5 Before you start this chapter1 A fair six-sided dice is rolled.

Work out the probability ofa rolling a 1 b rolling a number less than 7c rolling an even number d rolling a 12.

2 One letter is chosen at random from the wordP 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.

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

4 A fair three-sided spinner has sections labelled 1, 2 and 3. The spinner is spun and a fair six-sided dice is rolled at the same time. The two scores are added to give a total score.a Copy and complete the sample

space diagram to show all the possible total scores.

Dice1 1 2 3 4 5 6

Spin

ner 1

2

3 9

b What is the probability that the total score is 6?c What is the probability that the total score is

more than 6?

5 Copy and complete this table.

1

3

2

321

Probabilities can be written as fractions, decimals or percentages.

Fraction Decimal Percentage 1 __ 10

50%

3 _ 4

0.8

HELP Chapter 2

Probability Objectives

This chapter will show you how to• identify mutually exclusive events 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 for independent events B A

• draw and use tree diagrams for more complex problems A*

This chapter is about predicting the chance of things happening.

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

715.1 Mutually exclusive events

Exercise 5A1 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 has a toffee or a solid centre

b has a toffee or a soft centre

c has a toffee or a nut centre

d doesn’t have a toffee or a nut centre

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

5.1 Mutually exclusive events

Skills check1 Work out a 1 _ 5 1 1 _ 5 b 1 2 1 _ 4 c 1 2 2 _ 3 2 Work out a 0.4 1 0.3 b 1 2 0.82 c 0.4 4 2

Why learn this?It could help you win

if you remember what cards have already

been played.

ObjectivesD Understand and use the fact that the sum of the

probabilities of all mutually exclusive outcomes is 1

Mutually exclusive eventsMutually exclusive events cannot happen at the same time. When you roll a dice you cannot get a 1 and a 2 at the same time.

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

To calculate the probability of 1 or 2 you add the probabilities.

P(1 or 2) 5 1 _ 6 1 1 _ 6 5 2 _ 6 (or 1 _ 3 )

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

P(A or B) 5 P(A) 1 P(B)

Keywordsmutually exclusive, or, certain, add

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

This spinner has four sections numbered 5 to 8.

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

What is the probability that the spinner lands on 8?

Example 1

765

8

Number 5 6 7 8

Probability 0.2 0.2 0.2 ?

P(8) = 1 – 0.2 – 0.2 – 0.2 = 0.4 The events 5, 6, 7 and 8 are mutually exclusive, so P(5) 1 P(6) 1 P(7) 1 P(8) 5 1, as you are certain to get one of them.

D

D

72 Probability

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?

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 in the bag?

Biscuit Probability

digestive 0.4

wafer

cookie 0.15

ginger 0.25

Cosmetic Probability

eyeliner 0.3

lipgloss 0.3

eyeshadow

blusher

D

L5.2 Expectation

Why learn this?Knowing the expected

number of 6s in a number of rolls could help you

work out if a dice is fair.

Skills check1 Alice rolls a fair six-sided dice.

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

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

ObjectivesD Predict the likely number of successful events given

the probability of any outcome and the number of trials or experiments

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 3 number of trials

Keywordslikely, estimate, trial

HELP Section 2.1

AO3

D

AO2

D

735.2 Expectation

Exercise 5B1 A fair dice is rolled 60 times.

How many times would you expect it to land on 2?

2 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 get

a a red card

b a heart

c a king

d the king of hearts?

3 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?

4 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 150 times.

How many times would you expect her to take a blue counter from the bag?

5 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 profit? Give a reason for your answer.

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 2

a 1 __ 6 × 30 = 5

b 3 __ 6 × 30 = 15

P(6) 5 1 _ 6 , and 1 _ 6 3 30 5 30 4 6 5 5

P(even) 5 3 _ 6 5 1 _ 2 , and 1 _ 2 3 30 5 30 4 2 5 15

D

D

AO2

D

74 Probability

6 In a bag there are 20 counters. Ten of the counters are red, five 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 neither a red nor a blue counter?

7 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?

8 This table shows the probability of selecting coloured balls from a bag.

Colour Probability

black 0.2

white 0.3

yellow ?

pink ?

a The probability of choosing yellow is four times the probability of choosing pink. Work out the missing values.

b 400 balls are taken at random from the bag, one at a time, and replaced each time. Estimate how many of these are white.

9 John bought 20 tickets for a raffle. The probability of him winning is 0.04.How many raffle tickets were sold?

FUNC

TIONALFUNC

TION

AL8

12

3456

7

D

AO2

D

AO3

D

755.3 Relative frequency

10 Brightspark School is putting on a concert.

The weather on the day of the concert is bad – frequent snow showers and very cold.

Statistics show that in bad weather some parents will not come, even though they have bought tickets.

The table shows the number of tickets sold and the probabilities that parents will come to the concert.

Distance of parents from school Number of tickets sold Probability of coming

2 miles or less 870 0.9

more than 2 miles 240 0.6

The school kitchen staff are providing refreshments for the concert.

The cost per head of providing food is estimated at 85p.

Work out the likely cost of providing food for the concert.

FUNC

TIONALFUNC

TION

AL

Skills check1 Write each of these fractions as a decimal.

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

2 Work out a 0.2 3 100 b 0.3 3 200 c 0.6 3 700

5.3 Relative frequency

ObjectivesC Estimate probabilities

from experimental data

Why learn this?Estimating probabilities from

real data on asteroids can help scientists to predict future

asteroid collisions with the Earth.

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

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 a 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.

AO3

D

76 Probability

Example 3Anil carries out an experiment to work out the probability that when he drops a drawing pin it will land ‘point up’.The table shows his results at different stages of his 2000 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

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

b What do you think is the probability of this drawing pin landing ‘point up’?

c If 15 000 of these drawing pins were dropped, how many would you expect to land ‘point up’?

d Draw a graph of number of trials against relative frequency to illustrate the results.

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

The relative frequency is calculated using the formula

relative frequency 5 number of successful trials _____________________ total number of trials

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

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

The expected number is calculated using the formula

expectednumber 5

probability of event 3

number of trials

Number of trials

Rel

ativ

e fr

equ

ency

0

0.40.50.60.70.80.9

1

100

200

300

400

500

600

700

800

90010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

00

d

C

775.3 Relative frequency

Exercise 5C1 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 of blue

Number of red discs

Relative frequency of red

20 15 5

50 29 21

100 56 44

200 124 76

500 341 159

750 480 270

1000 631 369

1500 996 504

2000 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 relative frequency against number of trials 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 130 000 drivers in Swansea. How many of these do you estimate will have parked their car on double yellow lines?

3 Maleek thinks his dice is biased, as he never gets a 6 when he wants to.To test this theory, 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 6s 1 11 14 24 32 84

Relative frequency

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

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

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

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

C

Use a horizontal scale as in Example 3 and a vertical scale of 1 cm for 0.1, with the vertical axis going from 0 to 1.

AO2

C

78 Probability

4 Hollie 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 Hollie’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?

7 A fair six-sided dice is repeatedly rolled 10 times.The number of 6s is counted for each set of 10 rolls. Here are the results.

Set of 10 rolls Number of 6s Total number of 6s Total rolls Relative frequency

1 2 2 10 0.2

2 2 4 20 0.2

3 5 9 30 0.3

4 5 14 40 0.35

5 2 16 50 0.32

6 4 20 60 0.33

7 1 21 70 0.3

8 3

9 3

10 2

a Complete the table.

b Do these results suggest that the dice is biased towards the number 6? Explain your answer.

345

12

AO2

C

AO3

C

795.4 Independent events

5.4Objectives

Calculate the probability of twoindependent events happening at the same

time

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.

C

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 3 1 _ 3 b 1 _ 4 3 3 _ 5 c 2 _ 3 3 5 _ 9

Keywordsindependent, multiply, and

Independent events

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) 3 P(B)

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 4

a P(6) = 1 __ 6

P(6 and another 6)

= 1 __ 6 × 1 __ 6 = 1 ___ 36

The individual probability of getting a 6.

The events are independent, so P(A and B) 5 P(A) 3 P(B)

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

1 2 3 4 5 6

1 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 5 total number of outcomes for event A 3 total number of outcomes for event B

C

80 Probability

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

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

The individual probabilities of getting a head and a 1.

P(A and B) 5 P(A) 3 P(B)

Exercise 5D1 An ordinary coin is flipped twice. What is the probability of getting two heads?

2 Box A contains 10 identical looking chocolates. Four of them are truffles.Box B contains 20 identical looking chocolates. Ten of them are truffles.Carol takes one chocolate from each box.What is the probability that she gets two truffles?

3 When Lynn and Sally go to the shop, the probability of Lynn buying a bag of crisps is 1 _ 3 , and a muesli bar is 1 _ 4 . The probability of Sally buying a bag of crisps is 1 _ 2 , and a muesli bar is 1 _ 4 . The girls choose independently of each other.Calculate the probability that

a both girls buy a bag of crisps

b both girls buy a muesli bar

c Lynn buys a bag of crisps and Sally buys a muesli bar.

4 Sam plays the National Lottery ‘Thunderball’ every week.The probability of winning a £5 prize is 0.03.The probability of winning a £10 prize is 0.009.Work out the probability that

a Sam wins £5 one week and £10 the next week

b Sam wins £5 two weeks in a row.

5 Zoe takes two exams: history and French.

The probability that she passes history is 3 _ 4 .

The probability that she passes French is 1 _ 3 .

a What is the probability that she passes both exams?

b What is the probability that she passes one exam?

6 A box contains 5 yellow and 7 green tennis balls.A ball is chosen, its colour noted, then it is replaced.A second ball is chosen.

a What is the probability that both balls are yellow?

b What is the probability that both balls are the same colour?

7 Ethan and Jerry go to the canteen for lunch.Main meals come with either chips or salad.The probability that Ethan chooses chips is 0.7.The probability that Jerry chooses chips is 0.4.

a What is the probability that they both choose chips?

b What is the probability that they both choose salad?

c What is the probability that one chooses chips and the other chooses salad?

d What do you notice about your answers to parts a, b and c?

C

CAO2

C

815.5 Tree diagrams

8 Joe spins a spinner with five equal sectors numbered from 1 to 5.Sarah rolls a fair dice numbered from 1 to 6.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 How will a total of 5 arise?You will need to add some probabilities.

d Sarah’s score is twice Joe’s score

e they both obtain an even number.

9 Bernie has an ordinary pack of 52 cards.She shuffles the pack then selects a card at random.She replaces the card, shuffles the pack again and selects another card.What is the probability that

a both cards are red

b neither card is a spade

c both cards are queens

d exactly one card is a queen?

In d, which card will be a queen, the first or the second?

5.5 Tree diagrams

Skills check

1 Write ‘true’ or ‘false’ for each of these.a 1 _ 3 3 1 _ 3 5 1 _ 6 b 1 _ 4 3 1 _ 5 5 1 __ 20 c 3 _ 4 3 1 _ 2 5 3 _ 8 d 4 _ 5 3 2 _ 7 5 6 __ 12

2 Work out a 0.2 3 0.2 b 0.4 3 0.8 c 0.3 3 0.3 3 0.5

L

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

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.

ObjectivesB Use and understand tree diagrams in simple

contextsA A* Use the ‘and’ and ‘or’ rules in tree diagrams

Keywordstree diagram, combined events

AO2

C

82 Probability

Imagine you have a bag of discs. Each disc is either red or 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 the possible results by drawing a tree diagram.

Example 5

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 both ‘red, blue’ and ‘blue, red’ fit 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 ends of the branches.

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

Suppose there are 3 red discs and 2 blue discs. The probability of picking a red on the first 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.

Example 6

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

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

The sum of the probabilities of the final outcomes is 9 __ 25 1 6 __ 25 1 6 __ 25 1 4 __ 25 5 25

__ 25 5 1 This is always the case since the final outcomes represent everything that can possibly happen.

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

B

B

835.5 Tree diagrams

Use the tree diagram in Example 6 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 7

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) 1 P(RB) 1 P(BR) 5 9 __ 25 1 6 __ 25 1 6 __ 25

5 21 __ 25

… but the first method is much quicker.

You can also do this usingP(different colours)

5 1 2 P(same colour) 5 1 2 13

__ 25 5 12

__ 25

B

Exercise 5E1 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.

pass 0.8

0.2

0.2fail

pass

fail

pass

fail

StatisticsGeometry

b Work out the probability that Siobhan passes both tests.

c Work out the probability that Siobhan passes neither test.

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

B

84 Probability

2 On Megan’s way to school there are two sets 14

13

of traffic lights. The bus driver knows that the probability of the first set being green is 1 _ 4 and the probability of the second set being green is 1 _ 3 .a Copy and complete the tree diagram to show all

the possible outcomes.b Work out the probability that the bus gets held up at both sets of traffic lights.c Work out the probability that the bus only gets held up at the second set of

traffic lights.

3 a David flips a coin twice. Complete a tree diagram to show all the possible outcomes.

b Work out the probability that David flips i two heads ii a head and then a tail iii a head and a tail in any order iv at least one head.

4 At a fishing lake the probability of catching a trout is 0.2. The probability of catching a carp is 0.8. Adam and Tabitha both catch one fish each.

a Complete a tree diagram to show all the possible outcomes.

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

5 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 at least one card is a jack.

6 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.On any day, what is the probability that

a all three are late for work

b all three are on time

c Sophie is late but Luke and Matthew are on time

d at least one of them is late?

A

AO2

B

AO2

A

855.5 Tree diagrams

7 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.What is the probability thata all three pass b Tabitha and Oscar pass but Matilda failsc all three fail d at least one of them passese any two of them pass?

8 A bag contains 10 coloured beads.4 are red, 5 are white and 1 is blue.Simon takes a bead from the bag at random, records its colour then replaces it in the bag.He then takes out another bead and records its colour.What is the probability thata the first bead is white and the second is redb both beads are bluec both beads are the same colourd neither bead is rede at least one is red?

9 Amy, Beth and Clare take their cars for an MOT.

The probabilities of their cars passing on Lights Brakes Tyres

Amy’s car 0.8 0.4 0.9

Beth’s car 0.7 0.8 0.6

Clare’s car 0.9 0.7 0.3

lights, brakes and tyres are shown in the table.

What is the probability thata all three cars pass on lightsb all three cars fail on tyresc Beth’s car passes all three testsd two of the cars pass on brakes but one car fails on brakese Amy’s car passes only one of the three tests?

10 At a fairground you can play ‘Spin and Win’.

2

34

5

1The game involves spinning a spinner with five equal sections numbered from 1 to 5.This poster on the fairground stall shows how you can win.

Spin and Win£2 a go

Spin the spinner three timesThree 5s = £15Two 5s = £10

Throughout the day, 250 people play this game.

a Estimate how much profit the stallholder will make if he charges £2 a go.

b Estimate the minimum that the stallholder should charge if he is not to make a loss.

AO2

A

AO2

A*

AO2

A*

86 Probability

Skills check1 Work out a 1 __ 15 1 7 __ 15 b 1 2 (  2 _ 7 1 4 _ 7 ) c 1 2 (  2 __ 13 1 7 __ 13 1 1 __ 13 )

2 Cancel each fraction to its lowest terms.a 9 __ 45 b 8 __ 12 c 36

__ 48

5.6 Conditional probability

LObjectivesA* Use and understand

tree diagrams with two or more events for conditional probabilities

Why learn this?It’ll help with your socks!

If you have 8 black and 8 white socks in a drawer, how

many socks do you have to pull out to be sure of

getting a pair?

Keywordswithout replacement, dependent, conditional

Tree diagrams when one outcome affects the nextIn real life, the outcome of one event often affects the outcome of the next.

When you pick a card from a pack, P(black) 5 26 __ 52 .

If you don’t replace it, the probability of picking a black card next time depends on what colour you picked the first time.

The probability of the second event is conditional on (dependent on) the outcome of the first event.

Not replacing the card changes the probability of picking a black. Mathematicians call this a ‘without replacement’ problem.

Oscar picks a card at random from a normal pack.

He does not replace it.

He then picks another card at random. Work out the probability that he picks

a two black cards

b one red and one black card.

Example 8

a

b 26 _____ 102 1 26 _____ 102 5 52 _____ 102

black

red P(BR) �

P(RB) �

red

black

black

2652

red

2551

2651

2651

2551

2652

2651

26102

2652

� �

2652

2651

26102� �

P(RR) � 2652

2551

25102

25102

� �

P(BB) � 2652

2551� �

Once a black card is picked, there are 51 cards left, and 25 of these are black.

Multiply the probabilities.

Once a black card is picked, there are 51 cards left, and 26 of these are red.

P(RB or BR) 5 P(RB) 1 P(BR)

A*

875.6 Conditional probability

Exercise 5F1 Sandra has three blue mugs and four green mugs

in her cupboard.She takes out two mugs at random.

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

b What is the probability that i Sandra takes two blue mugs ii Sandra takes one mug of each colour?

2 John has four red and two green socks in a drawer.He takes two socks out at random.

a Draw a tree diagram to show all the possible outcomes.

b What is the probability that i John takes a matching pair of socks ii John doesn’t take a matching pair of socks?

3 A tin contains 12 shortbread and 10 coconut biscuits.Brendan takes two biscuits at random and eats them.Work out the probability that

a both biscuits are coconut

b the first biscuit is shortbread and the second biscuit is coconut.

4 Tina has these cards.

2 3 4 5 6

She takes two of the cards at random.

This is just like taking the first one, then the other.

Without drawing a tree diagram, work out the probability that

a both cards are yellow

b both cards have an even number

c the first card is blue and the second card is yellow.

5 Jane has a bag containing 15 identical cubes. Eight are red, four are white and three are blue.Jane takes two cubes at random from the bag.

a Draw a tree diagram to show all the possible outcomes.

b What is the probability that i both cubes are the same colour ii the cubes are different colours iii at least one cube is white?

6 Debbie has 10 tins of food in her cupboard, but all the labels are missing.She knows that six of them contain soup and four contain baked beans.She opens three tins at random.What is the probability that she opens one tin of soup and two tins of baked beans?

A*B3

7

G

B

G

B

G

2nd mug1st mug

26

88 Probability

7 A bag contains two black and three white discs.Ali takes a disc at random from the bag, then Mahmoud takes a disc at random from the bag.Mahmoud wins if both the discs are the same colour.Who has the better chance of winning? Give a reason for your answer.

8 Ian and Steve play three sets of tennis.The probability that Ian wins the first set is 0.6.When Ian wins a set, the probability that he wins the next set is 0.7.When Steve wins a set, the probability that he wins the next set is 0.5.The first person to win two sets wins the match.

a Copy and complete the tree diagram to show all the probabilities.

Ian wins

0.6

0.4

Steve wins

Ian wins

Steve winsIan wins

Steve wins

Ian wins

Steve winsIan wins

Steve wins

2nd set 3rd set1st set

b Calculate the probability that Ian wins the match.

9 At the end of a training course students must pass a test to gain a diploma.The probability of passing first time is 0.8.Those who fail first time are allowed just one more attempt.The probability of passing the re-sit is 0.65.

Two friends, Sam and Tim, follow the training course and take the test.

What is the probability that they both gain a diploma?

10 Bag A contains 5 red and 3 blue counters. Bag B contains 2 red and 6 blue counters.

Bag BBag A

Step 1 A counter is taken, at random, from Bag A and placed in Bag B.

Step 2 A counter is taken, at random, from Bag B and placed in Bag A.

Calculate the probability that Bag A has more red counters than blue counters after these two steps.

AO2

A*

AO2

A*

A

AO2

A*

89Chapter 5 Review exercise

Review exercise1 Donna designs a game of chance to help raise money

at her school fête.Donna uses a normal dartboard with sections labelled

20

3197

168

172

1510

512

914

11

1 18413

6

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]

2 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.

Number of throws

Relative frequency

50 0.63100 0.64300 0.66500 0.67

How many times did Katarina’s dart land in blue after 300 throws? [2 marks]

3 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 number of different types of meals it will need to cook.Previous data shows that the probability of a person choosing chicken is 1 _ 2 , beef is 1 _ 3 and vegetarian is 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

C

B

V

RN

RN

RN

12

14

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]

FUNC

TIONALFUNC

TION

AL

AO3

D

AO2

C

AO2

B

90 Probability

4 Nick always has cereal or toast for breakfast. Item Probabilitytoast 1 _ 4

cereal 3 _ 4

tea 1 _ 4

coffee 5 _ 8

fruit juice 1 _ 8

He also has tea, coffee or fruit juice.

The probabilities of what he chooses forbreakfast are given in the table.a What is the probability that Nick has toast

and coffee? [2 marks]

b What is the most likely combination Nick will choose? [1 mark]

c What is the probability of the combination in part b? [2 marks]

5 For breakfast, Magda always has cereal or toast followed by orange juice or apple juice.The probability that Magda has cereal is 1 _ 3 .When Magda has cereal, the probability that she then has apple juice is 5 _ 6 .

When Magda has toast, the probability that she then has orange juice is 7 _ 8 .Work out the probability that Magda has toast and apple juice for breakfast. [3 marks]

6 Alice takes maths and further maths at sixth-form college.The probability that she will pass maths is 0.7.If she passes maths, the probability that she will pass further maths is 0.6.If she fails maths, the probability that she will pass further maths is 0.2.Work out the probability thata she passes both [2 marks]

b she only passes one [2 marks]

c she passes at least one. [2 marks]

7 Steven and Rob have these cards.Steven shuffles the cards then selects one at L I V E R P O O L random. He does not replace it.Rob then selects a card at random.Steven wins if both the cards are vowels or both the cards are consonants.Who has the better chance of winning? Explain your answer. [3 marks]

8 Rachel has a credit card but sometimes has trouble remembering the PIN.The probability that she gets it right first time is 0.8.If she gets it wrong the first time, the probability that she gets it right the second time is 0.65.If she gets it wrong twice, the probability that she gets it right the third time is 0.15.A cash machine will keep her card after three wrong attempts.What is the probability that the machine doesn’t keep Rachel’s card? [3 marks]

In this chapter you have learned how to

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

• predict the likely number 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

• use the ‘and’ and ‘or’ rules in tree diagrams A A*

• use and understand tree diagrams with two or more events for conditional probabilities A*

Chapter summary

AO2

A

AO2

A*

AO3

A*