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).

Start practising Read the lesson

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 theme park has three rides: a roller coaster ( $C$ ), a log flume ( $L$ ) and a big wheel ( $W$ ). The Venn diagram below shows the probabilities that a visitor chosen at random goes on each of these rides during a visit, where $p$ and $q$ are unknown probabilities.(a) Find the probability that a randomly selected visitor goes on the big wheel but does not go on the roller coaster.

(1 mark)

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

(4 marks)
(c) Hence find the value of $q$

(2 marks)
(d) Find

(i) $\mathrm{P}(C' \cap (L \cup W))$

(ii) $\mathrm{P}(W \mid (C \cup L))$

(3 marks)

The same company owns a much larger theme park with the same three rides. On one day, 96 visitors to the larger park went on both the log flume and the big wheel.
(e) Assuming that the probabilities in the Venn diagram also apply to visitors to the larger park, estimate the number of visitors to the larger park that day who went on none of the three rides.

(3 marks)
Level 5 · 13 marksTry it
2
A fruit farm has 1200 trees, growing in two orchards, North and South. Every tree on the farm is an apple tree, a pear tree or a plum tree. The table shows the number of trees of each kind in each orchard.
North South
Apple 420 300
Pear 100 120
Plum 110 150
One of the trees on the farm is selected at random.

Find the probability that the selected tree
(a) is a pear tree,

(1 mark)
(b) is not an apple tree and is in the South orchard.

(1 mark)

A disease has affected some of the trees. The proportion of trees with the disease depends on the kind of tree but not on the orchard.
- In each orchard, 15% of the apple trees have the disease
- In each orchard, 30% of the pear trees have the disease
- In each orchard, 60% of the plum trees have the disease
The events $A$ $A$ , $D$ $D$ and $N$ $N$ are defined as follows
- $A$ is the event that the tree is an apple tree
- $D$ is the event that the tree has the disease
- $N$ is the event that the tree is in the North orchard
The Venn diagram shows the number of trees in four of the regions.(c) Complete the Venn diagram by finding the number of trees in each of the regions $A \cap D' \cap N$ $A \cap D^{'} \cap N$ , $A \cap D \cap N'$ $A \cap D \cap N^{'}$ , $A' \cap D \cap N'$ $A^{'} \cap D \cap N^{'}$ and $A' \cap D' \cap N$ $A^{'} \cap D^{'} \cap N$ . Give your four answers in this order.

(4 marks)
(d) Find $\mathrm{P}(A \cap N')$ $P (A \cap N^{'})$

(1 mark)
(e) Find $\mathrm{P}([A \cup D]')$ $P ([A \cup D]^{'})$

(1 mark)
(f) Find $\mathrm{P}(A \mid D)$ $P (A ∣ D)$

(2 marks)
Level 5 · 10 marksTry it
3
Four events, $A$ , $B$ , $C$ and $D$ , are drawn in the Venn diagram below, and each region is marked with its probability. The letters $p$ , $q$ and $r$ stand for unknown probabilities.(a) Using the diagram, state a pair of events that are mutually exclusive.

(1 mark)

Given that the events $B$ and $C$ are independent,
(b) find the value of $p$

(2 marks)
(c) Find the greatest possible value of $\mathrm{P}(D \mid C')$

(3 marks)

In addition, $\mathrm{P}(C \mid D') = 0.44$
(d) Find the value of $q$ and the value of $r$

(3 marks)
(e) Find $\mathrm{P}([C \cup D]' \cap B)$

(1 mark)
(f) Use set notation to describe the region with probability $p$

(1 mark)
Level 5 · 11 marksTry it
4
A hotel has $40$ $40$ rooms.
- $5$ of the rooms have a balcony and a sea view.
- $13$ of the rooms have neither a balcony nor a sea view.
- There are $6$ more rooms with a balcony than rooms with a sea view.
(a) Draw a Venn diagram to show this information, and determine the probability that a room chosen at random has a sea view.

(4 marks)

Two of the hotel's rooms are chosen at random.
(b) Exactly one of the two rooms chosen has a balcony. Determine the probability that at least one of them has a sea view.

(6 marks)
Level 5 · 10 marksTry it

	North	South
Apple	420	300
Pear	100	120
Plum	110	150