A-level · Probability

Venn diagramsExam-style questions

Using a Venn diagram to organise events: the intersection A∩B ("and"), the union A∪B ("or"), and the complement A′ ("not"); filling in the regions from the numbers in a problem (starting with the overlap); reading probabilities off the diagram; and the addition formula P(A∪B) = P(A) + P(B) − P(A∩B).

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18 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
A museum asked 150 visitors which of three galleries they had visited: the fossil gallery ( $F$ $F$ ), the map room ( $M$ $M$ ) and the sculpture hall ( $S$ $S$ ).
- 12 visitors had visited all three galleries
- 21 visitors had visited the fossil gallery and the map room
- 24 visitors had visited the fossil gallery and the sculpture hall
- 7 visitors had visited the map room and the sculpture hall but not the fossil gallery
- 56 visitors had visited the fossil gallery
- 18 visitors had visited the map room only
- 38 visitors had visited none of the three galleries
The Venn diagram below, where $r$ $r$ , $s$ $s$ , $t$ $t$ , $u$ $u$ , $v$ $v$ , $w$ $w$ , $y$ $y$ and $z$ $z$ are the numbers of visitors in each region, represents this information.(a) Using this information, find the values of $r$ $r$ , $s$ $s$ , $t$ $t$ , $u$ $u$ , $v$ $v$ , $w$ $w$ , $y$ $y$ and $z$ $z$

(3 marks)

One of these visitors is selected at random.
(b) Find the probability that the visitor

(i) had visited all three galleries,

(ii) had not visited the fossil gallery.

(3 marks)
(c) Given that the visitor had visited the map room, find the probability that the visitor had also visited the sculpture hall.

(2 marks)
Level 4 · 8 marksTry it
2
Three events $A$ , $B$ and $C$ are represented in the Venn diagram below, which gives the probability of each region; $q$ , $r$ and $s$ are unknown probabilities.(a) Write down

(i) $\mathrm{P}(C)$

(ii) $\mathrm{P}(A \cap C)$

(2 marks)

Given that $\mathrm{P}(B \mid A) = 0.4$ and $\mathrm{P}(A \mid B) = 0.4$
(b) find the value of $q$ and the value of $r$

(5 marks)
(c) Find the value of $s$

(1 mark)
(d) Determine whether or not the events $B$ and $C$ are statistically independent. You must show your working clearly.

(2 marks)
Level 4 · 10 marksTry it
3
A car dealer sold $160$ $160$ new cars last year. Each buyer could choose to have any of three optional extras fitted:
- satellite navigation ( $N$ )
- heated seats ( $H$ )
- a towbar ( $T$ )
The number of cars fitted with each extra, or with each combination of extras, is given in the table.
Extras fitted Number of cars
$N$ $87$
$H$ $60$
$T$ $31$
$N$ and $H$ $30$
$N$ and $T$ $11$
$H$ and $T$ $15$
$N$ , $H$ and $T$ $6$
(a) Represent this information by fully completing the Venn diagram below.(3 marks)
(b) A car is selected at random from the $160$ $160$ cars.
(b) (i) Find the probability that satellite navigation was the only extra fitted to the car.

(ii) Find the probability that the car had two or more of the extras fitted.

(iii) Given that the car did not have a towbar, find the probability that it had satellite navigation or heated seats or both.

(5 marks)
(c) Determine whether the events $N$ $N$ and $T$ $T$ are independent.

Fully justify your answer.

(2 marks)
Level 4 · 10 marksTry it
4
A library looked at the borrowing records of $120$ $120$ of its members for the last month, noting whether each of them had borrowed:
- printed books ( $B$ )
- e-books ( $E$ )
- audiobooks ( $A$ )
The table gives the number of these members who borrowed each type of item or combination of types.
Borrowed Number of members
$B$ $60$
$E$ $48$
$A$ $30$
$B$ and $E$ $24$
$B$ and $A$ $15$
$E$ and $A$ $12$
$B$ , $E$ and $A$ $5$
(a) Represent this information by fully completing the Venn diagram below.(3 marks)
(b) A member is picked at random from the $120$ $120$ members.
(b) (i) Find the probability that e-books were the only type of item the member borrowed.

(ii) Find the probability that the member borrowed exactly one of the three types.

(iii) Find the probability that the member borrowed printed books, given that the member borrowed e-books or audiobooks or both.

(5 marks)
(c) Determine whether borrowing printed books and borrowing audiobooks are independent.

Fully justify your answer.

(2 marks)
Level 4 · 10 marksTry it
5
Three events, $A$ , $B$ and $C$ , are shown in the Venn diagram below. The number in each region is its probability; the probabilities $p$ and $q$ are not known.(a) Find $\mathrm{P}(A)$

(1 mark)

The events $B$ and $C$ are independent.
(b) Find the value of $p$ and the value of $q$

(3 marks)
(c) Find $\mathrm{P}(A \mid B')$

(2 marks)
Level 4 · 6 marksTry it

Extras fitted	Number of cars
$N$	$87$
$H$	$60$
$T$	$31$
$N$ and $H$	$30$
$N$ and $T$	$11$
$H$ and $T$	$15$
$N$ , $H$ and $T$	$6$

Borrowed	Number of members
$B$	$60$
$E$	$48$
$A$	$30$
$B$ and $E$	$24$
$B$ and $A$	$15$
$E$ and $A$	$12$
$B$ , $E$ and $A$	$5$

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