GCSE Statistics · Higher · Chapter 11: Probability · Lesson 2

Venn diagrams and the addition law

About 55 minutesCalculator allowed44 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/stats-venn-diagrams-and-the-addition-law/

Date

Do Now

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A biased dice is rolled 100 times and shows a six 25 times. It is rolled 1200 more times. Estimate how many sixes it shows.
[2 marks]
Answer
A2
From chapter 6
Kemi has the lengths of some phone calls in a grouped table only. Kemi works out an estimate of the mean, $40.8$ , and says it is exactly the mean of these values. Is Kemi right? Give a reason.
[2 marks]
A3
From chapter 9
For a ski shop's sales, the trend line reads $345$ at Q4 of year 2 and rises by $2$ every quarter. The mean seasonal variation for Q3 is $- 34$ . Predict the sales for Q3 of year 3.
[2 marks]
Answer
A4
From chapter 10
Two sets of data on temperature and sales have PMCC $0.45$ (set A) and $- 0.35$ (set B). Jon says set A shows the stronger correlation. Is Jon right? Give a reason.
[2 marks]

Example, then your turn

Your teacher works through each example with you. Then try the one beside it.

Example 1

Spinner $A$ is numbered 1 to 4 and spinner $B$ is numbered 1 to 3. Both are spun and the scores are added. Work out the probability that the total is 5.

Hint List every pair of scores in a table: 4 rows and 3 columns.

B1
Two fair dice are rolled. Work out the probability that the difference between the two scores is 3 or more.
[2 marks]
Answer

Example 2

In a town, $P (dog) = 0.35$ , $P (cat) = 0.25$ and $P (dog and cat) = 0.08$ for a household. Work out $P (dog or cat)$ and $P (neither)$ .

Hint Households with both are counted twice if you just add.

B2
At a shop, the probability that a customer buys bread is $0.55$ , buys milk is $0.4$ and buys both is $0.22$ . Work out the probability that a customer buys neither bread nor milk.
[3 marks]
Answer

Example 3

A gym has 120 members. 58 use the pool ( $S$ ), 47 go to classes ( $C$ ) and 34 do neither. (a) Complete the Venn diagram. (b) Work out $P (C \cap S^{'})$ . (c) Describe in words the members in $S \cap C^{'}$ .

Hint Start with neither:

120 - 34 = 86

members are inside the circles.

B3
A vet sees 90 dogs. 52 are vaccinated ( $V$ ), 31 are microchipped ( $M$ ) and 23 are neither.
(a)
Complete the Venn diagram.
[2 marks]
(b)
One of the dogs is chosen at random. Work out $P (M \cap V^{'})$ .
[1 mark]
Answer
Total for B3: 3 marks

Example 4

The Venn diagram shows the probabilities of three events $A$ , $B$ and $C$ . (a) Which two events are mutually exclusive? Give a reason. (b) Are $A$ , $B$ and $C$ exhaustive? Give a reason. (c) Work out $P (A \cup C)$ .

Hint Mutually exclusive events have no overlap. Exhaustive events cover every outcome.

B4
The Venn diagram shows the probabilities of three events $X$ , $Y$ and $Z$ .
(a)
Which two events are mutually exclusive? Give a reason.
[2 marks]
(b)
Are $X$ , $Y$ and $Z$ exhaustive? Give a reason.
[1 mark]
(c)
Work out $P (X \cup Z)$ .
[2 marks]
Answer
Total for B4: 5 marks

Practice

C1
The Venn diagram shows 50 library members and whether they borrowed fiction ( $F$ ) or non-fiction ( $N$ ). One is chosen at random. Work out $P (F \cup N^{'})$ .
[2 marks]
Answer
C2
For two events, $P (A) = 0.45$ , $P (B) = 0.3$ and $P (A \cup B) = 0.75$ . Are $A$ and $B$ mutually exclusive? Are they exhaustive? Give a reason for each answer.
[2 marks]

C3
The Venn diagram shows the probabilities of two events $A$ and $B$ . Work out $P (B)$ .
[3 marks]
Answer
C4
Two fair four-sided dice, each numbered 1 to 4, are rolled and the scores are multiplied. Work out the probability that the result is a square number.
[2 marks]
Answer

Exam-style questions

D1
In a club, the probability that a member plays football is $0.42$ , plays tennis is $0.31$ and plays both is $0.12$ . Ola says, "The probability that a member plays football or tennis is $0.42 + 0.31 = 0.73$ ."
Is Ola correct? You must show your working.
[2 marks]

D2
Ivan asks 75 people on a train platform whether they listen to music ( $M$ ) and whether they read ( $R$ ) on the train. 49 listen to music, 31 read, 19 do both and the rest do neither.
(a)
Complete the Venn diagram.
[2 marks]
(b)
One of the 75 people is chosen at random. Work out $P (M^{'} \cap R)$ .
[2 marks]
Answer
(c)
Ivan says, "More than half of all train passengers listen to music." Comment on Ivan's statement.
[2 marks]
Total for D2: 6 marks
D3
The Venn diagram shows the 60 students at a music school who play the piano ( $P$ ), the guitar ( $G$ ) or the drums ( $D$ ).
(a)
Describe in words the students in the region that holds 4.
[1 mark]
(b)
A student is chosen at random. Work out the probability that the student plays at least two of the instruments.
[2 marks]
Answer
Total for D3: 3 marks

Extension

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
For two events, $P (A) = 2 P (B)$ and $P (A \cap B) = 0.1$ . The probability that neither $A$ nor $B$ happens is $0.2$ . Work out the probability that exactly one of $A$ and $B$ happens.
[3 marks]
Hint 1 · What to try
Write $P (A \cup B)$ in two ways.
Hint 2 · The first line
$P (A \cup B) = 1 - 0.2 = 0.8$ , and also $2 P (B) + P (B) - 0.1$ .
Hint 3 · The full method
$3 P (B) = 0.9$ , so $P (B) = 0.3$ . Exactly one is $P (A \cup B) - P (A \cap B) = 0.8 - 0.1 = 0.7$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can complete a Venn diagram, including the region outside the circles, and use it.
I can use the addition law and decide whether events are mutually exclusive or exhaustive.
I can read and write set notation and describe a region in words.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers