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E
M
PL
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Probability of winning
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The probability of wining first division
in Lotto, which requires the choice of
six numbers chosen from 40, is 1 in
3 838 380. This means that you would
expect to win first division once every
three million eight hundred thirty-eight
thousand three hundred and eighty
attempts. This is calculated by counting
the number of possible ways of
winning, as well as the total number of
ways that six numbers can be drawn
from 40 numbers. If you played 1
game per week every week of the year,
you would expect to win once every
738 centuries. The mathematics of
probability and counting can be used
in a similar way to analyse the
chances of success for all sorts of
games and events.
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Chapter 8.qxd
New Zealand
Curriculum
Level 3 Probability
Investigate simple situations that
involve elements of chance by
comparing experimental results with
expectations from models of all the
outcomes, acknowledging that samples
vary
Level 4 Probability
Investigate situations that involve
elements of chance by comparing
experimental distributions with
expectations from models of the
possible outcomes, acknowledging
variation and independence
Use simple fractions and percentages
to describe probabilities
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llshe
ki
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If you throw a fair coin it will come up heads.
If you throw a six-sided die a 4 will turn up.
The temperature on Christmas day in Wellington would be ⫺10°.
You win a prize if you buy 80 tickets out of a possible 100 tickets in a raffle.
If you throw a 10-sided die an even number will turn up.
all the numbers that could come up
even numbers that could come up
multiples of three that could come up
hearts
b
clubs
c
red cards
2
4
2
6
10
15
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7
8
d
aces
e
picture cards
Simplify the following fractions:
a
6
integers between 3 and 9
multiples of 4 less than 30
How many of each of the following types of cards are there in a pack of 52 playing cards?
a
5
b
d
List the following for the throw of a six-sided die:
a
b
c
4
natural numbers less than 10
prime numbers less than 20
factors of 24
1
Decide whether the chance of each of the following happening is less than , equal
2
1
1
to or greater than :
2
2
a
b
c
d
e
3
CH E
List the following:
a
c
e
2
R
S
1
Do now
et
T
EA
b
c
d
5
5
e
13
52
f
4
1000
Add these fractions and simplify:
1
3
1
2
13
13
3
4
1
3
a
b
c
d
e
⫹
⫹
⫹
⫹
⫹
5
5
4
4
52
52
100
100
10
10
Multiply these fractions and simplify:
1
3
1
3
1
3
2
3
7
3
a
b
c
d
e
⫻
⫻
⫻
⫻
⫻
5
5
5
4
10
4
3
7
10
100
Write down the probability of landing on yellow for the following spinners.
a
b
c
d
Prior knowledge
Prime number
Outcome
Equally likely outcomes
Experiment
292
Factor
Event
Probability
Multiple
Trial
Sample space
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Experiments in probability
Doing these experiments will help you understand probability.
Spinner
b
c
d
e
2
Make a spinner similar to the one shown here. Cut out a
cardboard circle of radius 5 cm and divide it into six equal
sectors. Each sector should form an angle of 60° at the centre
60°
of the circle. Colour three of the sectors green, two yellow and
one orange. Secure the centre of the circle to a larger piece of
cardboard with a split pin paper fastener and, on the backing
cardboard, 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.
Spin the spinner 50 times and record the results in a table.
Construct a column graph to show the number of times the spinner pointed to each colour.
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.
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?
E
a
Counters
You need 20 counters: 10 of one colour (A) and 10 of another colour (B). Place all 20
counters in a container. Copy the table below into your workbook then complete the table.
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a
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1
2
4
6
8
10
12
14
16
18
20
A
B
Proportion
Decimal
b
c
d
e
Select two counters at random. (No looking!) Record how many of them are colour A
and how many are colour B. Replace the counters you selected. Repeat the selection
procedure but this time select four counters. Continue this procedure, selecting 6, 8, 10,
12, 14, 16, 18 and 20 counters, and recording your selections.
Calculate the proportion of counters of colour A in each sample and write this
proportion:
i
as a fraction
ii as a decimal, rounded to two decimal places if necessary
Is there a pattern to the answers you got for the proportion of colour A as the number
of counters selected increases?
Compare your results with those of others in your class. Comment on your results.
Chapter 8 — Probability
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Rolling a six-sided die
a
Roll a six-sided die six times and record the outcomes in a table similar to the one
below. Record your outcomes in the table. Repeat the process 10 times.
Die
1
2
3
4
5
6
1
2
3
E
4
5
Throw
6
7
9
10
b
c
4
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8
How do your results compare with the results you expected?
Compare your results with those of another student in your class.
Rolling two dice and noting the sum
Work with a partner.
List all possible totals you can roll with two normal dice.
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a
b
Roll two dice 100 times and record the total of the numbers on the dice. Record your
results in a table like the one shown.
Outcome
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Tally
Frequency
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d
e
f
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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?
What happens when you add your answers from parts iv and v?
Compare your experimental results with those of others in your class. What do you
notice?
Sam says you should have a 1 as an outcome. Do you agree? Explain your answer.
If a die is rolled 1000 times, how many times do you think a total of 7 would result?
Change the 1 and 2 on the die to a 6. Repeat parts b and c and compare your results.
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Chapter 8.qxd
Drawing a card
a
b
c
Record the results in a table.
Calculate the experimental probability of drawing each
number card from your results.
What is the theoretical probability of drawing each number? Compare your
experimental results with the theoretical probability. What do you notice?
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 with
the values you calculated in part b?
If you repeated the experiment 100 times how many times would you expect a 10 to be
selected?
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PL
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 the
experiment 30 times.
e
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:
Outcome
Tally
Frequency
Right way up
Upside down
b
c
d
From your results, what is the experimental probability of the bottle top landing:
i
right way up?
ii upside down?
Did you expect the results you obtained? Why? Why not?
If you repeated the experiment 100 times, how many times would you expect the bottle
top to land upside down?
Chapter 8 — Probability
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Tossing a coin on a chess board
Work with a partner. You will need a chessboard and a 10 cent coin.
Roll the 10 cent coin onto the chessboard 50 times. In a table record whether the coin
lands 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?
What is the experimental probability of the coin landing:
i
inside a square?
ii on a line?
In the long run would you expect the coin to have an equal chance of landing inside a
square or on a line?
Do you think the experimental probailities would change if you used a smaller or
larger coin?
d
e
PL
c
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a
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Explain your answer.
Using technology: Simulating random events
Casio ClassPad 300 series
a Rolling a die
To roll a die 10 times and list the results can
be achieved using randList(n, a, b), where n
is the number of trials, a is the lowest integer
and b is the highest integer,
296
TI-Nspire
a Rolling a die
To roll a die 10 times and list the results can
be achieved using randInt(low, up, n), where
low is the lowest integer, up is the largest
integer and n is the number of trials.
Mathematics and Statistics Year 9
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b Tossing a coin
Define a list of possible outcomes.
Define a second list based on the random
selections from the first list using randSamp
(list, n), where list is the name of the list to
use and n is the number of selections. Then
display the list by entering its name.
PL
E
b Tossing a coin
Assume 1 ⫽ heads and 2 ⫽ tails.
Use the randList command with a ⫽ 1 and
b ⫽ 2.
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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 a
unique value. Use the randList command
with a ⫽ 1 and b ⫽ 5.
c Selecting a playing card from a pack
Define list 1 ⫽ {ace of clubs, two of
clubs, . . .}* This document can be downloaded from the www.mathsandstats.co.nz
website.
Define a second list, randSamp (list, n),
based on the random selections from the
first list, where list is the name of the list to
use and n is the number of selections.
Then display the list by entering its name.
Chapter 8 — Probability
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Long run proportion and expected value
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Experimental probability from an experiment repeated a large number of times can be used
to make predictions about events. The proportion of times that an event occurs in the long
run is a good estimate of the probability of that event.
As the number of experiments increases, the experimental probability gets closer to the
true probability.
The expected number of times an event happens is equal to the number of trials
multiplied by the probability of the event occuring in any one trial.
Throw a coin 100 times. Use a table to record how many times
the coin lands on ‘head’. After every 10 throws calculate the
proportion of heads so far.
a
b
c
Record the totals after every 10 tosses in a table like the
one below.
Calculate the proportions as decimals, correct to two
decimal places.
Draw a graph of the
number of throws vs the
Number of Number of Proportion of
proportion of heads.
tosses
heads
heads
What do you notice
ⵧ
about the proportion of
10
⫽ __
10
heads thrown as the
ⵧ
number of coin tosses
20
⫽ __
20
increases?
ⵧ
How many heads would
30
⫽ __
30
you expect to get when
ⵧ
you throw the coin
40
⫽ __
40
200 times?
ⵧ
Compare your results
50
⫽ __
with the theoretical
50
probability of throwing
ⵧ
60
⫽ __
a head.
60
Compare your results
ⵧ
70
⫽ __
with others in you class.
70
What do you notice?
ⵧ
80
⫽ __
80
ⵧ
90
⫽ __
90
ⵧ
⫽__
100
100
Proportion as
a decimal
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d
PL
Combine your results with those of other students so you
have results for 200 throws of the coin.
e
f
g
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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.
P (event) ⫽
number of times event E occurs
total number of trials of E
Probability of all possible outcomes always add to 1.
E
The experimental probability of an event happening is the proportion of times it occurs:
The estimate improves as the number of repeated trials gets larger.
Long run proportion is particularly useful for events that are not equally likely.
PL
If the experiment is repeated a large number of times, the experimental probability gets
closer to the theoretical probability.
Expected number ⴝ probability of the event E ⴛ number of trials of event E
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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 experiment
was carried out 200 times. The results are shown below.
Colour
Frequency
Red
Green
52
34
Yellow Orange Brown
38
44
32
Use these results to estimate the probability that a marble randomly selected from the jar is:
a
red
b
green
Solution
52
200
13
or 0.26
⫽
50
a
P(red) ⫽
b
P(green) ⫽
34
200
17
or 0.17
⫽
100
c
not green
d
yellow or orange
Explanation
There are 52 red marbles out of a total of 200.
There are 34 green marbles out of a total 200.
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166
200
83
⫽
or 0.83
100
c
P(not green) ⫽
d
P(yellow or orange) ⫽
82
200
41
or 0.41
⫽
100
There are 34 green marbles, so 166 out of a
total of 200 are not green. This could also be
calculated as
34
1⫺
.
200
There are 38 yellow and 44 orange marbles out
of a total of 200.
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Example 2
a
b
PL
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 sticks
tested, 15 were found to be faulty.
Using this information, estimate the probability that a randomly selected USB
memory stick from this factory will not be faulty.
A store receives 350 USB memory sticks from the factory. Predict how many of these
will not be faulty.
Solution
P(not faulty USB memory sticks)
985
⫽
1000
⫽ 0.985
15 in 1000 USB memory sticks were found to be
faulty, so 1000 ⫺ 15 ⫽ 985 USB memory sticks
are not faulty.
Expected number of not-faulty
USB memory sticks
= 0.985 ⫻ 350
⬇345
From 350 USB memory sticks we expect a
proportion of 0.985 to be non faulty.
Round your answer to the nearest whole number.
⬇ means ‘approximately equal to’.
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a
Explanation
b
Exercise 8A
Example
1
1
One wrapped lolly was selected at random from a bag containing a large number of
red, green, orange, yellow and purple wrapped lollies. Its colour was noted and the
lolly returned to the bag. This was done 50 times. The results are shown below.
Colour
Frequency
Red
11
Green
19
Orange
6
Yellow
4
Purple
10
Estimate the probability that a lolly selected at random from the bag will be:
a
300
red
b
green
c
orange or yellow
d
not purple
Mathematics and Statistics Year 9
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Page 301
A factory in Hamilton makes MP3 players. Quality controllers found that for every
300 players they inspected two were faulty.
a
b
Estimate the probability that a randomly selected MP3 player will be faulty.
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 would
you 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 taken
from a vat and the sizes noted.
Size
Frequency
Medium
16
Large
48
Determine how many medium pop rivets you would expect in a bag containing:
i 100
ii 1000
iii 250
Determine how many pop rivets would not be large in a bag containing:
i 50
ii 700
iii 550
b
5
Small
26
PL
a
Tiny
10
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How many times would you expect a 4 to appear in:
a
c
30 rolls of a fair die?
100 rolls of a fair die?
b
d
50 rolls of a fair die?
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 were
coloured red, blue, yellow or green. The graphs below show the frequencies of the
different colours in 100 spins of each spinner. Use the graphs to determine how many
of the sectors were coloured red, blue, yellow and green in each spinner.
Frequency
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Example
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50
50
40
40
30
20
10
0
8
30
20
10
16 33 17 34
17 18 15 50
0
R B Y G
B Y G
Colour
Colour
Spinner A
Spinner B
A spinning wheel has the numbers 1 to 20 equally spaced around it. If the wheel is
spun 25 times, determine the expected number of:
a
d
9
Frequency
Chapter 8.qxd
R
even numbers
factors of 36
b
e
numbers over 10
numbers greater than 30
c
f
multiples of 5
prime numbers
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
b
10
Page 302
Use the information in the table to estimate the probability that:
i
a 1 will be obtained on the next throw
ii a 3 will be obtained on the next throw
iii a 6 will be obtained on the next throw
Do you think that the outcomes are equally likely or is the die biased? Explain
your answer.
Two companies, Brightspark and Solar, each produce light bulbs at the rate of 5000
bulbs per day. On a particular day a random sample of 60 bulbs from Brightsparks and
40 from Solar are inspected. The number of defective light bulbs was recorded.
Sample size
Number of defective globes
60
40
2
3
Brightspark
Solar
b
c
Calculate the experimental probability of choosing a defective bulb from:
i
Brightspark
ii
Solar
Calculate the expected number of defective bulbs produced in one day by:
i
Brightspark
ii
Solar
Calculate the expected number of defective bulbs produced in a week by:
i
Brightspark
ii
Solar
PL
a
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A bag contains six red marbles, five blue marbles and one silver marble. Four children
took turns to select a marble at random from the bag and then replaced it. All four
children 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 16
white cars.
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11
a
b
Calculate the probability that the next car to leave
the school carpark is:
i red
ii green
iii blue
iv white
Jane used these results to estimate the number of
blue cars in New Zealand. She says that one-sixth
of 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 they
landed on were recorded in lists i to vi. The lists are not in the same order as the
spinners. Match each spinner to the most likely list. Is there more than one correct
answer?
a
b
4
2
8
6
302
c
6
4
8
3
2
2
4
1
4
4
2
4
1
4
8
3
4
5
7
6
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d
e
3
f
5
1
2
2
i
iii
v
14
7
3
1, 2, 8, 6, 4, 3, 3, 7, 5, 4
1, 4, 3, 2, 4, 4, 2, 4, 1, 4
2, 2, 8, 4, 8, 2, 4, 6, 6, 8
ii
iv
vi
8
4
1, 1, 4, 2, 1, 2, 3, 2, 4, 2
8, 8, 2, 2, 8, 2, 8, 8, 2, 8
7, 1, 5, 5, 1, 3, 1, 5, 3, 3
E
1
A bag has 10 counters of four different colours. Some students took turns to select
and replace one counter at a time. They did this 80 times.
Colour
Blue
Red
Green
Yellow
Colour
PL
Use the given information to try to estimate how many counters of each colour were in
the bag. Explain your method.
Total
Chris
26
17
29
8
Total
Barry
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Chapter 8.qxd
Blue
Red
Green
Yellow
15
24
19
31
6
Colour
Total
Blue
Red
Green
Yellow
30
13
27
10
Colour
Total
Blue
Red
Green
Yellow
25
14
34
7
Mary
Susan
A box has 12 wrapped chocolates. They are all the same size and shape but have
different 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.
Centre
Strawberry
Caramel
Coconut
Nut
Mint
Total
11
14
9
19
7
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Equally likely outcomes
For many situations or experiments involving chance, we can use the mathematics of
probability to determine the level of chance of a particular event occurring. Probability is
the mathematical term used to describe chance.
When outcomes of an event are equally likely, each outcome has an equal chance of
occurring, i.e. their probabilities are the same.
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PL
E
Make a spinner like the one shown below. (See section 8–1 for
2 3
instructions for making a spinner.)
1
1
a Spin the spinner 50 times and record in a table the number of times
each number comes up.
2
1
4 4
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.
Key ideas
If A is a particular event then:
number of outcomes in A
P (A) ⫽
total number of possible outcomes
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’.
P(not A) ⫽1 ⫺P(A)
All probabilities are between 0 and 1, therefore:
0 ⭐ P(A) ⭐ 1
The sample space is a list of all the possible outcomes.
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).
304
1
2
3
7
3
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Solution
Explanation
a
Sample space is 1, 2, 3, 3, 7
b
P(3) ⫽
c
P(not a 3) ⫽ 1 ⫺ P(3)
2
⫽ 1⫺
5
3
⫽ or 0.6
5
2
or 0.4
5
e
P(3) ⫹ P(not a 3) ⫽
Alternatively count the number of sectors which
are not 3.
P(not 3) ⫽ P(1, 2, 7)
⫽
1
1
1
3
⫹ ⫹ ⫽
5
5
5
5
There are two 3s and one 7.
PL
P(a 3 or a 7) ⫽
number of sections labelled 3
number of equal sections
E
1
2
⫹
5
5
3
⫽ or 0.6
5
d
1
2
⫹
5
5
5
⫽
5
⫽ 1
Example 4
The sample space is a list of all possible
outcomes.
P(3) ⫽
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There are two 3s and there are three numbers
that are not a 3.
P(3) ⫹ P(not a 3) is all of the outcomes in the
sample space.
A fair six-sided die is rolled.
a
b
List the outcomes in the sample space.
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)
Solution
a
Sample space = {1, 2, 3, 4, 5, 6}
b
i
ii
2
4
⫽
6
3
2
1
⫽
6
3
Explanation
This is the probability of throwing a 1, 2, 3 or 4.
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0
iv
1
6
v
2
4
⫽
6
3
Page 306
Mulitples of 12 are 12, 24, 36 . . . None of these
numbers 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.
Exercise 8B
3
1
Find the probability of each of the following events:
a
b
c
d
e
4
2
obtaining a head when a fair coin is tossed
obtaining the number 1 when six-sided die is rolled
selecting a heart when a card is selected at random
from a pack of 52 playing cards
selecting an ace when a card is selected at random
from a pack of 52 playing cards
the spinner with the letters CHANCE written around
the wheel landing on C when it is spun
H
A
C
E
N
C
A fair eight-sided die is rolled.
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Example
PL
Example
E
Chapter 8.qxd
a
b
3
Find the probability of each of the following events:
a
b
c
d
e
f
g
306
List the outcomes in the sample space.
Find the following probabilities:
i
P(number greater than 5)
ii P(even number)
iii P(multiple of 3)
Choosing a female captain of a hockey team when the captain is chosen at
random from six girls and eight boys
Choosing a Wednesday when a day is chosen at random from the days of the
week
Choosing the month of May when a month is chosen at random from the months
of the year
Winning a raffle if 100 tickets were sold and you bought one ticket
Guessing the correct answer in a multiple-choice question with answers A, B, C,
D and E
Choosing a male class captain when the captain is chosen at random from a class
of 15 boys and 17 girls
Choosing a red ball when a ball is chosen at random from a box containing one
yellow, two blue and three red balls
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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
b
c
5
Page 307
List the outcomes in event A.
List the outcomes not in A.
Calculate P(A) and P(A⬘).
A fair six-sided die is rolled. What is the probability of obtaining each of the following?
a 2
d a prime number
a
b
7
If a prize is selected at random, what is the probability of choosing the following?
i
a bag of marbles
ii a chocolate
iii a yoyo
iv a yoyo or a chocolate
List the prizes in order from the most likely to the least likely.
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
8
E
A lucky dip at a fete contains 10 yoyos, 12 bags of marbles, 15 chocolates and
13 packets of chips.
PL
6
b a number greater than 1 c 2 or 6
e 7
f a number less than or equal to 8
O
b
H
c
a vowel
a
c
e
g
i
k
9
d
a letter in the word CAT
One card is selected at random from a pack of 52 playing cards. Calculate the
following probabilities:
P(queen)
b
P(black card)
d
P(not a queen)
f
P(queen and a heart) h
P(queen or a heart)
j
P(not a queen and a heart)
P(heart)
P(picture card)
P(not a picture card)
P(black card and a picture card)
P(black card or a picture card)
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The probability that a person sitting their drivers licence test passes the first time is
0.85. What is the probability that Rose fails the first time she sits her driving test?
10
3
We are told that the probability that the Rovers will win a football match is . If this is
4
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
c
List the outcomes in event A.
Calculate P(A) and P(A⬘).
b
List the outcomes in event A⬘.
12
Terry has 36 buttons in a jar. Twelve are red and the rest are black. If he selects a
button 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
1
1
it lands on red is and blue is . What is the probability that it lands on green?
3
2
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 bases
painted red, five blue, two green and one yellow. Every ball thrown lands in a box.
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The table below shows the prizes associated with each colour.
Yellow
Plasma TV
a
b
Blue
Red
MP3 player
$50
No prize
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?
Another six red boxes and three blue boxes and one green box are added. Has her
chance of winning a prize changed, explain your answer.
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.
E
15
Green
a
b
c
d
16
PL
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.
What is the probability that Allanah wins the major prize?
What is the probability that Brian wins the major prize?
What is the probability that Chi wins the major prize?
How many tickets did Damian buy?
Draw a spinner that will land on the given colour with the following probabilities:
1
, P(white) ⫽
6
1
P(blue) ⫽ , P(black) ⫽
4
P(blue) ⫽
1
1
and P(red) ⫽
2
3
1
1
3
, P(white) ⫽
and P(red) ⫽
4
8
8
SA
M
a
b
Enrichment: Scrabble
17
Scrabble is a game in which we take lettered tiles and try to arrange them to spell words.
a
b
c
d
e
308
If one tile of each letter of the alphabet and two blank tiles are placed in a bag
and one tile is selected, determine the probability of selecting:
i a blank
ii a vowel
iii an X, J or Q
iv an S
If you replace the tile and select another, what is the probabilty of repeating the
events in part a?
If you do not replace the tile and select another, what is the probabilty of
repeating the events in part a?
If you continue choosing tiles one at a time without
replacing them, what happens to the probabilities?
Investigate.
If you continue choosing tiles one at a time without
replacing them, will the probability of choosing
1
a particular letter ever be ? Explain your answer.
2
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Probability trees
Tree diagrams are useful for listing outcomes of experiments that have two or more
successive events.
E
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 the
second branch etc.
The outcomes for the combined events are listed on the right-hand side.
L
WWL
W
WLW
L
WLL
W
LWW
L
LWL
W
LLW
L
LLL
PL
W
Outcomes
WWW
W
W
SA
M
L
W
L
L
Example 5
A coin is thrown and its result recorded. Then the coin is thrown again and the result
recorded.
a
b
c
Complete a tree diagram to show all possible outcomes.
What is the total number of outcomes?
Find the probability of tossing:
i
two tails
ii one tail
iii at least one head
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Solution
a
Explanation
Toss 1
The tree diagram results in 2 ⫻ 2 ⫽ 4 outcomes.
Toss 2 Outcome
H
HH
H
T
HT
H
TH
T
TT
T
The total number of outcomes is 4.
c
i
P(TT) ⫽
ii
P(1 tail) ⫽
iii
P(at least one head) ⫽
There are four possibilities in the outcomes
column.
E
b
1
4
One of the four outcomes is TT.
Two outcomes have one tail: {HT, TH}
1
2
⫽
4
2
Three outcomes have at least one head:
{HH, HT, TH}
PL
3
4
Exercise 8C
Example
5
1
The following tree diagram shows all the possible outcomes of having two children in
a family.
Complete this tree diagram to show all the possible outcomes.
What is the total number of outcomes?
Find the probability of having:
i
two girls
ii one girl
iii at least one girl
iv at least one boy
SA
M
a
b
c
Toss 1
Toss 2 Outcome
B
BB
B
G
2
...
...
...
...
...
A spinner is numbered 1, 2, 3 and each number is equally likely to occur. The spinner
is spun twice.
a
b
c
310
...
List the set of possible outcomes as a tree diagram.
What is the total number of possible outcomes?
Find the probability of spinning:
i
two 3s
ii at least one 3
iii no more than one 2
iv two odd numbers
Mathematics and Statistics Year 9
Cambridge University Press • Uncorrected Sample Pages • 2008 © Brookie, Halford, Lawrence, Tiffen, Wallace
An experiment involves tossing three coins and noting the outcome.
a
b
4
Complete a tree diagram to list all the possible outcomes.
Find the following probabilities:
i
P(three heads)
ii P(two heads)
iii
iv P(no tails)
v P(no more than two tails)
P(at least two tails)
A restaurant is offering a three course meal for $25. There is a choice of shrimp or
spring rolls for a starter; steak, chicken or lamb for main course; and either chocolate
cake or apple pie for desert. Assume that each choice is equally likely to be chosen.
a
b
5
Page 311
Draw a probability tree to show all possible combinations of orders.
Calculate the probability that a customer orders:
i
shrimp, steak and apple pie
ii chicken
iii steak and chocolate cake
E
3
3:07 PM
A first-aid test includes three multiple-chioce questions. Tessa decides to guess. There
are three choices of answer (A, B and C) for each question.
PL
7/22/08
If only one of the possible choices (A, B or C) is correct for each question, find the
probability that Tessa guesses:
a
c
1 correct answer
3 correct answers
Guess 1
b
d
Guess 2
B
C
...
A
A
6
...
C
...
Complete a tree diagram for tossing a coin four times to find the probability that you
toss:
a
7
B
2 correct answers
no correct answers
Guess 3 Outcome
A
AAA
B
...
C
...
...
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Chapter 8.qxd
0 tails
b
1 tail
c
2 tails
d
3 tails
e
4 tails
Four coins were tossed 50 times. The results are shown:
HTHH
HTHT
THTT
HTTH
THHT
HTTH
HHTH
THTT
HTTT
TTHT
HHTH
TTTT
HHHT
TTTT
HHHH
HHHT
HHHT
TTTH
THHT
HHTT
TTTH
THTH
HHTT
HHHT
TTTH
THTH
HTHT
THTH
HHTT
HTTT
THTH
THTT
THTH
TTTH
TTTH
HHHT
HHHT
HHHT
HTHT
TTTH
HTHT
THHT
HTHH
THHT
HHHT
THTH
TTTH
THTT
HHTT
THTT
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a
Page 312
Copy and complete the following frequency table.
Number
of heads
Tally
Frequency
c
d
e
Determine the experimental probability of obtaining:
i
0 heads
ii 1 head
iii 2 heads
iv 3 heads v 4 heads
Use a tree diagram to represent all possible outcomes in tossing four coins.
Determine the theoretical probability of tossing:
i
0 heads
ii 1 head
iii 2 head
iv 3 heads v 4 heads
Compare the theoretical probabilities with the experimental probabilities. What
do you notice?8
PL
b
E
0
1
2
3
4
Enrichment
Hohepa randomly selects his clothing. He chooses one pair of shoes from his
collection of one black and two red pairs, a shirt from a collection of one white and
two blue shirts, and either red or black jeans.
SA
M
8
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
c
e
g
i
9
b
d
f
h
no red items
two black items
black jeans
white shirt or red jeans
Use a tree diagram to investigate the probabilities involved in selecting two counters
from a bag of 3 black and 2 white counters:
a
b
312
black shoes, white shirt and black jeans
one red item
at least two red items
red shoes and black jeans
not a black item
A counter is selected from a bag and then replaced before a second counter is
selected.
A counter is selected from a bag and not replaced before a second counter is
selected.
Is there any difference? Explain your answer.
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W O R K
I
Mathematically
N G
Probability
For this activity you will need:
a bag or jacket pocket
different-coloured counters
paper and pen
Five counters
c
E
b
Work with a partner. Without watching, have a third
person place five counters of two different colours
into the bag or pocket. An example of five counters
could be two red and three blue. It is important that
you do not know what the counters are.
Without looking, one person selects a counter from
the bag while the other person records its colour.
Replace the counter in the bag.
Repeat part b for a total of 100 trials. Record the
results in a table similar to this one.
PL
a
Colour
Frequency
|||| |||
|||| |||| ||
100
100
SA
M
Red
Blue
Total
Tally
d
e
Find the experimental probability for each colour.
Use these experimental probabilities to guess how many counters of each colour there
are in the bag.
Use this table to help
Colour
Total
f
Frequency
Experimental
probability
Closest multiple
of 15 or 0.2, e.g.
0.2, 0.4 . . .
Guess of how many
counters of this
colour
100
1
1
5
Now take the counters out of the bag and see if your guess is correct.
More colours and counters
a
b
Repeat the steps above but this time use three colours and 8 counters.
Repeat the steps above but this time use four colours and 12 counters.
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PL
E
Long run proportion and expected value
Long 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:
number of times event E occurs
P(event) ⫽
total number of trials of E
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 gets
closer to the theoretical probability.
Expected number ⫽ probability of the event E ⫻ number of trials of event E.
Equally likely outcomes
If A is a particular event then:
number of outcomes in A
• P(A) ⫽
total possible number of outcomes
• 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’.
• P(not A) ⫽ 1 ⫺ P(A)
All probabilities are between 0 and 1, therefore: 0 ⭐ P(A) ⭐ 1
The sample space is a list of all the possible outcomes.
SA
M
Review
Chapter summary
Probability trees
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 the second branch etc.
The outcomes for the combined events are listed on the right-hand side.
Game 1 Game 2 Outcomes
W
WW
W
WL
L
W
LW
L
LL
L
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Short-answer questions
Review
SA
M
PL
E
1 Jack forgets his homework diary one day per week (school day). During the 40-week
school 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)? Write
your 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
3
chocolate. If the probability of selecting a chocolate frog is how many strawberry
7
frogs 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 tails
12 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
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Page 316
13 An unusual six-sided die is biased and has an unknown number of 6s on its sides. The
die is tossed many times and the number of 6s is recorded at different stages of the
experiment, as shown in this table.
5
5
Number of tosses
Number of 6s
10
5
20
8
40
21
PL
E
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 P(red) ⫽ P(queen)
b P(ace) ⫽ P(king)
c P(heart) ⫽ P(club)
d P(ace of spades) ⫽ P(black jack)
15 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?
3
4
3
4
ii What is the highest possible score?
b List all the ways in which a total of eight can be
5
5
obtained.
c What is the chance of getting a total score of:
spinner A
spinner B
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.
SA
M
Review
Chapter 8.qxd
Outcome
1
2
3
4
5
6
Number of times
34
44
35
30
29
28
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 6
Write your answers as a:
i fraction
ii decimal
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Extended-response questions
E
The data below shows the winners of the American NBA finals from 1987 to 2006:
1987 LA Lakers
1988 LA Lakers
1989 Detroit
1990 Detroit
1991 Chicago
1992 Chicago
1993 Chicago
1994 Houston
1995 Houston
1996 Chicago
1997 Chicago
1998 Chicago
1999 San Antonio
2000 LA Lakers
2001 LA Lakers
2002 LA Lakers
2003 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.
Number of NBA
final wins
Estimated
probability of
a win
LA
Lakers
Total
PL
San
Antonio Chicago Detroit Houston Miami
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.
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.
SA
M
2
Review
1
Tom selected one ball at random 80 times, with replacement, and the results of his
experiment are displayed in the table below.
Colour
Number in 80 selections
3
Yellow
Red
3
24
Brown Purple
32
Blue
13
8
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 blue
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.
Number on die
1
2
3
4
5
6
Frequency
18
12
22
13
25
10
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.
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