KS3 Maths · Year 7 · Chapter 3: Probability · Lesson 2

Using a sample space to calculate probabilities

Year 7About 60 minutesNo calculator48 marks

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Date

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Quick questions to start.

A1
Needed today
Write $\frac{15}{36}$ in its simplest form.
[1 mark]
Answer
A2
Needed today
Write $6$ out of $20$ as a fraction in its simplest form.
[1 mark]
Answer
A3
Last lesson
P(red) $= 0.35$ . Find P(not red).
[1 mark]
Answer
A4
From chapter 2
A quadrilateral has angles $8 5^{\circ}$ , $11 0^{\circ}$ , $7 0^{\circ}$ , $x$ . Find $x$ .
[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 coin is tossed and a spinner numbered $1$ to $4$ is spun. The sample space shows every outcome. Work out (i) P(tail and $3$ ) (ii) P(head and an odd number) (iii) P(tail and less than $4$ ).

Hint Count the outcomes in the table: there are

2 \times 4 = 8

, all equally likely.

B1
A coin is tossed and a spinner numbered $1$ to $5$ is spun. Complete the sample space. Then work out (i) P(head and $4$ ) (ii) P(tail and an even number) (iii) P(head and a number greater than $2$ ).
[3 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$ (iii) $\dots\dots$

Example 2

Two spinners, each numbered $1$ to $4$ , are spun and the scores are added. (a) Which total is most likely? (b) Work out P(total is $6$ or more). (c) Work out P(total is odd).

Hint The grid has

16

equally likely totals. Count how many times each total appears.

B2
A fair dice and a spinner numbered $1$ to $3$ are used. The scores are multiplied. Dice scores are across the top. Complete the grid. Then work out (i) P(product is even) (ii) P(product is greater than $10$ ) (iii) the most likely product.
[3 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$ (iii) $\dots\dots$

Example 3

A club has $40$ members. $18$ are adults. $12$ members swim, and $7$ of the swimmers are children. Complete the two-way table. Work out the probability that a member chosen at random is an adult who does not swim.

Hint Fill in the cells you can find straight away, then use the totals to find the rest.

B3
$36$ pupils go on a school trip. $20$ are girls. $9$ pupils bring a packed lunch, and $4$ of these are boys. Complete the two-way table. Then work out (i) P(a girl with a school lunch) (ii) P(a boy with a packed lunch).
[3 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Practice

Year 7

C1
$80$ cakes are each chocolate or vanilla. $45$ are iced, including $28$ chocolate cakes. $12$ vanilla cakes are not iced. Use the Venn diagram to find P(a chocolate cake that is not iced).
[3 marks]
Answer
C2
Both spinners are spun and the two scores are multiplied. Work out the probability that the product is greater than $0$ .
[2 marks]
Answer

C3
A game uses a grid with some winning squares. Grid A is $5$ by $5$ with $6$ winning squares. Grid B is $4$ by $8$ with $7$ winning squares. Grid C is $10$ by $10$ with $23$ winning squares. A square is picked at random. Which grid gives the best chance of a win? Show how you decide.
[2 marks]
Answer
C4
Three coins are tossed. List all the outcomes. Work out the probability of exactly two heads.
[2 marks]
Answer
C5
Two fair dice are rolled and the scores are added. Work out the probability that the total is a prime number.
[2 marks]
Answer
C6
Drawer 1 has $2$ red socks, $1$ blue and $1$ green. Drawer 2 has $1$ red, $2$ blue and $2$ green. One sock is taken from each drawer. Find P(both socks are the same colour).
[3 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Zoe tosses two fair coins. She says, "There are three outcomes: two heads, two tails, or one of each. So the probability of one head and one tail is $\frac{1}{3}$ ."
Is Zoe correct? Explain your answer and give the correct probability.
[3 marks]

D2
$60$ people were asked how they travel to work. The two-way table shows some of the results.
(a)
Complete the two-way table.
[3 marks]
(b)
One person is chosen at random. Work out the probability that this person is a child who travels by car.
[1 mark]
Answer
(c)
Work out the probability that the person chosen does not walk.
[1 mark]
Answer
Total for D2: 5 marks
D3
Spinner A is numbered $1$ , $2$ , $3$ , $4$ . Spinner B is numbered $1$ , $3$ , $5$ . Both are spun. The score is the larger number take away the smaller number. The grid shows some of the scores, with spinner A across the top.
(a)
Complete the grid.
[2 marks]
(b)
Leo says, "The most likely score is $2$ ." Is Leo correct? Give a reason for your answer.
[2 marks]
(c)
Work out the probability that the score is $3$ or more.
[1 mark]
Answer
Total for D3: 5 marks
D4
A café sells a meal deal. You choose a starter (soup or melon), a main (chicken, fish or pie) and a dessert (cake or fruit).
(a)
How many different meal deals are there? Show how you work it out.
[2 marks]
Answer
(b)
A meal deal is chosen at random. Work out the probability that it has melon and does not have fish.
[2 marks]
Answer
Total for D4: 4 marks

Extension

Stretch

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

E1
A fair dice is rolled and a fair spinner numbered $1$ to $n$ is spun. The probability that the dice and the spinner show the same number is $\frac{1}{8}$ . Work out the value of $n$ .
[4 marks]
Hint 1 · What to try
The sample space has $6 n$ equally likely outcomes.
Hint 2 · The first line
If $n$ is $6$ or more, the same number can happen in $6$ ways. If $n$ is less than $6$ , it can happen in $n$ ways.
Hint 3 · The full method
With $n$ less than $6$ the probability is $\frac{n}{6 n} = \frac{1}{6}$ , which is not $\frac{1}{8}$ . So $\frac{6}{6 n} = \frac{1}{n} = \frac{1}{8}$ and $n = 8$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw a sample space that lists every outcome of two events.
I can complete a two-way table from information given in words.
I can work out a probability from a sample space, a table or a Venn diagram.
I can count equally likely outcomes when items come in different numbers.

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