A-level · Probability

Conditional probabilityExam-style questions

The probability of A given B, P(A|B) = P(A ∩ B) / P(B), as restricting the sample space to B — reading it off a Venn diagram, the multiplication formula P(A ∩ B) = P(B)P(A|B), and testing independence with P(A|B) = P(A).

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28 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
For events $A$ and $B$ , $P(A) = 0.6$ , $P(B) = 0.5$ and $P(A \cup B) = 0.8$ .
(a) Find $P(A \cap B)$ .
(b) Find $P(A \mid B)$ .
(c) Determine whether $A$ and $B$ are independent.
Level 4 · 4 marksTry it
2
A weather station records the total rainfall and the mean wind speed every day. A researcher selects $200$ days at random from the station's records and notes, for each day, whether any rainfall was recorded ( $R$ ) and whether the mean wind speed exceeded $10$ knots ( $W$ ). Her results are summarised in the table.
$W$ not $W$ Total
$R$ $48$ $42$ $90$
not $R$ $32$ $78$ $110$
Total $80$ $120$ $200$
One of the $200$ days is selected at random.
(a) Determine, with a reason, whether the events $R$ and $W$ are independent. (2 marks)
(b) Find the probability that the mean wind speed did not exceed $10$ knots on the selected day, given that no rainfall was recorded. (2 marks)

Two different days are now selected at random from the $200$ days.
(c) Find the probability that on exactly one of the two days rainfall was recorded and the mean wind speed exceeded $10$ knots. (3 marks)
(d) Given that rainfall was recorded on both of the selected days, find the probability that the mean wind speed exceeded $10$ knots on exactly one of them. (3 marks)
Level 4 · 10 marksTry it
3
Members of a leisure centre can attend three activities: aerobics ( $A$ ), badminton ( $B$ ) and circuit training ( $C$ ). The Venn diagram shows the probabilities that a randomly chosen member attends each combination of the activities, where $x$ and $y$ are probabilities.The events $A$ and $B$ are independent.
(a) Find the values of $x$ and $y$ . (4 marks)
(b) Find the probability that a randomly chosen member attends badminton or circuit training, given that they do not attend aerobics. (2 marks)
(c) Determine, with a reason, whether the events $B$ and $C$ are independent. (2 marks)
(d) Given that a randomly chosen member attends exactly two of the three activities, find the probability that aerobics is one of them. (2 marks)
Level 4 · 10 marksTry it
4
A factory uses three machines, $A$ , $B$ and $C$ , to produce components. Machine $A$ produces $25\%$ of the components, machine $B$ produces $35\%$ and machine $C$ produces the rest.

Of the components produced by machine $A$ , $4\%$ are defective. The corresponding figures for machines $B$ and $C$ are $2\%$ and $5\%$ .

A component is chosen at random from the factory's output.
(a) Find the probability that the component is defective. (3 marks)
(b) Given that the component is defective, find the probability that it was produced by machine $C$ . Give your answer to $3$ significant figures. (3 marks)
(c) Two components are chosen at random from the factory's output. Find the probability that exactly one of them is defective. Give your answer to $3$ significant figures. (3 marks)
(d) State an assumption you have made in answering part (c). (1 mark)
Level 4 · 10 marksTry it
5
A leisure centre has a pool, a sauna and a programme of fitness classes. A survey of its members finds that $40\%$ of members use the pool and that, of those who use the pool, $35\%$ also use the sauna. $33\%$ of members use neither the pool nor the sauna.

A member is chosen at random. The events $A$ and $B$ are defined as follows.

$A$ : the member uses the pool $B$ : the member uses the sauna
(a) Find $\mathrm{P}(B)$ . (3 marks)
(b) Determine, giving a reason, whether the events $A$ and $B$ are statistically independent. (2 marks)
(c) Given that the member does not use the sauna, find the probability that they use the pool. Give your answer to $3$ significant figures. (2 marks)

The survey also finds that $30\%$ of members attend fitness classes, that attending fitness classes is independent of using the pool, and that no member who attends fitness classes uses the sauna.
(d) Find the probability that a randomly chosen member uses at least one of the three facilities. (3 marks)
Level 4 · 10 marksTry it

	$W$	not $W$	Total
$R$	$48$	$42$	$90$
not $R$	$32$	$78$	$110$
Total	$80$	$120$	$200$

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