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Cambridge University Press • Uncorrected Sample Pages • 2008 © Brookie, Halford, Lawrence, Tiffen, Wallace

SAMPLE

New ZealandCurriculum

Level 3 ProbabilityInvestigate simple situations thatinvolve elements of chance bycomparing experimental results withexpectations from models of all theoutcomes, acknowledging that samplesvary

Level 4 ProbabilityInvestigate situations that involveelements of chance by comparingexperimental distributions withexpectations from models of thepossible outcomes, acknowledgingvariation and independenceUse simple fractions and percentagesto describe probabilities

Probability of winningThe probability of wining first divisionin Lotto, which requires the choice ofsix numbers chosen from 40, is 1 in 3 838 380. This means that you wouldexpect to win first division once everythree million eight hundred thirty-eightthousand three hundred and eightyattempts. This is calculated by countingthe number of possible ways ofwinning, as well as the total number ofways that six numbers can be drawnfrom 40 numbers. If you played 1game per week every week of the year,you would expect to win once every738 centuries. The mathematics ofprobability and counting can be usedin a similar way to analyse thechances of success for all sorts ofgames and events.

Chapter 8.qxd 7/22/08 3:07 PM 91


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Do now

Skillsheet

T EACHE R

Mathematics and Statistics Year 9292

1 List the following:

a natural numbers less than 10 b integers between 3 and 9c prime numbers less than 20 d multiples of 4 less than 30e factors of 24

2 Decide whether the chance of each of the following happening is less than , equal

to or greater than :

a If you throw a fair coin it will come up heads.b If you throw a six-sided die a 4 will turn up.c The temperature on Christmas day in Wellington would be �10°.d You win a prize if you buy 80 tickets out of a possible 100 tickets in a raffle.e If you throw a 10-sided die an even number will turn up.

3 List the following for the throw of a six-sided die:

a all the numbers that could come upb even numbers that could come upc multiples of three that could come up

4 How many of each of the following types of cards are there in a pack of 52 playing cards?

a hearts b clubs c red cards d aces e picture cards

5 Simplify the following fractions:

a b c d e f

6 Add these fractions and simplify:

a b c d e

7 Multiply these fractions and simplify:

a b c d e

8 Write down the probability of landing on yellow for the following spinners.

a b c d

Prior knowledgePrime number Factor MultipleOutcome Event TrialEqually likely outcomes Probability Sample spaceExperiment

710

�3

10023

�37

110

�34

15

�34

15

�35

310

� 1

103

100�

4100

1352

� 1352

14

� 24

15

� 35

41000

1352

55

1015

26

24

12

12

12

Chapter 8.qxd 7/22/08 3:07 PM Page 292


SAMPLE

Chapter 8 — Probability 293

Doing these experiments will help you understand probability.

1 Spinner

a Make a spinner similar to the one shown here. Cut out acardboard circle of radius 5 cm and divide it into six equalsectors. Each sector should form an angle of 60° at the centreof the circle. Colour three of the sectors green, two yellow andone orange. Secure the centre of the circle to a larger piece ofcardboard with a split pin paper fastener and, on the backingcardboard, draw an arrow pointing to the centre of the circle.Or use a pen as an anchor and a paper clip as the spinner.When the spinner is spun, the arrow will point to one colour.

b Spin the spinner 50 times and record the results in a table.c Construct a column graph to show the number of times the spinner pointed to each colour.d Use the results in your table to calculate the experimental probability of the spinner

landing on a green sector, a yellow sector and an orange sector.e Compare your results with those of the other members of your class. What do you

notice when comparing your results with those of others in your class?What is similar to other results in your class and what is different?

2 Counters

a You need 20 counters: 10 of one colour (A) and 10 of another colour (B). Place all 20counters in a container. Copy the table below into your workbook then complete the table.

8-1 Experiments in probability

60°

b Select two counters at random. (No looking!) Record how many of them are colour Aand how many are colour B. Replace the counters you selected. Repeat the selectionprocedure but this time select four counters. Continue this procedure, selecting 6, 8, 10,12, 14, 16, 18 and 20 counters, and recording your selections.

c Calculate the proportion of counters of colour A in each sample and write thisproportion:i as a fractionii as a decimal, rounded to two decimal places if necessary

d Is there a pattern to the answers you got for the proportion of colour A as the numberof counters selected increases?

e Compare your results with those of others in your class. Comment on your results.

2 4 6 8 10 12 14 16 18 20

A

B

Proportion

Decimal



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3 Rolling a six-sided die

a Roll a six-sided die six times and record the outcomes in a table similar to the onebelow. Record your outcomes in the table. Repeat the process 10 times.

1 2 3 4 5 6

1

2

3

4

5

6

7

8

9

10

b How do your results compare with the results you expected?c Compare your results with those of another student in your class.

4 Rolling two dice and noting the sum

Work with a partner.

a List all possible totals you can roll with two normal dice.

b Roll two dice 100 times and record the total of the numbers on the dice. Record yourresults in a table like the one shown.

Outcome Tally Frequency

Die

Throw



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c From your table what is the experimental probability of rolling:i a six?ii a one?iii a number less than 4?iv an even number?v not an even number?

d What happens when you add your answers from parts iv and v?e Compare your experimental results with those of others in your class. What do you

notice?f Sam says you should have a 1 as an outcome. Do you agree? Explain your answer.g If a die is rolled 1000 times, how many times do you think a total of 7 would result?h Change the 1 and 2 on the die to a 6. Repeat parts b and c and compare your results.

5 Drawing a card

Work with a partner. You will need 10 cards numbered 1 to 10.Shuffle the cards well and ask your partner to select one.

Replace the card, reshuffle and select again. Repeat theexperiment 30 times.

a Record the results in a table.b Calculate the experimental probability of drawing each

number card from your results.c What is the theoretical probability of drawing each number? Compare your

experimental results with the theoretical probability. What do you notice?d Combine your results with those of two other people in your class and recalculate the

experimental probability for drawing each number. How do these results compare withthe values you calculated in part b?

e If you repeated the experiment 100 times how many times would you expect a 10 to beselected?

6 Tossing a bottle top

Work with a partner. You will need a bottle top.

a Toss a bottle top 50 times and record the way it lands. Use a table like the one below:

b From your results, what is the experimental probability of the bottle top landing:i right way up?ii upside down?

c Did you expect the results you obtained? Why? Why not?d If you repeated the experiment 100 times, how many times would you expect the bottle

top to land upside down?

Outcome Tally Frequency

Right way up

Upside down



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7 Tossing a coin on a chess board

Work with a partner. You will need a chessboard and a 10 cent coin.

a Roll the 10 cent coin onto the chessboard 50 times. In a table record whether the coinlands inside a square or on a line.

b What do you notice about the number of times the coin landed:i inside a square?ii on a line?

c What is the experimental probability of the coin landing:i inside a square?ii on a line?

d In the long run would you expect the coin to have an equal chance of landing inside asquare or on a line?

e Do you think the experimental probailities would change if you used a smaller or larger coin?

Explain your answer.

Using technology: Simulating random events

a Rolling a die

To roll a die 10 times and list the results canbe achieved using randList(n, a, b), where nis the number of trials, a is the lowest integerand b is the highest integer,

a Rolling a die

To roll a die 10 times and list the results canbe achieved using randInt(low, up, n), wherelow is the lowest integer, up is the largestinteger and n is the number of trials.

Casio ClassPad 300 series TI-Nspire



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b Tossing a coin

Assume 1 � heads and 2 � tails.

Use the randList command with and.

c Selecting a playing card from a pack

Assume the ace of clubs � 1, the 2 of clubs � 2 etc. until all 52 cards have aunique value. Use the randList commandwith a � 1 and b � 5.

b � 2a � 1

b Tossing a coin

Define a list of possible outcomes.

Define a second list based on the randomselections from the first list using randSamp(list, n), where list is the name of the list touse and n is the number of selections. Thendisplay the list by entering its name.

c Selecting a playing card from a pack

Define list 1 � {ace of clubs, two of clubs, . . .}* This document can be down-loaded from the www.mathsandstats.co.nzwebsite.

Define a second list, randSamp (list, n),based on the random selections from thefirst list, where list is the name of the list touse and n is the number of selections.

Then display the list by entering its name.



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Experimental probability from an experiment repeated a large number of times can be usedto make predictions about events. The proportion of times that an event occurs in the longrun is a good estimate of the probability of that event.

As the number of experiments increases, the experimental probability gets closer to thetrue probability.

The expected number of times an event happens is equal to the number of trialsmultiplied by the probability of the event occuring in any one trial.

8-2 Long run proportion and expected value

Throw a coin 100 times. Use a table to record how many timesthe coin lands on ‘head’. After every 10 throws calculate theproportion of heads so far.

Combine your results with those of other students so youhave results for 200 throws of the coin.

a Record the totals after every 10 tosses in a table like theone below.

b Calculate the proportions as decimals, correct to twodecimal places.

c Draw a graph of thenumber of throws vs theproportion of heads.

d What do you noticeabout the proportion ofheads thrown as thenumber of coin tossesincreases?

e How many heads wouldyou expect to get whenyou throw the coin200 times?

f Compare your resultswith the theoreticalprobability of throwinga head.

g Compare your resultswith others in you class.What do you notice?

Number of Number of Proportion of Proportion astosses heads heads a decimal

10 __

20 __

30 __

40 __

50 __

60 __

70 __

80 __

90 __

100 __�

100�

�

90�

�

80�

�

70�

�

60�

�

50�

�

40�

�

30�

�

20�

�

10�



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Key ideas

Long run proportions can be obtained by repeating the experiment a number of times.

Probabilities can be written as fractions, decimals or percentages.

Probabilites are always between 0 and 1.

The sample space is a list of all possible outcomes.

The experimental probability of an event happening is the proportion of times it occurs:

Probability of all possible outcomes always add to 1.

The estimate improves as the number of repeated trials gets larger.

Long run proportion is particularly useful for events that are not equally likely.

If the experiment is repeated a large number of times, the experimental probability getscloser to the theoretical probability.

EExxppeecctteedd nnuummbbeerr � probability of the event E � number of trials of event E

P(event) �number of times event E occurs

total number of trials of E

Example 1

A jar contains a large number of marbles coloured red, green, yellow, orange and brown. A marble was chosen at random, its colour noted and the marble replaced. This experimentwas carried out 200 times. The results are shown below.

Use these results to estimate the probability that a marble randomly selected from the jar is:

a red b green c not green d yellow or orange

ExplanationSolution

a

b

�17

100 or 0.17

P(green) �34200

�1350

or 0.26

P(red) �52

200

There are 52 red marbles out of a total of 200.

There are 34 green marbles out of a total 200.

Colour Red Green Yellow Orange Brown

Frequency 52 34 38 44 32



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c

d

�41100

or 0.41

P(yellow or orange) �82

200

�83

100 or 0.83

P(not green) �166200

There are 34 green marbles, so 166 out of a total of 200 are not green. This could also be calculated as

.

There are 38 yellow and 44 orange marbles outof a total of 200.

1 �34

200

Example 2

USB memory sticks are made in a factory in South Auckland. Inspectors tested samples of USB memory sticks and the factory’s records show that out of 1000 USB memory stickstested, 15 were found to be faulty.

a Using this information, estimate the probability that a randomly selected USBmemory stick from this factory will not be faulty.

b A store receives 350 USB memory sticks from the factory. Predict how many of thesewill not be faulty.

ExplanationSolution

a

� 0.985

b Expected number of not-faulty USB memory sticks = 0.985 � 350�345

�985

1000

P(not faulty USB memory sticks) 15 in 1000 USB memory sticks were found to befaulty, so 1000 � 15 � 985 USB memory sticksare not faulty.

From 350 USB memory sticks we expect aproportion of 0.985 to be non faulty.Round your answer to the nearest whole number.

means ‘approximately equal to’.�

8AExercise

1 One wrapped lolly was selected at random from a bag containing a large number ofred, green, orange, yellow and purple wrapped lollies. Its colour was noted and thelolly returned to the bag. This was done 50 times. The results are shown below.

Colour Red Green Orange Yellow Purple

Frequency 11 19 6 4 10

Estimate the probability that a lolly selected at random from the bag will be:

a red b green c orange or yellow d not purple

1Example



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2 A factory in Hamilton makes MP3 players. Quality controllers found that for every300 players they inspected two were faulty.

a Estimate the probability that a randomly selected MP3 player will be faulty.b If 10 000 MP3 players are produced in a day, how many would you expect to be faulty?

3 A new make of car has a two in five chance of needing repairs in its first 12 months.Keystone company buys a fleet of 150 of these cars. How many of these cars wouldyou expect to be returned for repairs in the first year?

4 Pop rivets are produced in four sizes. A sample of 100 mixed pop rivets were takenfrom a vat and the sizes noted.

Size Tiny Small Medium Large

Frequency 10 26 16 48

a Determine how many medium pop rivets you would expect in a bag containing:i 100 ii 1000 iii 250

b Determine how many pop rivets would not be large in a bag containing:i 50 ii 700 iii 550

5 How many times would you expect a 4 to appear in:

a 30 rolls of a fair die? b 50 rolls of a fair die?c 100 rolls of a fair die? d 1000 rolls of a fair die?

6 If a $1 coin is tossed 120 times, how many times would you expect it to land head up?

7 Two spinners were each divided into six sectors of equal size. The sectors werecoloured red, blue, yellow or green. The graphs below show the frequencies of thedifferent colours in 100 spins of each spinner. Use the graphs to determine how manyof the sectors were coloured red, blue, yellow and green in each spinner.

2Example

50

40

30

20

10

0R

16 33 17 34

B GY

Freq

uenc

y

Colour

50

40

30

20

10

0R

17 18 15 50

B GY

Freq

uenc

y

ColourSpinner A Spinner B

8 A spinning wheel has the numbers 1 to 20 equally spaced around it. If the wheel isspun 25 times, determine the expected number of:

a even numbers b numbers over 10 c multiples of 5d factors of 36 e numbers greater than 30 f prime numbers

9 A six-sided die was rolled 40 times.The results are shown.

Number 1 2 3 4 5 6

Frequency 5 6 14 4 5 6



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a Use the information in the table to estimate the probability that:i a 1 will be obtained on the next throwii a 3 will be obtained on the next throwiii a 6 will be obtained on the next throw

b Do you think that the outcomes are equally likely or is the die biased? Explainyour answer.

10 Two companies, Brightspark and Solar, each produce light bulbs at the rate of 5000bulbs per day. On a particular day a random sample of 60 bulbs from Brightsparks and40 from Solar are inspected. The number of defective light bulbs was recorded.

Sample size Number of defective globes

Brightspark 60 2

Solar 40 3

a Calculate the experimental probability of choosing a defective bulb from:i Brightspark ii Solar

b Calculate the expected number of defective bulbs produced in one day by:i Brightspark ii Solar

c Calculate the expected number of defective bulbs produced in a week by:i Brightspark ii Solar

11 A bag contains six red marbles, five blue marbles and one silver marble. Four childrentook turns to select a marble at random from the bag and then replaced it. All fourchildren selected the silver marble. Is this what you would expect? Explain your answer.

12 A school carpark contains 15 red, 4 green, 7 blue and 16white cars.

a Calculate the probability that the next car to leavethe school carpark is:i red ii green iii blue iv white

b Jane used these results to estimate the number ofblue cars in New Zealand. She says that one-sixthof all cars in New Zealand are white. Do you agree? Explain your answer.

Enrichment: Spinners

13 The spinners below, labelled a to f, were each spun 10 times and the numbers theylanded on were recorded in lists i to vi. The lists are not in the same order as thespinners. Match each spinner to the most likely list. Is there more than one correctanswer?

a b c2 3

7 6

1

8

4

5

4 4

1 4

3

2

4

4

4 6

6 4

2

8

8

2



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Chris Mary

Barry Susan

Colour Total

Blue 26

Red 17

Green 29

Yellow 8

Colour Total

Blue 30

Red 13

Green 27

Yellow 10

Colour Total

Blue 25

Red 14

Green 34

Yellow 7

Colour Total

Blue 24

Red 19

Green 31

Yellow 6

d e f

i 1, 2, 8, 6, 4, 3, 3, 7, 5, 4 ii 1, 1, 4, 2, 1, 2, 3, 2, 4, 2iii 1, 4, 3, 2, 4, 4, 2, 4, 1, 4 iv 8, 8, 2, 2, 8, 2, 8, 8, 2, 8v 2, 2, 8, 4, 8, 2, 4, 6, 6, 8 vi 7, 1, 5, 5, 1, 3, 1, 5, 3, 3

14 A bag has 10 counters of four different colours. Some students took turns to selectand replace one counter at a time. They did this 80 times.

Use the given information to try to estimate how many counters of each colour were inthe bag. Explain your method.

2 8

1 2

3 4

3 5

1 7

Centre Total

Strawberry 11

Caramel 14

Coconut 9

Nut 19

Mint 7

15 A box has 12 wrapped chocolates. They are all the same size and shape but havedifferent flavoured centres. The results from selecting and replacing one chocolate at a time for 60 trials are shown in this table.

Use the given information to try to predict how many chocolates of each type there are in the box. Explain your answer.



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For many situations or experiments involving chance, we can use the mathematics ofprobability to determine the level of chance of a particular event occurring. Probability isthe mathematical term used to describe chance.

When outcomes of an event are equally likely, each outcome has an equal chance ofoccurring, i.e. their probabilities are the same.

Make a spinner like the one shown below. (See section 8–1 forinstructions for making a spinner.)a Spin the spinner 50 times and record in a table the number of times

each number comes up.b From your results write down the proportion of spinning a 1, 2, 3, 4.c What do your proportions from part b add to?d Compare your results with those of your neighbour. How are they the same? How are

they different?e Combine your results with those of your neighbour and recalculate the proportions of

spinning a 1, 2, 3, 4.f Calculate the theoretical probability of spinning a 1, 2, 3, 4.g How do the results of your experiment compare with the theoretical probability?h Design your own spinner. Divide your spinner into six equal parts and use numbers

from 1 to 4 inclusive. Repeat parts a to c.

8-3 Equally likely outcomes

Key ideas

If A is a particular event then:

P(A) means ‘the probability that event A occurs’.

The ccoommpplleemmeenntt or opposite of A includes all the possible outcomes not in A and iswritten A�. We say ‘A dash’.

All probabilities are between 0 and 1, therefore:

The ssaammppllee ssppaaccee is a list of all the possible outcomes.

0 � P(A) � 1

P(not A) �1 �P(A)

P(A) �number of outcomes in A

total number of possible outcomes

2 3

4 4

1

2

1

1

Example 3

This spinner has five equally divided sections.a List the sample space. b Find P(3).c Find P(not a 3). d Find P(a 3 or a 7).e P(3) � P(not a 3).

21

3 73



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ExplanationSolution

a Sample space is 1, 2, 3, 3, 7

b

c

d

e

� 1

�55

P(3) � P(not a 3) �25

�15

�35

or 0.6

P(a 3 or a 7) �25

�15

�35

or 0.6

� 1 �25

P(not a 3) � 1 � P(3)

P(3) �25

or 0.4

The sample space is a list of all possibleoutcomes.

Alternatively count the number of sectors whichare not 3.

There are two 3s and one 7.

There are two 3s and there are three numbers that are not a 3.

is all of the outcomes in thesample space.P(3) � P(not a 3)

�15

�15

�15

�35

P(not 3) � P(1, 2, 7)

P(3) �number of sections labelled 3

number of equal sections

Example 4

A fair six-sided die is rolled.

a List the outcomes in the sample space.b Find the following probabilities:

i P(number less than 5)ii P(rolling a 4 or a 5)iii P(multiple of 12)iv P(multiple of 2 and multiple of 3)v P(multiple of 2 or multiple of 3)

ExplanationSolution

a Sample space = {1, 2, 3, 4, 5, 6}

b i

ii26

�13

46

�23

This is the probability of throwing a 1, 2, 3 or 4.



SAMPLE

306 Mathematics and Statistics Year 9

iii 0

iv

v46

�23

16

Mulitples of 12 are 12, 24, 36 . . . None of thesenumbers is less than 6, so it is impossible.

Multiples of 2 are 2, 4, 6. Multiple of 3 are 3, 6.There is only one number that is a multiple of 2 and a multiple of 3, i.e. 6.

Multiples of 2 are 2, 4, 6. Multiples of 3 are 3, 6.Numbers that are a multiple of 2 or 3 are 2, 3, 4, 6.

8BExercise

1 Find the probability of each of the following events:

a obtaining a head when a fair coin is tossedb obtaining the number 1 when six-sided die is rolledc selecting a heart when a card is selected at random

from a pack of 52 playing cardsd selecting an ace when a card is selected at random

from a pack of 52 playing cardse the spinner with the letters CHANCE written around

the wheel landing on C when it is spun

2 A fair eight-sided die is rolled.

a List the outcomes in the sample space.b Find the following probabilities:

i P(number greater than 5)ii P(even number)iii P(multiple of 3)

3 Find the probability of each of the following events:

a Choosing a female captain of a hockey team when the captain is chosen atrandom from six girls and eight boys

b Choosing a Wednesday when a day is chosen at random from the days of theweek

c Choosing the month of May when a month is chosen at random from the monthsof the year

d Winning a raffle if 100 tickets were sold and you bought one tickete Guessing the correct answer in a multiple-choice question with answers A, B, C,

D and Ef Choosing a male class captain when the captain is chosen at random from a class

of 15 boys and 17 girlsg Choosing a red ball when a ball is chosen at random from a box containing one

yellow, two blue and three red balls

H

C N

A

E C

3Example

4Example



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4 A spinning wheel with the numbers 1 to 5 equally spaced around the wheel is spun. A is the event of obtaining a number greater than 1.

a List the outcomes in event A.b List the outcomes not in A.c Calculate P(A) and P(A�).

5 A fair six-sided die is rolled. What is the probability of obtaining each of the following?

a 2 b a number greater than 1 c 2 or 6d a prime number e 7 f a number less than or equal to 8

6 A lucky dip at a fete contains 10 yoyos, 12 bags of marbles, 15 chocolates and 13 packets of chips.

a If a prize is selected at random, what is the probability of choosing the following?i a bag of marbles ii a chocolateiii a yoyo iv a yoyo or a chocolate

b List the prizes in order from the most likely to the least likely.

7 The nine letters in the word CHOCOLATE are written on cards and the cards shuffled.One card is selected at random. Find the probability that the letter on it is:

a O b H c a vowel d a letter in the word CAT

8 One card is selected at random from a pack of 52 playing cards. Calculate thefollowing probabilities:

a P(queen) b P(heart)c P(black card) d P(picture card)e P(not a queen) f P(not a picture card)g P(queen and a heart) h P(black card and a picture card)i P(queen or a heart) j P(black card or a picture card)k P(not a queen and a heart)

9 The probability that a person sitting their drivers licence test passes the first time is0.85. What is the probability that Rose fails the first time she sits her driving test?

10 We are told that the probability that the Rovers will win a football match is . If this is

true, what is the probability that they will lose or draw the next match?

11 A coin is tossed. A is the event of obtaining a head.

a List the outcomes in event A. b List the outcomes in event A�.c Calculate P(A) and P(A�).

12 Terry has 36 buttons in a jar. Twelve are red and the rest are black. If he selects abutton at random, what is the probability that it is black?

13 A spinner is divided into three parts coloured blue, red and green. The probability that

it lands on red is and blue is . What is the probability that it lands on green?

14 Amy was a game show participant. She randomly threw one ball into one of 20 boxes.From the top each box appears to be identical but 12 of the boxes have their basespainted red, five blue, two green and one yellow. Every ball thrown lands in a box.

1

213

34



SAMPLE


The table below shows the prizes associated with each colour.

a What is the probability that Any:i wins a plasma TV? ii wins an MP3 player? iii wins $50?iv wins a prize? v does not win a prize?

b Another six red boxes and three blue boxes and one green box are added. Has herchance of winning a prize changed, explain your answer.

15 A school runs a raffle to raise money for a big screen TV. They print 400 tickets and sell them all at $2 each. One major prize is offered.

Allanah buys one ticket.

Brian buys five tickets.

Chi buys $50 worth of tickets.

Damian buys enough tickets to have a one in five chance of winning.

a What is the probability that Allanah wins the major prize?b What is the probability that Brian wins the major prize?c What is the probability that Chi wins the major prize?d How many tickets did Damian buy?

16 Draw a spinner that will land on the given colour with the following probabilities:

a , and

b , , and P(red) �18

P(white) �38

P(black) �14

P(blue) �14

P(red) �13

P(white) �12

P(blue) �16

Yellow Green Blue Red

Plasma TV MP3 player $50 No prize

Enrichment: Scrabble

17 Scrabble is a game in which we take lettered tiles and try to arrange them to spell words.

a If one tile of each letter of the alphabet and two blank tiles are placed in a bagand one tile is selected, determine the probability of selecting:i a blank ii a vowel iii an X, J or Q iv an S

b If you replace the tile and select another, what is the probabilty of repeating theevents in part a?

c If you do not replace the tile and select another, what is the probabilty ofrepeating the events in part a?

d If you continue choosing tiles one at a time withoutreplacing them, what happens to the probabilities?Investigate.

e If you continue choosing tiles one at a time withoutreplacing them, will the probability of choosing

a particular letter ever be ? Explain your answer.12



SAMPLE

Tree diagrams are useful for listing outcomes of experiments that have two or moresuccessive events.

8-4 Probability trees

Key ideas

A tree diagram is used for combined events.

The first event is at the end of the first branch, the second event is at the end of thesecond branch etc.

The outcomes for the combined events are listed on the right-hand side.

W

L

W

L

W WWW

L WWL

WLW

W

L

W

L WLL

W LWW

L LWL

W LLW

L LLL

Outcomes

Example 5

A coin is thrown and its result recorded. Then the coin is thrown again and the resultrecorded.

a Complete a tree diagram to show all possible outcomes.b What is the total number of outcomes?c Find the probability of tossing:

i two tails ii one tail iii at least one head




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ExplanationSolution

a

b The total number of outcomes is 4.

c i

ii

iii P(at least one head) �34

P(1 tail) �24

�12

P(TT) �14

HH HH

HT

TH

TT

Toss 1 OutcomeToss 2

T

T

H

T

The tree diagram results in outcomes.

There are four possibilities in the outcomescolumn.

One of the four outcomes is TT.

Two outcomes have one tail: {HT, TH}

Three outcomes have at least one head:{HH, HT, TH}

2 � 2 � 4

8CExercise

1 The following tree diagram shows all the possible outcomes of having two children ina family.

a Complete this tree diagram to show all the possible outcomes.b What is the total number of outcomes?c Find the probability of having:

i two girls ii one girliii at least one girl iv at least one boy

2 A spinner is numbered 1, 2, 3 and each number is equally likely to occur. The spinneris spun twice.

a List the set of possible outcomes as a tree diagram.b What is the total number of possible outcomes?c Find the probability of spinning:

i two 3s ii at least one 3iii no more than one 2 iv two odd numbers

Toss 1 Toss 2 Outcome

...

...

...

...

...

...

BB

BB

G

5Example



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3 An experiment involves tossing three coins and noting the outcome.

a Complete a tree diagram to list all the possible outcomes.b Find the following probabilities:

i P(three heads) ii P(two heads) iii P(at least two tails)iv P(no tails) v P(no more than two tails)

4 A restaurant is offering a three course meal for $25. There is a choice of shrimp orspring rolls for a starter; steak, chicken or lamb for main course; and either chocolatecake or apple pie for desert. Assume that each choice is equally likely to be chosen.

a Draw a probability tree to show all possible combinations of orders.b Calculate the probability that a customer orders:

i shrimp, steak and apple pieii chickeniii steak and chocolate cake

5 A first-aid test includes three multiple-chioce questions. Tessa decides to guess. Thereare three choices of answer (A, B and C) for each question.

If only one of the possible choices (A, B or C) is correct for each question, find theprobability that Tessa guesses:

a 1 correct answer b 2 correct answersc 3 correct answers d no correct answers

6 Complete a tree diagram for tossing a coin four times to find the probability that youtoss:

a 0 tails b 1 tail c 2 tails d 3 tails e 4 tails

7 Four coins were tossed 50 times. The results are shown:

HTHH TTHT THHT HTHT TTTH HTHH

HTHT HHTH HHTT THTH HHHT THHT

THTT TTTT TTTH HHTT HHHT HHHT

HTTH HHHT THTH HTTT HHHT THTH

THHT TTTT HHTT THTH HTHT TTTH

HTTH HHHH HHHT THTT TTTH THTT

HHTH HHHT TTTH THTH HTHT HHTT

THTT HHHT THTH TTTH THHT THTT

HTTT TTTH

Guess 1 Guess 2 Guess 3 Outcome

AAA

...

...

...

......

...

A

B

C

A

B

C

A

B

C



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a Copy and complete the following frequency table.

Number of heads Tally Frequency

0

1

2

3

4

b Determine the experimental probability of obtaining:i 0 heads ii 1 head iii 2 heads iv 3 heads v 4 heads

c Use a tree diagram to represent all possible outcomes in tossing four coins.d Determine the theoretical probability of tossing:

i 0 heads ii 1 head iii 2 head iv 3 heads v 4 headse Compare the theoretical probabilities with the experimental probabilities. What

do you notice?8

Enrichment

8 Hohepa randomly selects his clothing. He chooses one pair of shoes from hiscollection of one black and two red pairs, a shirt from a collection of one white andtwo blue shirts, and either red or black jeans.

Use a tree diagram to help find the probability that he selects a pair of shoes, shirt and jeans according to the following descriptions:

a black shoes, white shirt and black jeans b no red itemsc one red item d two black itemse at least two red items f black jeansg red shoes and black jeans h white shirt or red jeansi not a black item

9 Use a tree diagram to investigate the probabilities involved in selecting two countersfrom a bag of 3 black and 2 white counters:

a A counter is selected from a bag and then replaced before a second counter isselected.

b A counter is selected from a bag and not replaced before a second counter isselected.Is there any difference? Explain your answer.



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W O R K I N G

For this activity you will need:

a bag or jacket pocket

different-coloured counters

paper and pen

Five countersa Work with a partner. Without watching, have a third

person place five counters of two different coloursinto the bag or pocket. An example of five counterscould be two red and three blue. It is important thatyou do not know what the counters are.

b Without looking, one person selects a counter fromthe bag while the other person records its colour.Replace the counter in the bag.

c Repeat part b for a total of 100 trials. Record theresults in a table similar to this one.

d Find the experimental probability for each colour.e Use these experimental probabilities to guess how many counters of each colour there

are in the bag.

Use this table to help

f Now take the counters out of the bag and see if your guess is correct.

More colours and countersa Repeat the steps above but this time use three colours and 8 counters.b Repeat the steps above but this time use four colours and 12 counters.

Probability

Colour Tally Frequency

Red |||| |||

Blue |||| |||| ||

Total 100 100

Closest multiple Guess of how many Experimental of 15 or 0.2, e.g. counters of this

Colour Frequency probability 0.2, 0.4 . . . colour

Total 100 1 1 5

Mathematically



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Chapter summary

Long run proportion and expected valueLong run proportion uses an experiment repeated a number of times.Probabilities can be written as a fraction, decimal or percentage.Probabilites are always between 0 and 1.The sample space is a list of all possible outcomes.The estimated probability of an event happening is the proportion of times it occurs:

Probability of all possible outcomes always add to 1.The estimate is increasingly reliable as the number of repeated trials gets larger.Long-run proportion is particularly useful for events that are not equally likely.If the experiment is repeated a large number of times, the experimental probability getscloser to the theoretical probability.

Equally likely outcomesIf A is a particular event then:

•

• P(A) means ‘The probability that event A occurs’.• The complement or opposite of A includes all the possible outcomes not in A and is

written A�. We say ‘A dash’.•All probabilities are between 0 and 1, therefore: The sample space is a list of all the possible outcomes.

Probability treesA tree diagram is used for combined events. The first event is at the end of the firstbranch, the second event is at the end of the second branch etc.The outcomes for the combined events are listed on the right-hand side.

WW

WL

WW

LW

LL

Game 1 OutcomesGame 2

L

L

W

L

0 � P(A) � 1P(not A) � 1 � P(A)

P(A) �number of outcomes in A

total possible number of outcomes

Expected number � probability of the event E � number of trials of event E.

P(event) �number of times event E occurs

total number of trials of E



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Short-answer questions

1 Jack forgets his homework diary one day per week (school day). During the 40-weekschool year, on how many days would you expect Jack to forget his homework diary?

2 A number is selected from the numbers 1–10. What is the P(number less than 4)? Writeyour answer as a decimal and as a fraction.

3 What is the probability of selecting an A from the word ABRACADABRA?4 A pack of 52 cards is well shuffled. What is the probability of selecting a picture card?5 What is the probability of selecting an ace when one card is chosen at random from a

pack of 52 playing cards?6 What is the proability that a letter selected at random from the letters in the word

CAMBRIDGE is a vowel?7 A hat contains six $2 coins, three $1 coins and four 50c coins. If one coin is selected at

random what is the probability it is a $1 coin?8 A gumball machine contains 5 blue gumballs, 20 green gumballs and 15 white

gumballs. What is the probability of:

a P(green gumball)

b P(not a white gumball)

c P(a white or a blue gumball)

d P(a red gumball)9 Sam believes that he can hit the bullseye on a dart board 90% of the time. How many

times would you expect him to miss the bullseye if he threw a dart 50 times?10 Sarah has a bag containing 28 Freddo Frogs. Some are strawberry and some are

chocolate. If the probability of selecting a chocolate frog is how many strawberryfrogs are in the bag?

11 A coin is tossed three times.

a Use a tree diagram to show the sample space.

b Calculate the probability of obtaining:

i three tails ii no heads

iii at least one head iv at most two tails12 A spinner with 15 equal sectors, numbered 1 to 15, is spun.

a How many different outcomes are possible?

b Are you more likely to obtain an odd number or an even number? Explain your

answer.

c Calculate the probability of obtaining each of the following:

i 2 ii 2 or 3 iii a prime number

iv a factor of 10 v a multiple of 20 vi a number less than 20

37

Review

Short-answer questions




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13 An unusual six-sided die is biased and has an unknown number of 6s on its sides. Thedie is tossed many times and the number of 6s is recorded at different stages of theexperiment, as shown in this table.

a What is the experimental probabiltity of the number of 6s obtained after tossing the die:

i 5 times? ii 10 times? iii 20 times? iv 40 times?

b State the long run proportion for the experiment.

c As a percentage, what would you say is the approximate chance of obtaining a 6 on

one toss of this die?14 One card is chosen at random from a pack of 52 playing cards. Decide whether the

following statements are true or false. Justify your answer.

a b

c d15 Two identical spinners are spun. The number spun on spinner A and the number spun on

spinner B are added to obtain a total score.

a i What is the lowest possible score?

ii What is the highest possible score?

b List all the ways in which a total of eight can be

obtained.

c What is the chance of getting a total score of:

i five? ii nine?16 Jo and Sam were each investigating how many left-handed writers there were among the

480 students at their school.

a Jo observed the 20 students in her class and noted that four of them were left-handed

writers. Using Jo’s information, estimate:

i the probability that a student selected at random from the whole school will be a

left-handed writer

ii the number of left-handed writers there are in the school

b Sam interviewed 80 students in the canteen queue and found that 12 of them were

left-handed writers. Repeat parts a i and a ii using Sam’s information.

c Comment on your answers to a and b.17 Jessie spun a spinner numbered 1 to 6 two hundred times. The results are shown in the

table below.

Use the data in the table to calculate the experimental probability of obtaining a:

a 1 b 2 c 3 d 4 e 5 f 6Write your answers as a:

i fraction ii decimal

P(ace of spades) � P(black jack)P(heart) � P(club)

P(ace) � P(king)P(red) � P(queen)

Number of tosses 5 10 20 40

Number of 6s 5 5 8 21

Outcome 1 2 3 4 5 6

Number of times 34 44 35 30 29 28

3 4

5

3 4

5

spinner A spinner B



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Review


Extended-response questions

1 The data below shows the winners of the American NBA finals from 1987 to 2006:1987 LA Lakers 1988 LA Lakers 1989 Detroit 1990 Detroit1991 Chicago 1992 Chicago 1993 Chicago 1994 Houston1995 Houston 1996 Chicago 1997 Chicago 1998 Chicago1999 San Antonio 2000 LA Lakers 2001 LA Lakers 2002 LA Lakers2003 San Antonio 2004 Detroit 2005 San Antonio 2006 Miami

a Copy the table below into your workbook and summarise the data showing the

number of times each team won the finals over the given 20-year period.

b Use the data to calculate the experimental probability of each of the six teams

winning the finals in 2007. Which team is the most likely to win?

c Find out who won the NBA Finals in 2007 and compare this with your prediction.

Comment on the accuracy of your prediction.2 A bubblegum machine has 250 bubblegum balls of the colours yellow, red, brown,

purple and blue. It is not known how many of each colour are in the machine.

Tom selected one ball at random 80 times, with replacement, and the results of hisexperiment are displayed in the table below.

a Using the results in Tom’s table, estimate the probability of each of the following

colours being selected:

i yellow ii red iii brown iv purple v blue

b Using the probabilities you wrote for part a, estimate how many balls of each of the

following colours are in the machine:

i yellow ii red iii brown iv purple v blue3 In a game of chance at a fair, a six-sided die is rolled and players bet on the outcome. A

player wins if the number is even and loses if the number is odd.The results of 100 games are recorded and shown in the table below.

a Using the results in the table, calculate the number of times:

i an even number was rolled ii an odd number was rolled

b Is there an even chance of winning the game?

c Do you think a fair die is being used? Give a reason for your answer.

San LAAntonio Chicago Detroit Houston Miami Lakers Total

Number of NBAfinal wins

Estimatedprobability of a win

Colour Yellow Red Brown Purple Blue

Number in 80 selections 3 24 32 13 8

Number on die 1 2 3 4 5 6

Frequency 18 12 22 13 25 10



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sample - cambridge university press · chapter 8 — probability 293 doing these experiments will...

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