Probability (1)Objectives

This chapter will show you how to• describe probability using words G

• write a list of outcomes G F

• describe probability using numbers and words G F

• work out the probability of an event happening F

• work out the probability of an event happening, when it can happen in more than one way F

• work out the probability of an event not happening E

• work with mutually exclusive events. D

This chapter is about predicting the chance of things happening.

Jelly bean recipes challenge you to fi nd enough of the right fl avours to make lots of your favourite taste. With 65 beans in a 70 g bag, you need probability to work out your chances!

Before you start this chapter1 What fraction of each shape is coloured?

a b

2 Work out each missing number.

a 1 _ 7 � 1 _ 7 � 1 _ 7 � __ 7 b 2 _ 9 � __ 9 � 1 _ 9 � 5 _ 9

c 4 _ 7 � 1 _ 7 � 3 __

3 Copy and complete.

a 6 � 7 � b 15 � 3 �

c � 5 � 40 d 36 � � 6

e 4 � � 32 f � 3 � 18

4 True or false?

a 0.5 � 0.3 � 0.1 � 0.9

b 0.8 � 0.4 � 0.2 � 0.3

c 0.8 � 2 � 1.6

d 75% � 25% � 100%

e 100% � 66% � 44%

f 18% � 3 � 6%

5 Find the next two terms.

a 1 _ 5 , 2 _ 5 ,

3 _ 5 , , b 10

__ 10 , 8 __ 10 ,

6 __ 10 , ,

Video: ‘Everyone’s a winner?’

BB B C ACTIVEACTIVE

HELP Chapter X

UNCORRECTED PROOF – AQA GCSE MATHEMATICS FOR FOUNDATION SETS

100 Probability (1)

Keywordschance, likelihood, probability, certain, impossible

7.1 The language of probability

ObjectiveUnderstand and use some of

the basic language of probability

Why learn this?Probability helps

you understand your chances of winning

the lottery.

G

Skills check1 What does the word ‘certain’ mean?2 If something ‘might’ happen, does that mean it is ‘certain’ to

happen?3 What does the word ‘impossible’ mean?

Write down whether these things are certain to happen, might happen or are impossible.

a Newborn twins will both be boys.

b It will rain in Scotland this year.

c An athlete will run 100 m in two seconds.

Example 1G

G

HELP Section 2.3

What is probability?People often talk about the chance or likelihood that something might happen. For example, ‘What is the chance that it will snow tomorrow?’

Probability is about measuring the likelihood that something might happen.

Some things are certain to happen.For example, a baby will be born today.

Some things cannot happen.For example, it is impossible that you will live until you are 180 years old.

Some things might happen.For example, the next car you see might be red.

Exercise 7AWrite down whether these things are certain to happen, might happen or are impossible.

1 The sun will rise tomorrow.

2 It will snow on New Year’s Day.

a might happen

b certain to happen

c impossible

They might be both girls, or one girl and one boy.

Scotland has a lot of rain every year.

The current world record is 9.69 seconds.

1017.2 Outcomes of an experiment

3 You will see a shooting star if you look at the sky tonight.

4 When you roll a normal dice you will roll a 7.

5 The next car to pass the school gates will be blue.

6 The day after Wednesday will be Thursday.

7 When you roll a dice you will roll a 6.

8 It will get dark tonight.

9 You will swim the length of a 25 m pool in 5 seconds.

10 In a litter of nine puppies, half of them will be male.

G

Skills check1 When you fl ip a coin, what could happen?2 When you roll a dice, what could happen?

Writing outcomesAn experiment is something you do to fi nd out what happens.

Rolling a dice is an experiment. In probability it is also called an event.An experiment (or event) has outcomes. When you roll a dice you might get a 3. So 3 is one of the outcomes of this event.

The event ‘rolling a dice’ has six possible outcomes 1, 2, 3, 4, 5 or 6.

To write a list of outcomes, work systematically to make sure that you don’t miss any out.

When two things happen at the same time, such as rolling a dice and fl ipping a coin, they are called combined events.

Skills check

Keywordsexperiment, event, outcome, possible outcomes, combined events

7.2 Outcomes of an experiment

ObjectivesG List all possible outcomes for an

experiment

F List all possible outcomes for a combined event

Why learn this?Knowing the possible

outcomes helps you predict your chances

of winning.L

102 Probability (1)

Exercise 7B1 A coin is fl ipped. List all the possible outcomes.

2 A normal six-sided dice is rolled. List all the possible outcomes.

3 This spinner has three equal sectors coloured yellow, orange and pink.

The spinner is spun. List all the possible outcomes.

4 One item is selected from a bag containing 1 apple, 1 orange, 1 banana and 1 pear.

List all the possible outcomes.

5 This is the list of vegetables available in a school canteen.

Bryoni chooses two different vegetables.

List all the possible combinations of vegetables

that Bryoni could choose.

6 Alison, Bethany, Christine, David and Eddie go to a dance together.

One of the girls must dance with one of the boys.

List all the possible combinations of dance partners.

7 Sam has a spinner, a coin and a normal six-sided dice.

The spinner has four equal sectors coloured red, blue, green

and yellow.

a Sam fl ips the coin and spins the spinner at the same time.

List all eight possible outcomes.

b Sam fl ips the coin and rolls the dice at the same time.

List all 12 possible outcomes.

c Sam spins the spinner and rolls the dice at the same time.

Without writing them all down, work out how many possible outcomes there are.

Explain how you worked out your answer.

a An odd number is chosen from the numbers 1 to 10. List all the possible outcomes.

b A 10p coin and a 5p coin are fl ipped at the same time. List all the possible outcomes.

c A coin and a four-sided dice, numbered 1, 2, 3 and 4, are thrown at the same time.

List all the possible outcomes.

Example 2G

a 1,3,5,7,9

b HH, HT, TH, TT

c H1, H2, H3, H4, T1,

T2, T3, T4

List the numbers in order so you don’t miss any out.

H stands for Head and T stands for Tail, so HT means Head and Tail.

H1 means a Head on the coin and a 1 on the dice. There are 8 possible outcomes altogether.

G

F

AO2

Today’svegetablesBroccoliCarrotsPeas

Sweetcorn

F

1037.3 The probability scale

F

AO2

FUNC

TIONALFUNC

TION

AL

Skills check1 Write these in order, starting with the smallest.

2 What fraction of each shape is yellow?

Probability in numbers and wordsProbability uses numbers and words to describe the chance that an event will happen. It is measured on a probability scale from 0 to 1.

Probability 0 means that the event cannot happen – it is impossible.

Probability 1 means that the event is certain to happen.

12

34 1

14 0

8 A triangular spinner has sections labelled 1, 2 and 3.

A circular spinner has sections labelled 1 and 2.

The spinners are spun at the same time.

The numbers that the spinners land on are added to

give the score.

List all the possible scores.

9 The numbers on these two spinners have been rubbed out.

When the spinners are spun at the same time, the numbers

that the spinners land on are added to give the score.

The possible scores are 4, 6, 8 and 10.

What could the numbers on the spinners be?

10 Glyn is organising a 5-a-side football tournament for a PE lesson.

25 pupils have been put into fi ve teams, A, B, C, D and E.

Each team must play all the other teams.

The results will be put into a league table.

a How many games will be played?

The lesson lasts 80 minutes.

Glyn estimates there will be 1 minute intervals between games.

At the end, putting the results into a league table to decide the winners will take

about 8 minutes.

b How many minutes should Glyn allow for each game?

(All games must be the same length.)

Skills check

L7.3 The probability scale

ObjectivesG Understand and use the basic

language of probability

G Understand, draw and use a probability scale from 0 to 1

Why learn this?The scale helps give

accurate measures of probability.

Keywordsprobability scale, unlikely, even chance, likely, fair, unbiased, biased

?

???

?

3

211

2

14

0 1

certainlikelyunlikelyimpossibleeven

chance

12

34

104 Probability (1)

Exercise 7C1 Choose a word from the box below that describes the probability of each event

happening.

impossible unlikely even chance likely certain

a The sun will set tomorrow.

b Picking out a diamond from a shuffl ed pack of cards.

c Rolling an ordinary dice and getting a 7.

d A new born baby will be a boy.

e Your birthday in 2020 will be on a Friday.

f You will play a computer game tonight.

2 Write down two events of your own that would have a probability of ‘unlikely’.

3 Write down two events of your own that would have a probability of ‘certain’.

a Use words to describe the probability of getting a head when a coin is fl ipped.

b What is the probability of getting a tail when a coin is fl ipped?

c Use words to describe the probability that the sun will rise tomorrow.

d What is the probability that the sun will rise tomorrow?

e Which colour is this spinner most likely to land on?

f What is the probability that this spinner will land on blue?

Example 3

a even chance

b 1

__

2

c certain

d 1

e red

f 1

__

4

F

The coin has 1 head and 1 tail.

Certain means the probability is 1.

Even chance means the probability is 1 _ 2 .

3 _ 4 of the spinner is red, which is likely.

The sun will always rise.

1 _ 4 of the spinner is blue, which is unlikely.

GAO2

G

In a pack of cards there are diamonds, hearts, clubs and spades.

G

1057.3 The probability scale

4 Copy this probability

scale with arrows.

Label each arrow with an event from the list below. The fi rst one is done for you.

a Picking the ace of hearts from the four aces in a pack of cards.

b It will rain in Glasgow next year.

c Flipping a coin and getting a tail.

d You will meet a famous movie star next Monday.

e Picking a letter from the word TENBY, and the letter is a vowel.

5 Draw a probability scale. Put an arrow on the scale to show the probability of each

of these events happening.

a The next car you see will be red.

b It will rain tomorrow.

c You will have maths homework next week.

d Picking a letter from the word ABERDEEN, and the letter is a vowel.

e Picking out an ace from a shuffl ed pack of cards.

6 Copy this probability scale with arrows.

Work out the probability of each of these

spinners landing on red.

Label each arrow with the letter for

each spinner.

a b c d e

7 A fair six-sided spinner is numbered from 1 to 6.

The spinner is spun once.

Copy this probability scale. Put an arrow on the scale to

show the probability of each of these outcomes.

a The spinner lands on an odd number.

b The spinner lands on 1 or 2.

c The spinner lands on a number greater than zero.

8 Copy this spinner. Shade it so that the probability

of landing on a shaded section is 1 _ 2 .

9 Copy this spinner. Use red, blue and yellow to

colour your spinner so the probability of landing

on red is 1 _ 4 .

14

0 1

a

12

34

12

3

65

40 1

G

F

G

F

AO2

AO2

A, E, I, O, U are the vowels.

14

0 112

34

106 Probability (1)

Keywordsprobability, event, successful, possible

7.4 Calculating probabilities

ObjectiveTo fi nd the probability of an

outcome

Why learn this?Is it fair to fl ip a coin

to decide which team starts the match?

F

Skills check1 True or false?

a 2 _ 4 � 1 _ 2 b 1 _ 5 � 3 __ 15

c 4 ___ 100 � 1 __ 50 d 2 __ 20 � 1 __ 10

2 In a normal pack of playing cardsa How many spades are there?b How many Kings are there?c How many picture cards are there?

Working out the probabilityThe probability of an event happening is

Probability � number of successful outcomes _____________________________ total number of possible outcomes

a What is the probability of rolling a 4 on a fair six-sided dice?

b What is the probability of picking the Jack of diamonds from a shuffl ed pack of cards?

c You have 10 raffl e tickets. 300 raffl e tickets have been sold in total. What is the probability

that you will win?

Example 4

a 1

__

6

b 1

___

52

c 10

_____

300

There is one 4 on a dice.There are six numbers on the dice altogether.

There is one Jack of diamonds in a pack of cards.There are 52 cards in the pack altogether.

Any one of your 10 tickets could win.There are 300 tickets altogether.

F

F

3 Copy and complete this table.

Fraction Decimal Percentage

1 __ 10

50%

3 _ 4

0.8

Exercise 7D 1 A fair six-sided dice is rolled. Work out the probability of

a rolling a 1 b rolling a 2

c rolling an even number d rolling a 12.

1077.4 Calculating probabilities

2 300 Christmas raffl e tickets are sold. What is the probability of winning the raffl e if

a you have one ticket

b you have fi ve tickets

c you have ticket number 7

d you have tickets numbered 253 and 254

e you forget to buy a ticket?

3 The probability of Alan winning a school raffl e is 5

___ 500

.

a How many raffl e tickets were sold?

b Did Alan have ticket number 5?

c How many tickets did Alan buy?

d Alan’s Mum gives him two more tickets.

What is the probability of Alan winning the raffl e now?

4 Hitesh buys three charity raffl e tickets. Altogether 100 tickets are sold.

Hitesh wants to make his probability of winning 1 __ 10

.

How many more tickets does he need?

5 In roulette, a ball is spun around a roulette wheel.

Players bet on where the ball will land.

The roulette wheel has 37 sections numbered 0 to 36.

The zero is coloured green and the other numbers are coloured red or black, as

shown in the diagram below.

A £10 bet could win these amounts.

A £10 bet on Possible win

a single number e.g. 17 £360

a pair of numbers e.g. 14 and 17 £180

a group of four numbers e.g. 14, 17, 15 and 18 £90

a group of 12 numbers e.g. 13 to 24 £30

a red or a black number £20

an odd or an even number £20

A player places one £10 bet. What is the probability that she wins

a £360 b £90 c £30?

1 4 7 10 13 16 19 22 25 28 31 34

2 5 8 11 14 17 20 23 26 29 32 35

3 6 9 12 15 18 21 24 27 30 33 36

1st 12

1 to 18 EVEN

2nd 12 3rd 12

ODD 18 to 36Red Black

0

F

AO2

F

108 Probability (1)

Mixed exercise1 Choose a word from the box below that describes the probability of each event

happening.

impossible unlikely even chance likely certain

a Rolling a fair dice and getting a number less than 7. [1 mark]

b Picking out an ace from a shuffl ed pack of cards. [1 mark]

c Rolling a fair dice and getting an odd number. [1 mark]

2 Copy this probability scale with arrows.

Label each arrow with an event from the list below.

Spinning this four-sided spinner and getting:

a an odd number b an even number

c a number greater than 4 d a number less than 7. [3 marks]

3 This is the sweet menu in a restaurant.

With each sweet a customer can have either

ice cream (I) or custard (C).

List all the possible combinations of sweets

that a customer could choose. [2 marks]

4 A fair six-sided dice is rolled. Work out the probability of

a rolling a 5 [1 mark]

b rolling a number greater than 5 [1 mark]

c rolling an 8. [1 mark]

5 The probability of Keshia winning a raffl e is 6

___ 300

.

a How many raffl e tickets were sold? [1 mark]

b How many tickets did Keshia buy? [1 mark]

c Keshia gives one of her tickets to her Mum.

What is the probability of Keshia winning the raffl e now? [1 mark]

6 A triangular spinner has sections labelled 4, 5 and 6.

A circular spinner has sections labelled 2 and 3.

The spinners are spun at the same time.

The numbers that the spinners land on are

multiplied to give the score.

List all the possible scores. [2 marks]

7 a Copy this spinner.

b Using red and blue, colour the spinner to make the probability of landing on a

blue section equal to 1 _ 3 . [2 marks]

14

0 112

34

52

4

6

SWEET MENUSponge pudding (S)

Lemon cheesecake (L)

Raspberry trifle (R)

Pavlova (P)

6

542

3

G

F

F

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1097.5 Events that can happen in more than one way

Exercise 7E1 A card is picked at random from an ordinary pack of

playing cards. What is the probability that the card is

a a Jack b a Queen c a picture card?

2 Anton rolls a fair six-sided dice.

What is the probability that he rolls a number that is

a greater than 4 b less than 4 c at least 4?

7.5 Events that can happen in more than one way

ObjectiveF Work out the probability of an event

that can happen in more then one way

Why learn this?You can work out

the probability of winning a bet.

Skills check1 Cancel these fractions to their lowest terms.

a 2 __ 10 b 6 __ 12 c 15 __ 20 d 12

__ 15

2 Write down all the factors of 52.

3 Work out the missing numbers in this sequence. 0, 2 __ 10 , __ 10 , 6 __ 10 , 8 __ , 1

How many ways can it happen?Some events can happen in more than one way.

For example, if you wanted to pick an ace from a pack of cards, there are four ways this could happen.

You could get the ace of spades, the ace of hearts, the ace of clubs or the ace of diamonds.

F

A card is picked at random from an ordinary pack of playing cards.

What is the probability that the card is:

a an ace b a King c a red card?

Example 5

a 4

___

52

b 4

___

52 =

4 ÷ 4

________

52 ÷ 4 =

1

___

13

c 26

___

52 =

26 ÷ 26

_________

52 ÷ 26 =

1

__

2

F

There are 4 aces in a pack.There are 52 cards in the pack altogether.

There are 4 Kings in the pack of 52 cards.

4 __ 52 cancels down to 1 __

13 .

A red card could be a diamond or a heart.

In a pack of cards, 1 _ 2 are red and 1 _

2 are black.

Picture cards include all of the Jacks, Queens and Kings.

‘At least 4’ means you must include the 4.

L

110 Probability (1)

3 Lucy has fi ve raffl e tickets.

Altogether 500 raffl e tickets have been sold.

What is the probability that

a Lucy wins the raffl e?

b the winning ticket is a number greater than 400?

4 This fair spinner is spun.

What is the probability that it lands on an odd number?

5 One letter is chosen at random from the word

P R O B A B I L I T Y

Work out the probability that the letter is

a the letter B b a vowel c made up entirely of straight lines.

6 Lily has a bag containing 10 sweets.

Three are strawberry, one is cherry and six are raspberry fl avour.

Lily takes one sweet from the bag at random.

Copy the probability scale below. Put an arrow on the scale to show the probability

of each of these outcomes.

a The sweet is strawberry fl avour.

b The sweet is cherry fl avour.

c The sweet is raspberry fl avour.

7 Aster has a bag containing eight counters. Six of the counters are blue and two are red.

Aster takes a counter from the bag at random.

Draw a probability scale. Put an arrow on the scale to show the probability of each

of these outcomes.

a The counter is blue.

b The counter is red.

c The counter is orange.

8 Margery has three bags of counters. The bags contain red, blue and green counters.

a Which two bags should Margery mix together to give her the highest probability

of picking a red counter?

b Margery mixes together the two bags from part a.

What is the probability that she picks a red counter at random from this mixed

bag?

12

3

4

56

7

8

0 1

A B C

F

F

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1117.6 The probability that an event does not happen

9 Franz has three bags of counters. The bags contain red, green and yellow counters.

Franz mixes two of the bags together.

Franz says ‘If I mix bags B and C together, I have the best chance of picking out

a red counter. This is because in the mixed bag there are fewer green and yellow

counters than red counters’. Explain why Franz is wrong.

10 Eve bought 4 raffl e tickets. Pete said ‘400 tickets have been sold altogether’.

Later Eve bought another raffl e ticket. Pete said ‘Now 500 tickets have been sold

altogether’.

Eve said ‘Oh no! I had more chance of winning when I had 4 tickets and only 400

tickets had been sold’. Explain why Eve is wrong.

A B CF

AO3

a The probability of picking an ace from a pack of cards is 4 __ 52

.

What is the probability of not picking an ace from a pack of cards?

b The probability of picking a heart from a pack of cards is 0.25

What is the probability of not picking a heart from a pack of cards?

Example 6

a 48

___

52 =

12

___

13

b 0.75

E

1 � 4 __ 52 � 52

__

52 � 4 __ 52 � 48

__

52

1 � 0.25 � 0.75

Keywordsprobability, event, not, random

7.6 The probability that an event does not happen

ObjectiveE Work out the probability of an event not happening when

you know the probability that it does happen

Why learn this?If you know the

probability it won’t rain, you can decide whether

to take an umbrella. Skills check1 Work out a 1 � 0.2 b 1 � 0.752 Work out a 100% � 30% b 100% � 92%3 Work out a 1 � 1 _ 3 b 1 � 3 _ 5

Calculating the probability an event does notnot happenWhen you know the probability that an event will happen, you can calculate the probability that the event will not happen by using this fact:

(� Probability that an event

will not happen ) � 1 � (� Probability that an event

will happen

)

L

112 Probability (1)

Exercise 7F1 The probability of picking a King from a pack of cards is 1 __

13 .

What is the probability of not picking a King from a pack of cards?

2 Hamish is learning golf.

The probability that he hits the ball in the right direction is 1 in 10.

What is the probability that his next shot

a goes in the right direction

b doesn’t go in the right direction?

3 The probability that this spinner lands on 1 is 0.7

The probability that this spinner lands on blue is 0.85

What is the probability that the spinner

a does not land on 1

b does not land on blue?

4 The probability that this spinner lands on 1 is 28%.

The probability that this spinner lands on blue is 99%

What is the probability that the spinner

a does not land on 1

b does not land on blue?

5 The probability that Hazel misses her bus is 0.05

What is the probability that Hazel catches her bus?

6 The probability of winning a £5 prize on the National lottery thunderball is 3

___ 100

.

The probability of winning a £10 prize on the National lottery thunderball is 9

___ 1000

.

Work out the probability of

a not winning a £5 prize

b not winning a £10 prize.

7 Alan buys a special fi ve-sided spinner.

The spinner has equal sections numbered 1 to 5.

The probabilities of different scores are listed in the table.

Work out the probability of

a not spinning a 1

b not spinning a 3

c not spinning a 5.

d Explain what your answer to part c means.

8 Sage has a biased dice numbered 1 to 6.

The probability of getting a 6 with this dice is 1 _ 3 .

Sage says ‘There are 5 other numbers. So the probability of not getting a 6 with

this dice is 5 _ 6 ’. Explain why Sage is wrong.

1

3

2

1

3

2

E

E

AO2

Catching the bus means not missing the bus.

% means ‘out of 100’

Number 1 2 3 4

Probability 0.3 0.2 0.25 0.25

1137.7 Mutually exclusive events

9 Leanne has a box that contains 20 counters.

Leanne picks a blue counter at random from the box.

The probability that she picks a blue counter is 4 _ 5 .

a What is the probability that Leanne picks a counter

that is not blue?

b How many counters in the box are not blue?

10 Lee has 10 coins in his pocket. He picks one at random.

The probability Lee doesn’t pick a 10p coin is 2 _ 5 .

How many 10p coins does Lee have in his pocket?

E

E

AO2

AO3

Keywordsmutually exclusive, or, certain, add

7.7 Mutually exclusive events

ObjectiveD Understand and use the fact that the

sum of the probabilities of all mutually exclusive outcomes is 1

Why learn this?It could help you win

if you remember what cards have already

been played.Skills check1 Work out: a 1 _ 5 � 1 _ 5 b 1 � 1 _ 4 c 1 � 2 _ 3 2 Work out: a 0.4 � 0.3 b 1 � 0.82 c 0.4 � 2

Mutually exclusive eventsMutually exclusive events cannot happen at the same time.

When you roll a dice you cannot get a 1 and a 6 at the same time.

When you fl ip a coin you can get either a head or a tail, but not both at the same time.

For any two events, A and B, which are mutually exclusive

P(A or B) � P(A) � P(B)

For a dice, the probability of rolling a 2 is 1 _ 6

You can write this as P(2) � 1 _ 6

Also, P(1) � 1 _ 6 , and P(3) � 1 _ 6 , P(4) � 1 _ 6 , P(5) � 1 _ 6 and P(6) � 1 _ 6

Add together the probabilities of all the possible outcomes:

P(1) � P(2) � P(3) � P(4) � P(5) � P(6) � 1 _ 6 � 1 _ 6 � 1 _ 6 � 1 _ 6 � 1 _ 6 � 1 _ 6 � 1

This is because you are certain to roll either 1 or 2 or 3 or 4 or 5 or 6.

Rolling a 2 and not rolling a 2 are mutually exclusive events.

The total sum of their probabilities is 1 _ 6 � 5 _ 6 . This is because you are certain to get either ‘2’ or ‘not 2’.

Picking at random means that each counter is equally likely to be picked.

P(A) means the probability of event A occurring.

114 Probability (1)

Exercise 7G1 A box contains 15 chocolates.

Six of the chocolates have toffee centres, four are solid chocolate,

three have soft centres and two have nut centres.

One chocolate is taken from the box at random.

What is the probability that the chocolate

a doesn’t have a nut centre b doesn’t have a toffee centre

c has a toffee or a solid centre d has a toffee or a soft centre

e has a toffee or a nut centre f doesn’t have a toffee or a nut centre

g doesn’t have a soft or a nut or a toffee centre?

2 A tin contains biscuits.

One biscuit is taken from the tin at random.

The table shows the probabilities of taking each

type of biscuit.

a What is the probability that the biscuit is a

digestive or a cookie?

b What is the probability that the biscuit is a

wafer?

3 A bag contains cosmetics.

One cosmetic is taken from the bag at random.

The table shows the probabilities of taking each

type of cosmetic. There are three times as many

eyeshadows as blushers.

What is the probability that the cosmetic is an

eyeshadow?

a Work out the probability of rolling a 5 or a 6 with a fair dice.

b This spinner has four sections numbered 5 to 8.

The table shows the probability of the spinner landing on each number.

Number 5 6 7 8

Probability 0.2 0.2 0.2 ?

What is the probability that the spinner lands on 8?

Example 7D

a P(5) = 1

__

6 , P(6) =

1

__

6

P(5 or 6) = 1 __ 6 + 1

__ 6 = 2

__ 6

b P(8) = 1 – 0.2 – 0.2 – 0.2

= 0.4

7

65

8

Work out P(50) and P(6).

Rolling a 5 and rolling a 6 are mutually exclusive so add the probabilities together.

Subtract the probabilities of 5, 6 and 7 from 1.

E

D

D

D

AO2

Biscuit Probability

digestive 0.4

wafer

cookie 0.15

ginger 0.25

Cosmetic Probability

eyeliner 0.3

lipgloss 0.3

eyeshadow

blusher

115Chapter 7 Review exercise

4 David puts 15 CDs into a bag.

Elliot puts 9 computer games into the same bag.

Fern puts some DVDs into the bag.

The probability of taking a DVD from the bag at random is 1 _ 3 .

How many DVDs did Fern put in the bag?

Review exercise 1 Diego has two fair spinners.

Spinner A has four equal sections.

Two sections are blue,

one is brown and the other is pink.

Spinner B has eight equal sections.

Three are blue, one is brown and four are pink.

Diego spins each spinner once.

a Which colour is spinner B most likely to land on? [1 mark]

b Which spinner is more likely to land on brown, spinner A or spinner B? [1 mark]

Give a reason for your answer.

c Copy this probability scale with arrows.

Label each arrow with an event

from the list below.

i Spinner A lands on blue

ii Spinner A lands on pink

iii Spinner B lands on brown

iv Spinner B lands on blue [2 marks]

2 In a raffl e 400 tickets are sold. There is only one prize.

Ruth buys 10 tickets, Penny buys 5 tickets, Holly and Lilly buy 3 tickets each.

a Which of the four girls has the best chance of winning the prize? [1 mark]

b What is the probability that Penny wins the prize? [1 mark]

c What is the probability that none of them win the prize? [1 mark]

3 Stefan has this ten-sided dice.

It has the numbers 1 to 10 on it.

He rolls the dice once.

Work out the probability that Stefan

a rolls a 5 [1 mark]

b doesn’t roll a 5 [1 mark]

c rolls an even number [1 mark]

d rolls a number greater than 6. [1 mark]

4 A fair six-sided dice and a fair coin are thrown at

the same time.

The outcome T5 means a tail and a 5.

a Write a list of all the possible outcomes. [1 mark]

b What is the probability of getting a tail and an odd number? [1 mark]

c What is the probability of getting a head and a number less than three? [1 mark]

Spinner A Spinner B

FE

FAO2

F

AO2

F7

04

0 1

116 Probability (1)

5 Alfi e says ‘The probability that I don’t miss the bus in the morning is 0.85, so the

probability that I do miss the bus in the morning is 0.25’.

Is Alfi e’s statement correct? Give a reason for your answer. [1 mark]

6 A game of chance consists of turning over one card from each of two sets of cards.

The cards are

a The numbers on the cards are added together.

Complete this table to show all the possible outcomes.

Blue

Red

2 3 5 6 9

[2 marks]

1 3 4

4 6

7

8

b You win £1 if the total of your two cards is 10

i What is the probability of winning? [1 mark]

ii What is the probability of not winning? [1 mark]

c 50 people play this game.

i How many of them do you expect to win £1? [1 mark]

ii How much should you charge if you don’t want to lose money? [1 mark]

7 Derry designs a game of chance to help raise

money at his school fête.

Derry uses a normal dartboard with sections

labelled 1 to 20.

It costs £1 to throw one dart. Derry gives a £2

prize if the dart lands in a prime number section.

He gives a £3 prize if the dart lands in a square

number section.

During the fête, 100 people play the game.

a Do you expect Derry to make a profi t?

Assume there is an equal probability of hitting any number on the board. [3 marks]

b Suggest how Derry could change the rules to make more money. [1 mark]

20

3197

168

172

1510

512

914

11

1 18

413

6

EAO2

D

E

AO2

FUNC

TIONALFUNC

TION

AL

1 4 7 8 2 3 5 6 9

117Chapter 7 Summary

Chapter summaryWhat is probability? G

Probability is about measuring the likelihood that something might happen.

The probability scale G

Probability uses numbers and words to describe the chance that an event will happen.

It is measured on a probability scale from 0 to 1.

Writing outcomes G F

To write a list of outcomes, work systematically to make sure that you don’t miss any out.

Working out the probability F

The probability of an event happening is

Probability � number of successful outcomes _____________________________ total number of possible outcomes

Remember that some events can happen in more than one way.

Calculating the probability an event does notnot happen E

You can calculate the probability that the event will not happen by using this fact

(� Probability that an event

will not happen ) � 1 � (� Probability that an event

will happen

)

Mutually exclusive events D

Mutually exclusive events cannot happen at the same time.

For any two events, A and B, which are mutually exclusive

P(A or B) � P(A) � P(B)

Chapter 8 Probability (2) for foundation sets contains8.1 Frequency diagrams

8.2 Sample space diagrams

8.3 Two-way tables

8.4 Expectation

8.5 Relative frequency

8.6 Independent events