KS3 Maths · Year 7 · Chapter 3: Probability · Lesson 1

Mutually exclusive and exhaustive outcomes

Year 7About 60 minutesNo calculator46 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $1 - \frac{5}{12}$ .
[1 mark]
Answer
A2
Needed today
Work out $1 - (0.45 + 0.2)$ .
[1 mark]
Answer
A3
From chapter 1
Find the HCF of $18$ and $42$ .
[1 mark]
Answer
A4
Last lesson
An angle of $7 6^{\circ}$ is bisected. How big is each half?
[1 mark]
Answer

Example, then your turn

Year 7

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

Example 1

A fair dice is rolled. Event $A$ : a square number. Event $B$ : a prime number. Event $C$ : a number greater than $3$ . Which pairs of events are mutually exclusive? Give a reason for each.

Hint List the scores for each event. Two events are mutually exclusive if no score is in both lists.

B1
A card is taken from cards numbered $1$ to $15$ . $E$ : an even number. $F$ : a multiple of $5$ . $S$ : a square number. Is each pair mutually exclusive? Give a reason. (i) $E$ and $F$ (ii) $F$ and $S$ (iii) $E$ and $S$
[3 marks]

Example 2

Omar has a 10p coin, a 20p coin and a £1 coin in his pocket. Some money falls out: one, two or all three coins. (a) List the seven amounts that could fall out. (b) Each amount is equally likely. Is it more likely that more than 50p falls out, or less than 50p? Explain your answer.

Hint Work systematically: one coin, then two coins, then all three.

B2
Priya has a 5p coin, a 10p coin and a 20p coin. Some money falls out: one, two or all three coins. Each of the seven possible amounts is equally likely. Work out the probability that more than 18p falls out.
[3 marks]
Answer

Example 3

Cards are numbered $1$ to $10$ . Set $A$ is the multiples of $3$ . Set $B$ is the numbers greater than $5$ . Put every number in the Venn diagram. Then work out the probability that a card taken at random is a multiple of $3$ and greater than $5$ .

Hint Numbers in both sets go in the overlap. Numbers in neither set go outside both circles.

B3
Cards are numbered $1$ to $16$ . Set $A$ is the factors of $16$ . Set $B$ is the odd numbers. Complete the Venn diagram. Then work out the probability that a card taken at random is a factor of $16$ and odd.
[3 marks]
Answer

Example 4

A spinner can land on red, blue, green or yellow. The table shows the probabilities. (a) Work out the probability of green. (b) Work out the probability of red or blue. (c) Work out the probability of not yellow.

Hint The four colours cover every outcome and cannot happen together, so their probabilities add up to

1

B4
A tile is picked from a bag. Each tile shows P, Q, R or S. The table shows the probabilities. Work out (i) the probability of Q, (ii) the probability of P or R, (iii) the probability of not S.
[3 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$ (iii) $\dots\dots$

Practice

Year 7

C1
A bottle holds red, green and black beads. One bead is seen. Are the events "the bead is green" and "the bead is not red" mutually exclusive? Give a reason for your answer.
[2 marks]
C2
A bottle holds $30$ beads. Sam sees one bead at a time. In $20$ looks he sees $14$ green beads and $6$ red beads. Sam says, "There are only green and red beads in the bottle." Explain why Sam cannot be sure.
[1 mark]

C3
A set of $30$ cards each shows one animal. The table shows how many cards show each animal. A card is taken at random. Work out the probability that it shows a dog or a rabbit.
[2 marks]
Answer
C4
Both spinners are spun and the two scores are added. Is a positive total or a negative total more likely? Show your working.
[2 marks]
Answer
C5
A fair dice is rolled. Which pair of events is both mutually exclusive and exhaustive? (i) "less than $4$ " and "more than $4$ " (ii) "a factor of $6$ " and " $4$ or $5$ " (iii) "even" and "a multiple of $3$ ". Give a reason for your answer.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
A biased spinner can land on red, blue, green or yellow. The table shows some of the probabilities. The probability of green is the same as the probability of yellow.
(a)
Work out the value of $x$ .
[2 marks]
Answer
(b)
Work out the probability that the spinner lands on red or blue.
[1 mark]
Answer
(c)
Work out the probability that the spinner does not land on yellow.
[1 mark]
Answer
Total for D1: 4 marks

D2
Kai rolls a fair dice. He says, "The probability of an even number or a number greater than $4$ is $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ ."
Is Kai correct? Explain your answer and give the correct probability.
[3 marks]
D3
$ξ = {1, 2, 3, \dots, 15}$ . $A$ is the multiples of $3$ . $B$ is the factors of $30$ .
(a)
Complete the Venn diagram.
[3 marks]
(b)
A number is chosen at random from $ξ$ . Work out the probability that it is in $A$ and in $B$ .
[1 mark]
Answer
(c)
Are $A$ and $B$ mutually exclusive? Give a reason for your answer.
[1 mark]
Total for D3: 5 marks
D4
A bag contains only red, blue and green counters. The probability of taking a red counter is $0.4$ . There are twice as many blue counters as green counters.
(a)
Work out the probability of taking a green counter.
[3 marks]
Answer
(b)
There are $12$ red counters in the bag. Work out the total number of counters in the bag.
[2 marks]
Answer
Total for D4: 5 marks

Extension

Stretch

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

E1
Tara has four coins: 5p, 10p, 20p and 50p. Some money falls out of her pocket: one or more of the coins. Each possible set of coins is equally likely. Work out the probability that more than 40p falls out.
[4 marks]
Hint 1 · What to try
List every set of coins in an organised way: one coin, two coins, three coins, all four.
Hint 2 · The first line
There are $4$ sets of one coin, $6$ of two, $4$ of three and $1$ of four: $15$ sets.
Hint 3 · The full method
Totals over 40p: 50p, 55p, 60p, 70p, 65p, 75p, 80p, 85p. That is $8$ of the $15$ sets, so the probability is $\frac{8}{15}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can decide whether two events are mutually exclusive and give a reason.
I can put numbers in a Venn diagram and find a probability from it.
I can use the fact that the probabilities of exhaustive outcomes add up to $1$ .
I can list every outcome in an organised way.

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