mathrm{P}(X \geqslant 3) = 1 - \mathrm{P}(X \leqslant 2) = 1 - 0.5518 = 0.4482$.

④(c) Independent events: $0.3 \times 0.3 = 0.09$.

Everything in DfE content areas M and N that is examined at AS, and the page that covers it.

MProbability and Venn DiagramsMutually exclusive and independent events, the addition rule, Venn diagrams and tree diagrams.
NThe Binomial DistributionDiscrete random variables, the discrete uniform distribution, $\mathrm{B}(n, p)$, cumulative probabilities and modelling assumptions.

The binomial distribution is the engine of Hypothesis Testing at AS, and Year 2 adds conditional probability and the normal distribution.

Confusing mutually exclusive with independent. Mutually exclusive means no overlap; independent means $\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B)$. Events with non-zero probabilities cannot be both.

Forgetting to subtract the overlap. $\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B)$ only when $A$ and $B$ are mutually exclusive.

Missing the $\binom{n}{r}$. $(0.3)^2(0.7)^6$ is the probability of one particular order. There are $\binom82 = 28$ orders.

Off-by-one with inequalities. $\mathrm{P}(X \geqslant 3) = 1 - \mathrm{P}(X \leqslant 2)$, not $1 - \mathrm{P}(X \leqslant 3)$.

Generic assumptions. "The trials are independent" earns little; "whether one seed germinates does not affect whether another does" earns the mark.

Complement

$\mathrm{P}(A') = 1 - \mathrm{P}(A)$

Addition rule

Add, then subtract the overlap.

Mutually exclusive

$\mathrm{P}(A \cap B) = 0$

Independent

$\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B)$

Distribution

Probabilities sum to $1$.

Uniform

Each of $n$ values has probability $\tfrac1n$.

Binomial

Fixed $n$, two outcomes, independent, constant $p$.

$\mathrm{P}(X \geqslant r)$

$1 - \mathrm{P}(X \leqslant r - 1)$

At least one

$1 - \mathrm{P}(\text{none})$

Ten questions drawing on the whole Probability topic. Work each one through before opening the solution.

Question 1

$A$ and $B$ are mutually exclusive with $\mathrm{P}(A) = 0.3$ and $\mathrm{P}(B) = 0.5$. Find $\mathrm{P}(A \cup B)$.

▶ Show solution

$0.3 + 0.5 = 0.8$

Question 2

$A$ and $B$ are independent with $\mathrm{P}(A) = 0.4$ and $\mathrm{P}(B) = 0.5$. Find $\mathrm{P}(A \cap B)$ and $\mathrm{P}(A \cup B)$.

▶ Show solution

$\mathrm{P}(A \cap B) = 0.2$; $\mathrm{P}(A \cup B) = 0.4 + 0.5 - 0.2 = 0.7$.

Question 3

Of $30$ people, $12$ own a cat, $15$ own a dog and $5$ own both. Find the probability that a person chosen at random owns (a) a cat but not a dog, (b) neither.

▶ Show solution

Cat only $7$, dog only $10$, both $5$, so neither $= 30 - 22 = 8$.

(a) $\dfrac{7}{30}$ (b) $\dfrac{8}{30} = \dfrac{4}{15}$

Question 4

A bag has $3$ red and $2$ blue counters. A counter is taken, replaced, and a second is taken. Find the probability of one of each colour.

▶ Show solution

RB or BR: $2 \times \tfrac35 \times \tfrac25 = \dfrac{12}{25}$.

Question 5

$\mathrm{P}(X = x) = \dfrac{x}{c}$ for $x = 1, 2, 3, 4$. Find $c$ and $\mathrm{P}(X \geqslant 3)$.

▶ Show solution

$\dfrac{1 + 2 + 3 + 4}{c} = 1$, so $c = 10$.

$\mathrm{P}(X \geqslant 3) = \dfrac{3 + 4}{10} = 0.7$.

Question 6

A fair coin is tossed $6$ times. Find the probability of exactly $3$ heads.

▶ Show solution

$X \sim \mathrm{B}(6, 0.5)$: $\dbinom63 (0.5)^6 = \dfrac{20}{64} = 0.3125$.

Question 7

$X \sim \mathrm{B}(10, 0.3)$. Find $\mathrm{P}(X > 2)$.

▶ Show solution

$1 - \mathrm{P}(X \leqslant 2) = 1 - 0.3828 = 0.6172$

Question 8

$10\%$ of a population is left-handed. In a random sample of $25$, find the probability that (a) at most $2$, (b) more than $3$ are left-handed.

▶ Show solution

$L \sim \mathrm{B}(25, 0.1)$.

(a) $\mathrm{P}(L \leqslant 2) = 0.5371$

(b) $\mathrm{P}(L > 3) = 1 - \mathrm{P}(L \leqslant 3) = 1 - 0.7636 = 0.2364$

Question 9

Explain whether $\mathrm{B}(n, p)$ is suitable for: (a) the number of sixes in $20$ rolls of a fair die; (b) the number of days it rains in a week in March.

▶ Show solution

(a) Yes: $n = 20$ fixed, six or not six, rolls independent, $p = \tfrac16$ constant. $\mathrm{B}\left(20, \tfrac16\right)$.

(b) Doubtful: weather on consecutive days is not independent (wet spells last several days), and the probability of rain may vary through the week.

Question 10

A machine produces items of which $5\%$ are defective, independently. Items are packed in boxes of $20$.

(a) Find the probability that a box contains no defective items. (b) Find the probability that a box contains at least $2$ defective items. (c) A shop receives $10$ boxes. Find the probability that none of them contains at least $2$ defective items. (d) Give one reason why the binomial model might be unrealistic.

▶ Show solution

$D \sim \mathrm{B}(20, 0.05)$.

(a) $\mathrm{P}(D = 0) = 0.95^{20} = 0.3585$

(b) $\mathrm{P}(D = 1) = 20(0.05)(0.95)^{19} = 0.3774$, so $\mathrm{P}(D \leqslant 1) = 0.7358$ and $\mathrm{P}(D \geqslant 2) = 0.2642$.

(c) Each box independently has probability $0.7358$ of fewer than $2$ defectives: $0.73584^{10} = 0.0465$.

(d) Faults may come in runs — for example if the machine goes out of adjustment, several consecutive items will be defective — so items are not independent and $p$ may not be constant.