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Probability

5Objectives

This chapter will show you how to• work out the probability of an event not happening E

• identify mutually exclusive events D

• understand and use two-way tables E D

• predict the number of times an event is likely to happen D

• calculate relative frequencies and estimate probabilities C

• calculate the probability of two independent events happening at the same time C

• draw and use tree diagrams B

This chapter is about predicting the chance of things happening.

Jelly beans come in 60 different fl avours! If there are 65 beans in a bag, what is the chance of picking your favourite?

Before you start this chapter1 This is the list of

vegetables available in a school canteen.

Bryoni chooses two different types of vegetables.List all the possible combinations of vegetables that Bryoni could choose.

2 Here is a probability scale with arrows.

Work out the probability of each of these spinners landing on red.

Which arrow should be labelled with which spinner?

a b c d e

3 Sam rolls a fair six-sided dice. Work out the probability of

a rolling a 1

b rolling a number less than 7

c rolling an even number

d rolling a 12.

4 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.

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Today’svegetables

BroccoliCarrotsPeas

Sweetcorn You can write probabilities using this notation:P(1) P(7) P(even)

74 Probability

Exercise 5A1 Hamish is learning to play golf.

The probability that he hits the ball in the right direction is 1 __ 10 .What is the probability that his next shot

a goes in the right direction

b does not go in the right direction?

5.1Objectives

Work out the probability of an event not happeningwhen 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.

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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 2 _ 5 4 Copy and complete this table.

Fraction Decimal Percentage

1 __ 10

50%

3 _ 4

0.8

Calculating the probability an event does not 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 ) 5 1 (� probability that the event

will happen

)

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 1

a 48 ___ 52 = 12 ___ 13

b 0.75

1 4 __ 52 5 52

__ 52 4 __ 52 5 48

__ 52

1 0.25 5 0.75

E

Keywordsprobability, event, not

Probability that an event does not happen

E

Probabilities can be written as fractions, decimals or percentages.

When rolling a dice, the probability of getting a 4 is 1 _ 6 . So the probability of not getting a 4 is 1 1 _ 6 5 _ 6 .

HELP Section 2.4

L

755.1 Probability that an event does not happen

2 The probability of picking a King from a pack of cards is 1 __ 13 .a What is the probability of picking ‘not a King’?b What is the probability of picking a picture card?c What is the probability of picking ‘not a picture card’?

3 The probability that Hazel misses her bus is 0.05.What is the probability that Hazel catches her bus?

4 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 ofa not winning a £5 prize b not winning a £10 prize.

5 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 spinnera does not land on 1 b does not land on blue?

Give each answer as a fraction, a decimal and a percentage.

6 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 spinnera does not land on 1b does not land on blue?

7 Alan buys a special spinner.The spinner has sections numbered 1 to 5.The probabilities of different scores are listed in the table.Work out the probability ofa not spinning a 1b not spinning a 3c 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.

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 that Lee does not pick a 10p coin is 2 _ 5 .How many 10p coins does Lee have in his pocket?

1

3

2

1

3

2

A picture card is a Jack, Queen or King.

3

Catching the bus means not missing the bus.

% means ‘out of 100’

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

Number 1 2 3 4

Probability 0.3 0.2 0.25 0.25

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76 Probability

Objectives 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.

D

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) 5 P(A) P(B)

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

You can write this as P(2) 5 1 _ 6

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

Add together the probabilities of all the possible outcomes.

P(1) P(2) P(3) P(4) P(5) P(6) 5 1 _ 6 1 _ 6 1 _ 6 1 _ 6 1 _ 6 1 _ 6 5 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 5 1. This is because you are certain to get either ‘2’ or ‘not 2’.

5.2 Mutually exclusive events Keywordsmutually exclusive, or, add, certain

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

b This spinner has four sections numbered 5 to 8.The table shows the probability of the spinnerlanding 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 2

765

8

a P(5) = 1 __ 6 , P(6) = 1 __ 6

P(5 or 6) = 1 __ 6 + 1 __ 6 = 2 __ 6 = 1 __ 3

b P(8) = 1 – 0.2 – 0.2 – 0.2 = 0.4

P(5) is short for ‘the probability of getting a 5’.

The events ‘landing on 5’ and ‘landing on 6’ are mutually exclusive so add the probabilities together.

Subtract P(5), P(6) and P(7) from 1.

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

775.2 Mutually exclusive events

Exercise 5B1 A box of chocolates contains 15 identical looking 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 chocolate 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.

Biscuit Probability

digestive 0.4

wafer

cookie 0.15

ginger 0.25

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.

Cosmetic Probability

eyeliner 0.3

lipgloss 0.3

eyeshadow

blusher

What is the probability that the cosmetic is an eyeshadow?

4 A bag contains 36 marbles of three different colours, red (R), blue (B) and yellow (Y).

P(R) 5 5 __ 12 P(Y) 5 1 _ 4

a Work out the probability of picking a blue marble.b Work out the number of marbles of each colour in the bag.

5 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 into the bag?

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78 Probability

The two-way table shows the number of doors and the number of windows in each offi ce in a block.

Number of doors Total

1 2 3

Numberof

windows

1 3 0 0 3

2 7 3 0 10

3 5 5 3 13

4 1 1 2 4

Total 16 9 5 30

a How many offi ces have three doors?

b One offi ce is chosen at random. What is the probability that it hasi three doors ii two doors and three windows?

c An offi ce with two windows is selected. What is the probability that it will have two doors?

Example 3

a 5

b i P(3D) = 5 ___ 30

ii P(2D & 3W) = 5 ___ 30

c P(2D) = 3 ___ 10

5 offi ces have 3 doors. There are 30 offi ces in total.

5 offi ces have 2 doors and 3 windows. There are 30 offi ces in total.

7 3 0 5 10, so 10 offi ces have 2 windows. Out of these 10 offi ces, 3 have 2 doors.

0 0 3 2 5 5

Keywordstwo-way table

5.3 Two-way tables

Objectives Understand and use

two-way tables

Skills check1 There are 12 girls and 18 boys in a class. A pupil is selected at random.

What is the probability that the pupil selected is a girl?2 Copy and complete the following.

a 8 6 5 20 b 10 3 5 27

Probability from two-way tablesA two-way table shows two or more sets of data at the same time.

You can work out probabilities from a two-way table.

Why learn this?A two-way table can

show data in two categories, for example

breed and gender of cat.

E D

E

Including the totals helps with checking your working.

795.3 Two-way tables

Exercise 5C1 The two-way table shows the number of doors and the number of windows in each

room in a mansion.

Number of doors Total

1 2 3 4

Numberof

windows

1 6 1 0 0 7

2 2 3 2 0 7

3 2 4 0 1 7

4 5 4 1 1 11

5 0 2 3 3 8

Total 15 14 6 5 40

a How many rooms are there in the mansion?b How many of the rooms have one door?c How many of the rooms have one window?d One room is chosen at random.

What is the probability that the room i has one door ii has one door and three windows iii has the same number of doors as windows?

2 The table shows the number of Year 10 students at Brightspark High School who do or don’t have part-time jobs.One student is chosen at random.What is the probability that this studenta is a boy who has a part-time jobb is a girl who has a part-time jobc doesn’t have a part-time job?

3 The table shows the age and sex of a sample of 50 pupils in a school.

Age in years Total

11 12 13 14 15 16

Boys 3 3 6 3 5 4 24

Girls 3 5 3 5 4 6 26

Total 6 8 9 8 9 10 50

A pupil is chosen at random.What is the probability that this pupila is a 13-year-old boyb is a 16-year-old girlc is 14 years oldd is at least 14 years old e is not 11 years old?

There are 1000 pupils in the school altogether. How many of these are likely to bef 11 years oldg girls?

Year 10 students Total

Job No job

Boys 62 41 103

Girls 58 39 97

Total 120 80 200

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80 Probability

4 The table shows the number of walkers on a trek and whether they are wearing trainers or walking boots.

Wearing trainers

Wearing walking boots

Total

Boys 3 21 24

Girls 17 9 26

Total 20 30 50

a A walker is chosen at random. What is the probability that they are wearing walking boots?

b A boy is chosen at random.What is the probability that he is wearing trainers?

c A walker wearing trainers is chosen at random.What is the probability that it is a girl?

5 The table shows the history exam results of 50 students.

A student is chosen at random.

What is the probability that the studenta has passed the examb is malec is a male who has passed the exam?

6 Professor Newton has spilled tea on the table that shows the test results of his science class.Use Professor Newton’s notes below to work out the numbers underneath the tea stains.

7 The tables show the driving test results of Ace Driving School and UCan Driving School.

Ace

Pass Fail Total

UCan

Pass Fail Total

Male 12 4 16 Male 3 3 6

Female 10 4 14 Female 9 0 9

Total 22 8 30 Total 12 3 15

a A learner driver is chosen at random from Ace Driving School.What is the probability that they passed their driving test?

b A learner driver is chosen at random from UCan Driving School.What is the probability that they passed their driving test?

c One learner from one of the driving schools is chosen at random to be interviewed by local radio. What is the probability that the learner is from Ace Driving School and failed their test?

Pass Fail Total

Male 13 15 28

Female 14 8 22

Total 27 23 50

Pass Fail

Male

Female

Altogether 60 students took the test.

The probability that a student failed is 29 __ 60 .

The probability that the student is male is 1 _ 2 .

The probability that the student failed and is male is 1 _ 4 .

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815.4 Expectation

Exercise 5D1 A fair coin is fl ipped 500 times.

How many times would you expect to get ‘heads’?

2 A fair dice is rolled 60 times.How many times would you expect it to land on 2?

3 In an experiment, a card is drawn at random from a normal pack of playing cards.This is done 520 times.How many times would you expect to geta a red card b a heartc a King d the King of hearts?

In a game a fair six-sided dice is rolled 30 times.

a How many 6s would you expect to get?

b How many even numbers would you expect to get?

Example 4

a 1 __ 6 × 30 = 5

b 3 __ 6 × 30 = 15

P(6) 5 1 _ 6 , 1

_ 6 30 5 30 6 5 5

P(even) 5 3 _ 6 5 1 _ 2 , 1

_ 2 30 5 30 2 5 15

Skills check1 Alice rolls a fair six-sided dice.

What is the probability that she rollsa a 2 b an odd numberc a number less than 3?

2 Work outa 1 _ 2 40 b 1 _ 3 15 c 2 _ 5 30

5.4 Expectation

Skills check

LObjectives

D Predict the likely number of successful events given the probability of any outcome and the number of trials or

experiments

Why learn this?

Knowing the expected number of 6s in a number

of rolls could help you work out if a dice is fair.

The number of times an event is likely to happenSometimes you will want to know the number of times an event is likely to happen.

You can work out an estimate of the frequency using this formula.

Expected frequency 5 probability of the event happening number of trials

Predict the likely number of successful events given

Keywordslikely, estimate, trial

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82 Probability

4 Asif buys 60 scratch-cards. Each scratch-card costs £1.The probability of winning the £20 prize with each scratch-card is 1 __ 30 .a How many times is Asif likely to win?b How much money is Asif likely to win?c Overall, how much money is Asif likely to lose?

5 In a bag there are 15 red, 5 blue, 5 green and 5 orange counters.Moira takes a counter at random from the bag, notes the colour, then puts the counter back in the bag. She does this 1500 times.How many times would you expect her to take a blue counter from the bag?

6 The probability that a slot machine pays out its £10 jackpot is 1 __ 80 .The rest of the time it pays out nothing.Jimmy plays the slot machine 400 times.a How many times is Jimmy likely to win?b How much money is Jimmy likely to win?

Each game costs 20p to play.c How much does it cost Jimmy to play the 400 games?d Is Jimmy likely to make a profi t? Give a reason for your answer.

7 In a bag there are 20 counters. Ten of the counters are red, fi ve are blue, four are green and one is gold. A counter is taken at random from the bag then replaced. This is done 300 times.How many times would you expect to geta a gold counter b a red counterc not a blue counter d a white countere a blue or a green counter f not a red or a blue counter?

8 At a summer fête, Alun runs a charity ‘Wheel of fortune’ game.He charges £1 to spin the wheel.If the arrow lands on a square number he gives a prize of £2.Altogether 200 people play the game.How much money would you expect Alun to make for charity?

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TIONALFUNC

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5.5 Relative frequency

Why learn this?

Estimating probabilities from real data on asteroids can

help predict future asteroid collisions with the Earth.

Skills check1 Write each of these fractions as decimals.

a 3 __ 10 b 17 ___ 100 c 9 __ 20 d 8 __ 25

2 Work out a 0.2 100 b 0.3 200 c 0.6 700

ObjectivesC Estimate probabilities

from experimental data

Keywordstheoretical probability, experimental probability, estimated probability, relative frequency, successful trials, expect

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835.5 Relative frequency

Calculating relative frequencyFor a fair dice, the theoretical probability of getting a 3 is 1 _ 6 .

For some events, you don’t know the theoretical probability. For example, when you drop a drawing pin, what is the probability that it lands ‘point up’?

You could carry out an experiment. Drop the drawing pin many times and record the number of times it lands point up. Then work out the experimental or estimated probability. This estimated probability is called the relative frequency.

Relative frequency 5 number of successful trials _______________________ total number of trials

The theory (or idea) is that there are 6 possible outcomes and they are all equally likely, so each has probability 1 _ 6 .

As the number of trials increases, the relative frequency approaches the theoretical probability.

Anil carries out an experiment. He drops a drawing pin and records the number of times it lands ‘point up’.Here are his results at different stages of his 2000 trials.

a Calculate the relative frequency at each stage of the testing.

b Estimate the probability of this drawing pin landing ‘point up’.c 15 000 of these drawing pins are dropped. How many do you expect to land

‘point up’?d Draw a graph of number of trials against relative frequency to illustrate the results.

Example 5

b 0.6

c 0.6 × 15 000 = 9000

a Relative frequency

82 _____ 100 = 0.82

101 _____ 200 = 0.505

326 _____ 500 = 0.652

586 _______ 1000 = 0.586

882 _______ 1500 = 0.588

1194 _______ 2000 = 0.597

Use the formularelative frequency 5 number of successful trials ______________________

total number of trials

As more trials are carried out, the probability seems to be getting closer to 0.6.

The expected number is calculated using the formulaexpected 5 probability

number

number of event of trials

Number of times pin is dropped

Number of times pin lands ‘point up’

Relative frequency

100 82

200 101

500 326

1000 586

1500 882

2000 1194

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84 Probability

Plotting a graph of number of trials against relative frequency shows that as the number of trials increases the experimental probability approaches the theoretical probability (0.6 in this example).

Number of trials

Relative frequency of a drawing pin landing ‘point up’

Rel

ativ

e fr

equ

ency

0

0.40.50.60.70.80.91

10020030040050060070080090010001100120013001400150016001700180019002000

d

Exercise 5E1 A bag contains 50 coloured discs. The discs are either blue or red. Jake conducts an

experiment to see if he can work out how many of each colour there are.He takes out a disc, records its colour then replaces it in the bag.He keeps a tally of how many of each colour there are after different numbers of trials.The table shows his results.

Number of trials

Number of blue discs

Relative frequency (blue)

Number of red discs

Relative frequency (red)

20 15 5 50 29 21 100 56 44 200 124 76 500 341 159 750 480 2701000 631 3691500 996 5042000 1316 684

a Calculate the relative frequency for each colour at each stage of the experiment.

b Estimate the theoretical probability of obtaining i a blue disc ii a red disc.

c Work out how many of each colour there are in the bag.d Draw a graph of number of trials against relative

frequency to illustrate your results.Plot the graphs for blue and red discs on the same axes.

2 200 drivers in Swansea were asked if they had ever parked their car on double yellow lines.47 answered ‘yes’.a What is the relative frequency of ‘yes’ answers?b There are 230 000 drivers in Swansea.

How many of these do you estimate will have parked their car on double yellow lines?

Use a horizontal scale as in Example 5 and a vertical scale of 1 cm for 0.1 with the vertical axis going from 0 to 1.

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855.5 Relative frequency

3 Salib thinks his dice is biased as he never gets a 6 when he wants to.To test it, he rolls the dice and records the number of 6s he gets.The table shows his results.

Number of rolls 20 50 100 150 200 500

Number of sixes 1 11 14 24 32 84

Relative frequency

a Calculate the relative frequency of scoring a 6 at each stage of Salib’s experiment.

b What is the theoretical probability of rolling a 6 with a fair dice?

c Do you think that Salib’s dice is biased? Explain your answer.

d Salib rolls the dice 1200 times. How many 6s do you expect him to get?

4 Maleek thinks her dice is biased. To test this theory she rolls the dice 200 times and records the scores. Her results are shown in the table below.

Score on the dice 1 2 3 4 5 6

Frequency 35 22 25 27 51 40

Relative frequency

a Calculate the relative frequency of rolling each number.

b What is the theoretical probability of rolling each number on a fair dice?

c Do you think Maleek’s dice is fair? Explain your answer.

5 George and Zoe each carry out an experiment with the same four-sided spinner.

The tables show their results.

George’s results Zoe’s results

Number on spinner 1 2 3 4 Number on spinner 1 2 3 4

Frequency 3 14 10 13 Frequency 45 53 48 54

George thinks the spinner is biased. Zoe thinks the spinner is fair.Who is correct? Explain your answer.

6 Peter wants to test if a spinner is biased.

The spinner has five equal sections labelled 1, 2, 3, 4, 5.Peter spins the spinner 20 times. Here are his results.

2 1 3 1 5 5 1 4 3 54 2 1 5 1 4 3 1 2 1

a Copy and complete the relative frequency table.

Number 1 2 3 4 5

Relative frequency

b Peter thinks that the spinner is biased.Write down the number you think the spinner is biased towards.Explain your answer.

c What could Peter do to make sure his results are more reliable?

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86 Probability

5.6 Independent events

Why learn this?Understanding independent events

gives you a better idea of everyday probabilities. The numbers 1, 2, 3, 4,

5 and 6 are just as likely to come up together on the lottery as any other

set of six numbers between 1 and 49.

Skills check1 A fair dice is rolled once.

What is the probability of getting a number less than 3?

2 Work out a 1 _ 2 1 _ 3 b 1 _ 4 3 _ 5 c 2 _ 3 5 _ 9

ObjectivesCalculate the probability of two independent

events happening at the same timeCalculate the probability of two independent

Keywordsindependent, multiply, and

Independent eventsTwo events are independent if the outcome of one does not affect the outcome of the other. When you roll two dice at the same time, the number you get on one dice does not affect the number you get on the other. To calculate the probability of two independent events happening at the same time, you multiply the individual probabilities.

When A and B are independent events P(A and B) 5 P(A) P(B)

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C a A fair dice is rolled twice. What is the probability of getting two 6s?

b A coin and a dice are thrown together. What is the probability of getting a head and a 1?

Example 6

a P(6) = 1 __ 6

P(6 and another 6)

= 1 __ 6 × 1 __ 6 = 1 ___ 36

b P(H) = 1 __ 2 , P(1) = 1 __ 6

P(H and 1) = 1 __ 2 × 1 __ 6 = 1 ___ 12

The individual probability of getting a 6.

The combined probability. Here ‘and’ means multiply.

The combined probability. Here ‘and’ means multiply.

This is a sample space diagram. It lists all the possible outcomes when a dice is rolled twice. There are 36 possible outcomes and only one with two 6s, which confi rms that the probability 5 1 __ 36 .

1 2 3 4 5 61 1,1 1,2 1,3 1,4 1,5 1,6

2 2,1 2,2 2,3 2,4 2,5 2,6

3 3,1 3,2 3,3 3,4 3,5 3,6

4 4,1 4,2 4,3 4,4 4,5 4,6

5 5,1 5,2 5,3 5,4 5,5 5,6

6 6,1 6,2 6,3 6,4 6,5 6,6

Total number of outcomes total number of outcomes for event A total number of outcomes for event B

A sample space diagram is one way of showing all the possible outcomes of an experiment.

The individual probabilities of getting a head and a 1.

875.6 Independent events

Exercise 5F1 A fair six-sided dice is rolled twice.

a What is the probability of getting a 5 and then a 4?

b What is the probability of getting two 3s?

2 When Lynn and Sally go to the shop, the probability that Lynn buys a bag of crisps is 1 _ 3 . The probability that she buys a muesli bar is 1 _ 4 . The probability that Sally buys a bag of crisps is 1 _ 2 . The probability that she buys a muesli bar is 1 _ 4 . The girls choose independently of each other.Calculate the probability thata both girls buy a bag of crispsb both girls buy a muesli barc Lynn buys a bag of crisps and Sally buys a muesli bar.

3 A fair four-sided dice, numbered 1 to 4, and a fair 10-sided dice, numbered 1 to 10, are rolled together. Their scores are added.What is the probability of getting a total of 14?

4 In roulette, a ball is spun around a wheel. Players bet on where the ball will land.A 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.

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 two £10 bets. What is the probability that she wins

a £720 b £60?

5 Joe spins a spinner with fi ve equal sectors numbered from 1 to 5.Sarah rolls a fair dice numbered from 1 to 6. How will a total of 5 arise?

You will need to add some probabilities.Work out the probability that

a they both obtain a 3

b the total of their scores is 2 c the total of their scores is 5

d Sarah’s score is twice Joe’s score e they both obtain an even number.

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

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L5.7 Tree diagrams

Why learn this?

If you know the probability of your team winning, losing

or drawing matches, a tree diagram is an easy way to

see the possible outcomes of future matches.

Skills check1 Write true or false for each of these.

a 1 _ 3 1 _ 3 5 1 _ 6 b 1 _ 4 1 _ 5 5 1 __ 20 c 3 _ 4 1 _ 2 5 3 _ 8 d 4 _ 5 2 _ 7 5 6 __ 12

2 Work out a 0.2 0.2 b 0.4 0.8 c 0.3 0.3 0.5

ObjectivesB Use and understand tree

diagrams in simple contexts

Keywordstree diagram, combined events

Drawing tree diagramsA tree diagram can show all the possible outcomes of two or more combined events, and their probabilities.

Imagine you have a bag of discs, some are red and some are blue.You pick out a disc, record its colour then replace it in the bag.Then you pick out a second disc.

You can show all the possible results by drawing a tree diagram.

Example 7

There is only one way of obtaining the outcomes ‘red, red’ or ‘blue, blue’ (usually written ‘both red’ or ‘both blue’).

There are two ways of obtaining the outcome ‘one of each colour’ because either ‘red, blue’ or ‘blue, red’ fi t this description.

red

blue

red

blue

red

blue

RR

Outcomes2nd pick1st pick

RB

BR

BB

Notice that the events (red and blue) are written at the end of the branches.

It is useful to list the outcomes to the right of the tree diagram, in line with the branches.

895.7 Tree diagrams

Exercise 5GDraw a tree diagram to show all the possible outcomes in each question.For each one, think about the possible outcomes if you carried out the experiment.Label the branches with the appropriate outcomes.

1 A card is selected at random from a normal pack of 52 cards.Its colour is recorded then it is replaced.A second card is selected.

2 A 2p coin is fl ipped then a 10p coin is fl ipped.

3 On a certain day Tom takes his driving test. He either passes or fails.The next day his friend Marcus takes his driving test. He either passes or fails.

4 A bag contains red, blue and yellow discs.Tiffany picks out a disc, records its colour then replaces it.She then picks out a second disc.

5 For breakfast today I have a choice of cereal, toast or porridge.Tomorrow I can choose between the same three items.

6 A box contains black and white balls.Anisha picks out a ball, records its colour then replaces it.She picks out a second ball, records its colour then replaces it.She picks out a third ball.

Calculations using tree diagramsLook again at Example 7.

The bag contained red and blue discs.

Suppose there are 3 red discs and 2 blue discs.

The probability of picking a red on the fi rst pick is 3 _ 5 and picking a blue is 2 _ 5 .

Since the discs are replaced, these probabilities always remain the same.

You can write the probabilities on the appropriate branches, so the completed tree diagram looks like this.

red35

blue

red

blue

red

blue

RR �

Outcomes2nd pick1st pick

RB

BR

BB

25

35

35

35 � 9

25

�35

25 � 6

25

�25

35 � 6

25

�25

25 � 4

25

25

35

25

The events are independent so you can multiply the probabilities on the branches to calculate the probability of each fi nal outcome.

The sum of the probabilities of the fi nal outcomes is 9 __ 25 6 __ 25 6 __ 25 4 __ 25 5 25

__ 25 5 1

This is always the case since the fi nal outcomes represent everything that can possibly happen.

There are always 5 discs in the bag, 3 red and 2 blue.

B

90 Probability

Exercise 5H1 a David fl ips a coin twice.

Copy and complete the tree diagram to show all the possible outcomes and their probabilities.

b Work out the probability that David fl ips i two heads ii a head and then a tail iii a tail and then a head iv two tails.

2 At a fi shing lake the probability of catching a trout is 0.2. The probability of catching a carp is 0.8. Adam and Tabitha catch one fi sh each.

a Copy and complete the tree diagram to show all the possible outcomes.

b Work out the probability that i they both catch a trout ii they both catch a carp iii Adam catches a trout and Tabitha catches a carp iv one catches a trout and the other a carp.

B Use the tree diagram above to answer these questions.

a What is the probability that the two discs are the same colour?

b What is the probability that the two discs are different colours?

c What is the probability of picking at least one red disc?

Example 8

a P(same colour) = P(RR or BB)

= P(RR) + P(BB)

= 9 ___ 25 + 4 ___ 25

= 13 ___ 25

b P(different colours) = P(RB or BR)

= P(RB) + P(BR)

= 6 ___ 25 + 6 ___ 25

= 12 ___ 25

c P(at least one red) = 1 − P(two blue)

= 1 − 4 ___ 25

= 25 ___ 25 − 4 ___ 25

= 21 ___ 25

RR and BB are mutually exclusive events.

You can also do this usingP(at least one red) 5 P(RR or RB or BR) 5 P(RR) P(RB) P(BR) 5 9 __ 25 6 __ 25 6 __ 25

5 21 __ 25

… but the fi rst method is much quicker.

You can also do this usingP(different colours)

5 1 P(same colour)5 1 13

__ 25 5 12

__ 25

H12

T

H

T

H

T

2nd flip1st flip

12

12

trout0.2

carp

trout

carp

trout

carp

TabithaAdam

B

915.7 Tree diagrams

3 Zina has a pack of cards. She shuffles the pack then takes a card at random. She replaces the card, shuffles the pack, then takes another card.

a What is the probability that the first card Zina takes is a Jack?

b What is the probability that the first card Zina takes isn’t a Jack?

c Work out the probability that i both cards are a Jack ii neither card is a Jack iii the first card isn’t a Jack, but the second card is.

4 Aleksy has a 5p coin, a 10p coin and a 50p coin. He spins the coins at the same time and records whether they land heads up or tails up.

a Copy and complete the tree diagram to show all the possible outcomes.

H

b Calculate the probability that the 5p and the 10p coins land heads up and the 50p coin lands tails up.

c Calculate the probability that all coins land heads up.

d Calculate the probability that one coin lands heads up and two coins land tails up.

e Calculate the probability that at least one coin lands tails up.

5 Siobhan has two maths tests next week. She estimates the probability of her passing the geometry test is 0.8, but the probability of her passing the statistics test is only 0.2.

a Copy and complete the tree diagram to show all the possible outcomes.

b Work out the probability that Siobhan only passes the geometry test.

c Work out the probability that Siobhan passes neither test.

d Work out the probability that Siobhan passes only one test.

6 On Errol’s way to work there are two sets of traffic lights. The probability that the first set is green is 1 _ 4 . The probability that the second set is green is 1 _ 3 .

a Copy and complete the tree diagram to show all possible outcomes.

b Work out the probability that Errol is stopped at both sets of lights.

c Work out the probability that he is stopped at exactly one set of lights.

d Work out the probability that he is not stopped at either set of lights.

B

pass

green

AO2

B

92 Probability

7 A fair dice is rolled twice. You are only interested in whether a 6 is rolled.

a Copy and complete the tree diagram to show all possible outcomes.

b Work out the probability of getting i two 6s ii no 6s iii a 6 followed by not a 6.

8 Luke, Matthew and Sophie all work for the same company.The probability that Luke is late for work is 0.3.The probability that Matthew is late for work is 0.4.The probability that Sophie is late for work is 0.2.

a Copy and complete the tree diagram to show all possible outcomes.

b On any day, what is the probability that i all three are late for work ii all three are on time iii Sophie is late but Luke and Matthew are on time iv at least one of them is late?

9 Tabitha, Matilda and Oscar are taking an English exam.The probability that Tabitha passes is 0.9.The probability that Matilda passes is 0.7.The probability that Oscar passes is 0.6.

a Copy and complete the tree diagram to show all possible outcomes.

b What is the probability that i all three pass ii Tabitha and Oscar pass but Matilda fails iii all three fail iv at least one of them passes v any two of them pass?

Review exercise1 Colleen has these two four-sided spinners.

She spins the spinners at the same time.She works out the difference between the numbers that the spinners land on.

423

142

3

1

a Copy and complete the sample space diagram to show all the possible differences.

1 2 3 4

1 0 1 2

2 1

3 2

4 [1 mark]

b What is the probability that the difference is 0? [1 mark]

c What is the probability that the difference is an odd number? [1 mark]

6

late

pass

E

AO2

B

93Chapter 5 Review exercise

2 Donna designs a game of chance to help raise money at her school fête.Donna uses a normal dartboard with sections labelled 1 to 20.It costs £1.50 to throw one dart and Donna gives a prize of £10 if the dart lands in the number 20 section and £5 if the dart lands in the number 6 or number 11 sections.Altogether 240 people play the game during the day.Assume there is an equal probability of hitting any number on the board.How much profit should Donna expect to make? [4 marks]

3 As part of a council education report, 300 students in Dudley were asked if they had a job as well as being a student.78 answered ‘yes’ to the question.There are approximately 30 000 students in Dudley.Approximately how many of these do you estimate have a job? [2 marks]

4 A spinner has sections coloured red (R), yellow (Y) and brown (B). The spinner is spun 15 times and the colour it lands on each time is recorded.

a These are the results of the first 15 spins.

R Y R B B Y R B Y R B B B Y R

Copy and complete the relative frequency table.

Colour Red (R) Yellow (Y) Brown (B)Relative frequency [1 mark]

b This table shows the relative frequencies after the spinner has been spun 150 times.

Colour Red (R) Yellow (Y) Brown (B)

Relative frequency 48 ___ 150 45

___ 150 57 ___ 150

Which of the two relative frequencies for brown gives the better estimate of the probability of the spinner landing on brown?Give a reason for your answer. [1 mark]

5 Katarina has a dartboard with a blue section (B) and a green section (G).She throws a dart at the board 500 times.

a These are the results of her first 20 throws.

G G B B B G B B G B B B G G B G B G B B

Work out the relative frequency of blue after 20 throws. [1 mark]

b The table shows the relative frequency of blue after different numbers of throws. How many times did Katarina’s dart land in blue after 300 throws?

[2 marks]

20

3197

168

172

1510

512

914

11

1 18

413

6

Number of throws

Relative frequency

50 0.63100 0.64300 0.66500 0.67

AO3

D

AO2

C

94 Probability

6 A catering company produces hot meals for parties.It offers three main courses: chicken (C), beef (B) or vegetarian (V).It offers two types of potatoes: roast (R) or new (N).The company uses previous data to estimate the numbers of different types of meals that need to be cooked.Previous data shows that the probability of a person choosing chicken is 1 _ 2 , beef 1 _ 3 and vegetarian 1 _ 6 .The probability of a person choosing roast potatoes is 1 _ 4 and new potatoes is 3 _ 4 .

a Copy and complete the tree diagram to show all the possible outcomes.

[2 marks]

b Work out the probability that a person chooses chicken and new potatoes. [2 marks]

c Work out the probability that a person chooses meat and roast potatoes. [3 marks]

d At the next party, 120 guests are expected.Estimate the number of ‘vegetarian and new potato’ meals the company needs to cook. [2 marks]

7 A fair six-sided dice, numbered from 1 to 6, is thrown three times.What is the probability of getting

a three 6s b two 6s

c at least two 6s d no 6s

e at least one 6?

[2 marks each part]

C

B

V

RN

RN

RN

12

14

6

In this chapter you have learned how to

• work out the probability of an event not happening when you know the probability that it does happen E

• understand and use two-way tables E D

• understand and use the fact that the sum of the probabilities of all mutually exclusive outcomes is 1 D

• predict the likely numbers of successful events given the probability of any outcome and the number of trials or experiments D

• estimate probabilities from experimental data C

• calculate the probability of two independent events happening at the same time C

• use and understand tree diagrams in simple contexts B

Chapter summary

AO2

B