Download 13 • Probability

STATiSTicS And probAbiliTy
Topic 13
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Probability
13.1 Overview
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What do you know?
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Probability lies at the heart of nature. Think about all the events that
had to happen for you to be born, for example . . . the odds are
extraordinary.
Probability is that part of mathematics that gives meaning to the idea
of uncertainty, of not fully knowing or understanding the occurrence of
some event. We often hear that there is a good chance of rain, people
bet with different odds that a favourite horse will win at Caulfield, and
so on. In each case, we are making a guess as to what will be the
outcome of some event.
It is important to learn about probability so that you can understand
that chance is involved in many decisions that you will have to make in
your life and in everyday events.
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Why learn this?
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1 THinK List what you know about probability. Use a thinking
tool such as a concept map to show your list.
2 pAir Share what you know with a partner and then with a
small group.
3 SHArE As a class, create a thinking tool such as a large concept
map to show your class’s knowledge about probability.
Learning sequence
Overview
Theoretical probability
Experimental probability
Venn diagrams and two-way tables
Two-step experiments
Mutually exclusive and independent events
Conditional probability
Review ONLINE ONLY
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13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
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WATcH THiS vidEo
The story of mathematics:
What are the chances?
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STATiSTicS And probAbiliTy
13.2 Theoretical probability
The language of probability
Unlikely
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• The probability of an event is a measure of the likelihood that the event will take
place.
• If an event is certain to occur, then it has a probability of 1.
• If an event is impossible, then it has a probability of 0.
• The probability of any other event taking place is given by a number between 0 and 1.
• The higher the probability, the more likely it is for the event to occur.
• Descriptive words such as ‘impossible’, ‘unlikely’, ‘likely’ and ‘certain’ are commonly
used when referring to the chance of an event occurring. Some of these are shown on the
probability scale below.
Even
chance
Likely
0.25
0.5
0.75
1
50%
WorKEd EXAmplE 1
100%
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0%
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0
Certain
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Impossible
EV
On the probability scale given at right, insert each
0
of the following events at appropriate points.
a You will sleep tonight.
b You will come to school the next Monday during a school term.
c It will snow in Victoria this year.
THinK
Carefully read the given statement
and label its position on the
probability scale.
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1
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b
2
Provide reasoning.
1
Carefully read the given statement
and label its position on the
probability scale.
2
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Provide reasoning.
1
WriTE/drAW
a
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a
0.5
a
0
0.5
1
Under normal circumstances, I will
certainly sleep tonight.
b
b
0
0.5
1
It is very likely but not certain that I will
come to school on a Monday during
term. Circumstances such as illness or
public holidays may prevent me from
coming to school on a specific Monday
during a school term.
Maths Quest 9
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STATiSTicS And probAbiliTy
Carefully read the given statement
and label its position on the
probability scale.
1
c c (Summer)
0
Provide reasoning.
2
c (Winter)
0.5
1
It is highly likely but not certain that
it will snow in Victoria during winter.
The chance of snow falling in Victoria
in summer is highly unlikely but not
impossible.
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Key terms
WorKEd EXAmplE 2
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• The study of probability uses many special terms that must be clearly understood. Here
is an explanation of some of the more common terms.
Chance experiment: A chance experiment is a process, such as rolling a die, that can be
repeated many times.
Trial: A trial is one performance of an experiment to get a result. For example, each roll
of the die is called a trial.
Outcome: The outcome is the result obtained when the experiment is conducted. For
example, when a normal six-sided die is rolled the outcome can be 1, 2, 3, 4, 5 or 6.
Sample space: The set of all possible outcomes is called the sample space and is given
the symbol ξ. For the example of rolling a die, ξ = {1, 2, 3, 4, 5, 6}.
Event: An event is the favourable outcome of a trial and is often represented by a capital
letter. For example, when a die is rolled, A could be the event of getting an even number;
A = {2, 4, 6}.
Favourable outcome: A favourable outcome for an event is any outcome that belongs to
the event. For event A above (rolling an even number), the favourable outcomes are 2, 4
and 6.
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For the chance experiment of rolling a die:
a list the sample space
b list the events:
i rolling a 4
ii rolling an even number
iii rolling at least 5
iv rolling at most 2
c list the favourable outcomes for:
i {4, 5, 6}
ii not rolling 5
iii rolling 3 or 4
iv rolling 3 and 4.
THinK
a
b
WriTE
The outcomes are the numbers 1 to 6.
a
This describes only 1 outcome.
b
i
ii
The possible even numbers are 2, 4
and 6.
ξ = {1, 2, 3, 4, 5, 6}
i
ii
{4}
{2, 4, 6}
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STATiSTicS And probAbiliTy
iii
‘At least 5’ means 5 is the smallest.
iii
{5, 6}
iv
‘At most 2’ means 2 is the largest.
iv
{1, 2}
i
The outcomes are shown inside the
brackets.
i
4, 5, 6
ii
‘Not 5’ means everything except 5.
c
c
ii
1, 2, 3, 4, 6
The event is {3, 4}.
iii
3, 4
iv
There is no number that is both
3 and 4.
iv
There are no favourable outcomes.
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iii
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Theoretical probability
P(Tails) = 12.
and
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• When a coin is tossed, there are two possible outcomes, Heads or Tails. That is,
ξ = {H, T}.
• In ideal circumstances, the two outcomes have the same likelihood of occurring, so they
are allocated the same probability.
For example, P(Heads) = 12 (This says the probability of Heads = 12.)
number of favourable outcomes
.
total number of outcomes
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P(A) =
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• The total of the probabilities equals 1, as there are no
other possible outcomes.
• In general, if all outcomes are equally likely to occur
(ideal circumstances), then the probability of event A
occurring is given by
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WorKEd EXAmplE 3
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A die is rolled and the number uppermost is noted. Determine the probability of
each of the following events.
a A = {1}
b B = {odd numbers}
c C = {4 or 6}
THinK
WriTE
There are 6 possible outcomes.
a
A has 1 favourable outcome.
a
P(A) = 16
b
B has 3 favourable outcomes: 1, 3
and 5.
b
P(B) = 36
C has 2 favourable outcomes.
c
c
= 12
P(C) = 26
= 13
474
Maths Quest 9
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STATiSTicS And probAbiliTy
Exercise 13.2 Theoretical probability
individUAl pATHWAyS
⬛
prAcTiSE
⬛
Questions:
1–16
conSolidATE
⬛
Questions:
1–18
⬛ ⬛ ⬛ Individual pathway interactivity
rEFlEcTion
Write a sentence using the
word ‘probability’ that shows
its meaning.
mASTEr
Questions:
1–19
int-4534
WE1 On the given probability scale, insert each of the
0
0.5
following events at appropriate points.
a The school will have a lunch break on Friday.
b Australia will have a swimming team in the Commonwealth Games.
c Australia will host two consecutive Olympic Games.
d At least one student in a particular class will obtain an A for Mathematics.
e Mathematics will be taught in secondary schools.
f In the future most cars will run without LPG or petrol.
g Winter will be cold.
h Bean seeds, when sown, will germinate.
Indicate the chance of each event listed in question 1 using one of the following terms:
certain, likely, unlikely, impossible.
WE2a For each chance experiment below, list the sample space.
a Rolling a die
b Tossing a coin
c Testing a light bulb to see whether it is defective or not
d Choosing a card from a normal deck and noting its colour
e Choosing a card from a normal deck and noting its suit
WE2b A normal 6-sided die is rolled. List each of the following events.
a Rolling a number less than or equal to 3
b Rolling an odd number
c Rolling an even number or 1
d Not rolling a 1 or 2
e Rolling at most a 4
f Rolling at least a 5
WE2c A normal 6-sided die is rolled. List the favourable outcomes for each of the
following events.
a A = {3, 5}
b B = {1, 2}
c C = ‘rolling a number greater than 5’
d D = ‘not rolling a 3 or a 4’
e E = ‘rolling an odd number or a 2’
f F = ‘rolling an odd number and a 2’
g G = ‘rolling an odd number and a 3’
doc-6308
doc-6309
doc-6310
5
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doc-6307
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FlUEncy
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STATistics and probability
A card is selected from a normal deck of 52 cards and its suit is noted.
a List the sample space.
b List each of the following events.
i ‘Drawing a black card’
ii ‘Drawing a red card’
iii ‘Not drawing a heart’
iv ‘Drawing a black or a red card’
7 How many outcomes are there for:
a rolling a die
b tossing a coin
c drawing a card from a standard deck
d drawing a card and noting its suit
e noting the remainder when a number is divided by 5?
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6
A card is drawn at random from a standard deck of 52 cards.
Note: ‘At random’ means that every card has the same chance of being selected.
Find the probability of selecting:
a an ace b a king c the 2 of spades d a diamond.
WE3 9
A card is drawn at random from a deck of 52. Find the probability of each event
below.
a A = {5 of clubs}
b B = {black card}
c C = {5 of clubs or queen of diamonds}
d D = {hearts}
8
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STATiSTicS And probAbiliTy
E = {hearts or clubs}
f F = {hearts and 5}
g G = {hearts or 5}
h H = {aces or kings}
i I = {aces and kings}
j J = {not a 7}
10 A letter is chosen at random from the letters in the word PROBABILITY. What is the
probability that the letter is:
a B
b not B
c a vowel
d not a vowel?
11 The following coloured spinner is spun and the colour is noted.
What is the probability of each of the events given below?
a A = {blue}
b B = {yellow}
c C = {yellow or red}
d D = {yellow and red}
e E = {not blue}
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UndErSTAndinG
N
A bag contains 4 purple balls and 2 green balls. If a
ball is drawn at random, then what is the probability
that it will be:
i purple
ii green?
b Design an experiment like the one in part a but where
the probability of drawing a purple ball is 3 times that
of drawing a green ball.
13 Design spinners (see question 11) using red, white and
blue sections so that:
a each colour has the same probability of being spun
b red is twice as likely to be spun as either of the other
2 colours
c red is twice as likely to be spun as white and 3 times as likely to be spun as blue.
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12 a
Do you think that the probability of tossing Heads is the same as the probability of
tossing Tails if your friend tosses the coin? What are some reasons that it might not be?
15 If the following four probabilities were given to you, which two would you say were
not correct? Give reasons why.
0.725, −0.5, 0.005, 1.05
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int-0089
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problEm SolvinG
16
Consider this spinner.
Discuss whether the spinner has equal chance
of falling on each of the colours.
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STATistics and probability
A box contains two coins. One is a double-headed coin, and the other is a normal coin
with Heads on one side and Tails on the other. You draw one of the coins from a box
and look at one of the sides. It is Heads. What is the probability that the other side
shows Heads also?
18 ‘Unders and Overs’ is a game played with two normal six-sided dice. The two dice are
rolled, and the numbers uppermost added to give a total. Players bet on the outcome
being ‘under 7’, ‘equal to 7’ or ‘over 7’. If you had to choose one of these outcomes,
which would you choose? Explain why.
19 Justine and Mary have designed a new darts game for their Year 9 Fete Day. Instead of
a circular dart board, their dart board is in the shape of two equilateral triangles. The
inner triangle (bullseye) has a side length of 3 cm, while the outer triangle has side
length 10 cm.
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10 cm
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3 cm
Given that a player’s dart falls in one of the triangles, what is the probability that it
lands in the bullseye? Write your answer correct to 2 decimal places.
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13.3 Experimental probability
Relative frequency
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•• A die is rolled 12 times and the outcomes are recorded in the table below.
1
2
3
4
5
6
Frequency
3
1
1
2
2
3
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Outcome
In this chance experiment there were 12 trials.
The table shows that the number 1 was rolled 3 times out of 12.
3
= 14.
•• So the relative frequency of 1 is 3 out of 12, or 12
As a decimal, the relative frequency of 1 is equal to 0.25.
•• In general, the relative frequency of an outcome =
the frequency of the outcome
.
total number of trials
If the number of trials is very large, then the relative frequency of each outcome becomes
very close to the theoretical probability.
478 Maths Quest 9
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STATiSTicS And probAbiliTy
WorKEd EXAmplE 4
For the chance experiment of rolling a die, the following
outcomes were noted.
Outcome
1
2
3
4
5
6
Frequency
3
1
4
6
3
3
THinK
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How many trials were there?
How many threes were rolled?
What was the relative frequency for each number written as decimals?
WriTE
Adding the frequencies
will give the number of
trials.
a
1 + 3 + 4 + 6 + 3 + 3 = 20 trials
b
The frequency of 3 is 4.
b
4 threes were rolled.
c
Add a relative
frequency row to the
table and complete it.
c
1
Frequency
3
3
20
=
0.15
3
4
1
4
1
20
4
20
=
0.05
=
0.2
6
6
20
=
0.3
5
6
3
3
3
20
=
0.15
3
20
=
0.15
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Relative
frequency
2
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Outcome
O
a
N
a
b
c
Group experiment
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• Organise for the class to toss a coin at least 500 times. For example, if the class has
20 students, each one should record 25 outcomes and enter their information into a grid
as shown below.
B
C
Total T
Total H
Relative
frequency
(Tails)
17 8
17
8
25
17
25
= 0.68
8
25
= 0.32
15 10
32
18
50
32
50
= 0.64
18
50
= 0.36
75
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etc.
T
Relative
frequency
(Heads)
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Group
Total
outcomes
• Questions:
1. In an ideal situation, what would you expect the relative frequencies to be? Has this
occurred?
2. As more information was added to the table, what happened to the relative frequencies?
3. What do you think might happen if the experiment was continued for another
500 tosses?
• A rule called the law of large numbers indicates that as the number of trials increases,
then the relative frequencies will tend to get closer to the expected value (in this
case 0.5).
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STATiSTicS And probAbiliTy
Experimental probability
• Sometimes it is not possible to calculate theoretical probabilities and in such
cases experiments, sometimes called simulations, are conducted to determine the
experimental probability.
• The relative frequency is equal to the experimental probability.
Experimental probability =
the frequency of the outcome
total number of trials
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For example, the spinner shown at right (made from
light cardboard and a toothpick) is not symmetrical,
and the probability of each outcome cannot be
determined theoretically.
However, the probability of each outcome can
be found by using the spinner many times and
recording the outcomes. If a large number of trials is
conducted, the relative frequency of each outcome
will be very close to its probability.
WorKEd EXAmplE 5
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The spinner shown above was spun 100 times and the following results
were achieved.
Outcome
1
2
3
4
Frequency
7
26
9
58
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How many trials were there?
What is the experimental probability of each outcome?
What is the sum of the 4 probabilities?
THinK
E
a
b
c
WriTE
Adding the frequencies will determine the
number of trials.
a
7 + 26 + 9 + 58 = 100 trials
b
The experimental probability equals the
relative frequency.
b
7
P(1) = 100
= 0.07
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a
26
P(2) = 100
= 0.26
9
P(3) = 100
= 0.09
58
P(4) = 100
= 0.58
c
480
Add the probabilities (they should equal 1).
c
0.07 + 0.26 + 0.09 + 0.58 = 1
Maths Quest 9
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STATiSTicS And probAbiliTy
Exercise 13.3 Experimental probability
individUAl pATHWAyS
⬛
prAcTiSE
⬛
Questions:
1–4, 6–13, 19–21, 25
conSolidATE
⬛
Questions:
1–4, 6–11, 15, 16, 22–25
⬛ ⬛ ⬛ Individual pathway interactivity
mASTEr
int-4535
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FlUEncy
Each of the two tables below contains the results of a chance experiment (rolling a
die). For each table, find:
i the number of trials held
ii the number of fives rolled
iii the relative frequency for each outcome, correct to 2 decimal places
iv the sum of the relative frequencies.
WE4
2
4
1
4
5
6
49
40
46
2
3
4
Frequency
3
1
5
Number
1
2
3
Frequency
52
38
45
a
Outcome
H
Frequency
22
Outcome
H
T
Frequency
31
19
T
28
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b
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A coin is tossed in two chance experiments. The outcomes are recorded in the tables
below. For each experiment, find:
i the relative frequency of both outcomes
ii the sum of the relative frequencies.
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5
Number
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1
rEFlEcTion
What are the most important
similarities between theoretical
and experimental probability
calculations?
Questions:
1–3, 5–8, 12–14, 16–25
Construct an irregular spinner using cardboard and a toothpick. By carrying out a
number of trials, estimate the probability of each outcome.
4 WE5 An unbalanced die was rolled 200 times and the following outcomes were
recorded.
SA
3
Number
1
2
3
4
5
6
Frequency
18
32
25
29
23
73
Using these results, find:
P(6)
b P(odd number)
c P(at most 2)
d P(not 3).
a
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STATiSTicS And probAbiliTy
5
A box of matches claims on its cover to contain 100 matches.
A survey of 200 boxes established the following results.
Number of matches
95
96
97
98
99
100 101 102 103 104
Frequency
1
13
14
17
27
55
16
13
14
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If you were to purchase a box of these matches, what
is the probability that:
a the box would contain 100 matches
b the box would contain at least 100 matches
c the box would contain more than 100 matches
d the box would contain no more than 100 matches?
30
UndErSTAndinG
Here is a series of statements based on experimental
probability. If a statement is not reasonable, give a reason why.
a I tossed a coin 5 times and there were 4 Heads, so P(H) = 0.8.
b Sydney Roosters have won 1064 matches out of the 2045 that they have played, so
P(Sydney will win their next game) = 0.54.
c P(The sun will rise tomorrow) = 1.
d At a factory, a test of 10 000 light globes showed that 7 were faulty. Therefore,
P(faulty light globe) = 0.0007.
e In Sydney it rains an average of 143.7 days each year, so P(it will rain in Sydney on
the 17th of next month) = 0.39.
7 At a birthday party, some cans of soft drink were put in a container of ice. There were
16 cans of Coke, 20 cans of Sprite, 13 cans of Fanta, 8 cans of Sunkist and 15 cans
of Pepsi.
If a can was picked at random, what is the probability that it was:
a a can of Pepsi
b not a can of Fanta?
8 MC In Tattslotto, 6 numbers are drawn from the numbers 1, 2, 3, . . . 45. The number
of different combinations of 6 numbers is 8 145 060. If you buy 1 ticket, what is the
probability that you will win the draw?
1
8 145 060
b
1
45
c
45
8 145 060
d
1
6
E
6
8 145 060
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Maths Quest 9
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STATistics and probability
If a fair coin is tossed 400 times, how many Tails are expected?
10 If a fair die is rolled 120 times, how many threes are expected?
11 WE9 MC A survey of high school students asked ‘Should Saturday be a normal
school day?’ 350 students voted yes, and 450 voted no. What is the probability that a
student chosen at random said no?
9 WE8 A
7
9
7
B C
16
16
9
D
9
14
1
350
E
In a poll of 200 people, 110 supported party M, 60 supported party N and 30 were
undecided. If a person is chosen at random from this group of people, what is the
probability that he or she:
a supports party M b supports party N
c supports a party d is not sure what party to support?
13
A random number is picked from N = {1, 2, 3, . . . 100}. What is the probability of
picking a number that is:
a a multiple of 3
b a multiple of 4 or 5
c a multiple of 5 and 6?
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12
The numbers 3, 5 and 6 are combined to form a three-digit number such that no digit
may be repeated.
a i How many numbers can be formed?
ii List them.
b Find P(the number is odd).
c Find P(the number is even).
d Find P(the number is a multiple of 5).
15 MC In a batch of batteries, 2 out of every 10 in a large sample were faulty. At this rate,
how many batteries are expected to be faulty in a batch of 1500?
A 2B
150C
200
D 300
E
750
16 Svetlana, Sarah, Leonie and Trang are volleyball
players. The probabilities that they will score a point
on serve are 0.6, 0.4, 0.3 and 0.2 respectively. How
many points on serve are expected from each player
if they serve 10 times each?
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STATistics and probability
A survey of the favourite leisure activity of 200 Year 9 students produced the
following results.
17 MC Activity
Number of students
Playing sport Fishing Watching TV Video games Surfing
58
26
28
38
50
N
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N
LY
The probability (given as a percentage) that a student selected at random from this
group will have surfing as their favourite leisure activity is:
A 50%B
100%C
25%D
0%E
29%
18 The numbers 1, 2 and 5 are combined to form a three-digit number, allowing for any
digit to be repeated up to three times.
a How many different numbers can be formed?
b List the numbers.
c Determine P(the number is even).
d Determine P(the number is odd).
e Determine P(the number is a multiple of 3).
AT
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REASONING
John has a 12-sided die numbered 1 to 12 and Lisa has a 20-sided die numbered
1 to 20. They are playing a game where the first person to get the number 10 wins.
They are rolling their dice individually.
a Find P(John gets a 10). b Find P(Lisa gets a 10).
c Is this game fair? Explain.
20 At a supermarket checkout, the scanners have temporarily broken down and the
cashiers must enter in the bar codes manually. One particular cashier overcharged
7 of the last 10 customers she served by entering the incorrect bar code.
a Based on the cashier’s record, what is the probability of making a mistake with the
next customer?
b Should another customer have any objections with being served by this cashier?
c Justify your answer to part b.
21 If you flip a coin 6 times, how many of the possible outcomes could include a Tail on
the second toss?
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PL
problem solving
In a jar, there are 600 red balls, 400 green balls, and an unknown number of yellow balls.
If the probability of selecting a green ball is 15, how many yellow balls are in the jar?
23 In another jar there are an unknown number of balls, N, with 20 of them green. The
other colours contained in the jar are red, yellow and blue, with P(red or yellow) = 12,
P(red or green) = 14 and P(blue) = 13. Determine the number of red, yellow and blue
balls in the jar.
24 The gender of babies in a set of triplets is simulated by flipping 3 coins. If a coin lands
Tails up, the baby is a boy. If a coin lands Heads up, the baby is a girl. In the simulation,
the trial is repeated 40 times. The following results show the number of Heads obtained
in each trial:
0, 3, 2, 1, 1, 0, 1, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 2, 3, 2, 1, 3, 0, 2, 1, 2, 0, 3, 1, 3, 0, 1, 0, 1, 3,
2, 2, 1, 2, 1.
a Calculate the probability that exactly one of the babies in a set of triplets is female.
b Calculate the probability that more than one of the babies in the set of triplets is
female.
SA
22
484 Maths Quest 9
c13Probability.indd 484
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
A survey of the favourite foods of Year 9 students is recorded, with the following results.
Hamburger
45
Fish and chips
31
Macaroni and cheese
30
Lamb souvlaki
25
BBQ pork ribs
21
Cornflakes
17
T-bone steak
14
Banana split
12
9
Hot dogs
8
AT
IO
Corn-on-the-cob
Garden salad
8
Veggie burger
7
Fruit salad
6
5
AL
U
Smoked salmon
Muesli
N
LY
Tally
O
Meal
N
25
3
Estimate the probability that macaroni and cheese is the favourite food of a
randomly selected Year 9 student.
b Estimate the probability that a vegetarian dish is a randomly selected student’s
favourite food.
c Estimate the probability that a beef dish is a randomly selected student’s
favourite food.
EV
a
M
PL
E
doc-6313
13.4 Venn diagrams and two-way tables
The complement of an event
SA
• Suppose that a die is rolled: ξ = {1, 2, 3, 4, 5, 6}.
If A is the event ‘rolling an odd number’, then A = {1, 3, 5}.
• There is another event called ‘the complement of A’, or ‘not A’. This event contains all
the outcomes that do not belong to A. It is given the symbol A′.
• In this case A′ = {2, 4, 6}.
• A and A′ can be shown on a Venn diagram.
ξ
ξ
A
1
3 5
4
6
2
A is coloured.
A
1
3 5
4
6
2
A′ (not A) is coloured.
Topic 13 • Probability 485
c13Probability.indd 485
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
WorKEd EXAmplE 6
For the sample space ξ = {1, 2, 3, 4, 5}, list the complement of each of the
following events.
a A = {multiples of 3}
b B = {square numbers}
c C = {1, 2, 3, 5}
WriTE
The only multiple of 3 in the set is 3.
Therefore A = {3}. A′ is every other element of the set.
b
The only square numbers are 1 and 4.
Therefore B = {1, 4}. B′ is every other element of
the set.
c
C = {1, 2, 3, 5}. C′ is every other element of the set.
a
A′ = {1, 2, 4, 5}
b
B′ = {2, 3, 5}
O
a
N
LY
THinK
C′ = {4}
N
c
AT
IO
Venn diagrams and two-way tables
Venn diagrams
ξ
A
ξ
2
ξ
A
B
3
7
8
B
2
A
3
7
B
2
7
8
8
M
PL
E
3
EV
AL
U
• Venn diagrams convey information in a concise manner and are often used to illustrate
sample spaces and events. Here is an example.
– In a class of 20 students, 5 study Art, 9 study Biology, and 2 students study both
subjects.
– This information is shown on the diagram below, where
A = {students who study Art} and B = {students who study Biology}.
A contains 5 students.
B contains 9 students.
Note: In the case shown above, 8 students in the class study neither Art nor Biology.
SA
• The Venn diagram has 4 regions, each with its own name.
A∩B
ξ
A
3
ξ
B
2
A
3
7
8
There are 2 students who study Art and
Biology.
They occupy the region called ‘A and B’
or A ∩ B.
486
A ∩ B′
B
2
7
8
There are 3 students who study Art but not
Biology.
They occupy the region called ‘A and not
B’ or A ∩ B′.
Maths Quest 9
c13Probability.indd 486
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
ξ
A′ ∩ B
A
3
A′ ∩ B′
ξ
A
B
2
3
7
B
2
7
8
8
The remaining 8 students study neither
subject.
They occupy the region called ‘not A and
not B’ or A′ ∩ B′.
N
LY
There are 7 students who study Biology
but not Art.
They occupy the region called ‘not A and
B’ or A′ ∩ B.
• The information can also be summarised in a two-way table.
Total
5
15
9
11
20
Total
N
Not Biology
3
8
AT
IO
Art
Not Art
Biology
2
7
O
Two-way tables
Note: Nine students in total study Biology and 11 do not. Five students in total study Art
and 15 do not.
AL
U
Number of outcomes
• If event A contains 7 outcomes or members, this is written as n(A) = 7.
• So n(A ∩ B′) = 3 means that the event ‘A and not B’ has 3 outcomes.
EV
WorKEd EXAmplE 7
THinK
Identify the regions showing M and add the
outcomes.
M
6
11 15
4
WriTE/drAW
a
SA
a
M
PL
E
For the Venn diagram shown, write down the number of outcomes in each of the
following.
aM
b M′
c M∩N
ξ
N
d M ∩ N′
e M′ ∩ N′
ξ
M
6
N
11 15
4
n(M) = 6 + 11 = 17
b
Identify the regions showing M′ and add the
outcomes.
b
ξ
M
6
N
11 15
4
n(M′) = 4 + 15 = 19
Topic 13 • Probability 487
c13Probability.indd 487
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
c
M ∩ N means ‘M and N’. Identify the region.
ξ
c
M
6
N
11 15
4
n(M ∩ N) = 11
M ∩ N′ means ‘M and not N’. Identify the
region.
d
ξ
M
6
N
11 15
N
LY
d
4
n(M ∩ N′) = 6
M′ ∩ N′ means ‘not M and not N’. Identify the
regions.
e
ξ
N
11 15
O
e
M
4
N
6
AT
IO
n(M′ ∩ N′) = 4
AL
U
WorKEd EXAmplE 8
EV
Show the information from the Venn diagram on a
two-way table.
THinK
M
PL
SA
2
3
488
A
3
There are 7 elements in A and B.
There are 3 elements in A and ‘not B’.
There are 2 elements in ‘not A’ and B.
There are 5 elements in ‘not A’ and
‘not B’.
B
7 2
5
WriTE
Draw a 2 × 2 table and add the labels
A, A′, B and B′.
E
1
ξ
A
A′
A
A′
B
7
2
B′
3
5
B
B′
Add in a column and a row to show
the totals.
A
A′
Total
B
7
2
9
B′
3
5
8
Total
10
7
17
Maths Quest 9
c13Probability.indd 488
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
WorKEd EXAmplE 9
Left-handed
Right-handed
7
20
17
48
Blue eyes
Not blue eyes
n(L ∩ B) = 7
n(L ∩ B′) = 17
n(L′ ∩ B) = 20
n(L′ ∩ B′) = 48
ξ
L
17
7
B
20
48
N
2
Draw a Venn diagram that includes a sample
space and events L for left-handedness and
B for blue eyes. (Right-handedness = L′)
AT
IO
1
drAW
O
THinK
N
LY
Show the information from the two-way table on a Venn
diagram.
Event A or B
M
PL
E
EV
AL
U
• This Venn diagram illustrates the results of a survey of 20 people, showing whether they
drink tea and whether they drink coffee.
In all there are 19 people who drink tea or coffee.
They are found in the shaded region of the diagram.
ξ
This large group of people is written as C ∪ T and
T
called ‘C or T ’. Note that the people who drink both
tea and coffee, C ∩ T, are included in this group.
C
5
n(C ∪ T) = 19
12
A number of people drink tea or coffee, but not
2
both. This group contains the 2 people who drink
only tea and the 5 people who drink only coffee.
1
n(people who drink tea or coffee, but not both) = 7
WorKEd EXAmplE 10
SA
In a class of 24 students, 11 students play basketball, 7 play tennis, and 4 play both sports.
a Show the information on a Venn diagram.
b If one student is selected at random, then find the probability that:
i the student plays basketball
ii the student plays tennis or basketball
iii the student plays tennis or basketball but not both.
THinK
a
1
Draw a sample space with events B and T.
drAW
a
ξ
B
T
Topic 13 • Probability 489
c13Probability.indd 489
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
n(B ∩ T) = 4
n(B ∩ T′) = 11 − 4 = 7
n(T ∩ B′) = 7 − 4 = 3
So far, 14 students out of 24 have been
placed.
n(B′ ∩ T′) = 24 − 14 = 10
2
Identify the number of students who play
basketball.
i
ξ
B
7
B
7
T
4 3
10
b
i
number of students who play basketball
total number of students
n(B)
=
24
11
=
24
P(B) =
N
LY
b
ξ
T
4 3
number of favourable outcomes
total number of outcomes
Identify the number of students who play
tennis or basketball.
ξ
B
7
ii
T
4 3
AL
U
10
Identify the number of students who play
tennis or basketball but not both.
ξ
B
7
T
4 3
iii
=
M
PL
E
10
n(B ∩ T′) + n(B′ ∩ T) = 3 + 7
= 10
P(tennis or basketball but not both)
10
24
5
=
12
EV
iii
n(T ∪ B)
24
14
=
24
7
=
12
P(T ∪ B) =
AT
IO
ii
N
P B =
O
10
Exercise 13.4 Venn diagrams and two-way tables
individUAl pATHWAyS
⬛
SA
rEFlEcTion
How will you remember the
difference between when one
event and another occurs and
when one event or another
occurs?
prAcTiSE
Questions:
1–5, 7, 9, 11, 13, 15–17
⬛
conSolidATE
⬛
Questions:
1–4, 6, 8, 10, 13–15, 17–19
⬛ ⬛ ⬛ Individual pathway interactivity
mASTEr
Questions:
1, 3, 4, 6, 8, 10, 12, 14, 16–20
int-4536
FlUEncy
1
490
If ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, list the complement of each of the
following events.
a A = {multiples of 3}
b B = {numbers less than 20}
c C = {prime numbers}
d D = {odd numbers and numbers greater than 16}
WE6
Maths Quest 9
c13Probability.indd 490
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
For the Venn diagram shown, write down the number of
outcomes in:
a ξ
b S
c T
d S∩T
e T∩S
f T ∩ S′
g S′ ∩ T′.
3 WE8 Show the information from question 2 on a two-way table.
4 WE9 Show the information from this two-way table on a Venn diagram.
W′
V
21
7
V′
2
10
S
5
6
T
7
9
doc-6311
N
LY
W
ξ
doc-6312
O
For each of the following Venn diagrams, use set notation to write the name of the
region coloured in:
i purple
ii pink.
a
b
ξ
W
ξ
c
A
B
ξ
N
5
WE7
AT
IO
2
A
B
The membership of a tennis club consists of 55 men and 45 women.
There are 27 left-handed people, including 15 men.
a Show the information on a two-way table.
b Show the information on a Venn diagram.
c If one member is chosen at random, find the probability that the
person is:
i right-handed
ii a right-handed man
iii a left-handed woman.
7 Using the information given in the Venn diagram, if one outcome is chosen at random, find:
a P(L)
b P(L′)
c P(L ∩ M)
d P(L ∩ M′).
WE10
8
Using the information given in the table, if one family is chosen at random, find the
probability that they own:
SA
M
PL
E
EV
AL
U
6
ξ
L
3
5
M
7
10
Pets owned by families
Cat
No cat
Dog
4
11
No dog
16
9
a cat
a cat and a dog
c a cat or a dog or both
d a cat or a dog but not both
e neither a cat nor a dog.
a
b
Topic 13 • Probability 491
c13Probability.indd 491
18/05/16 4:29 PM
STATistics and probability
ξ
group of athletes was surveyed and the results were shown on
S
a Venn diagram.
L
3 5
S = {sprinters} and L = {long jumpers}.
2
a How many athletes were included in the survey?
b If one of the athletes is chosen at random, what is the probability
that the athlete competes in:
i long jump
ii long jump and sprints
iii long jump or sprints
iv long jump or sprints but not both?
9A
N
LY
UNDERSTANDING
6
If ξ = {children}, S = {swimmers} and R = {runners}, describe in words each of the
following.
a S′
b S ∩ R c R′ ∩ S′ d R ∪ S
11 A group of 12 students was asked
C
H
whether they liked hip hop (H) and
Ali
✓
✓
whether they liked classical music
Anu
(C). The results are shown in the
Chris
✓
table below.
a Show the results on:
George
✓
i a Venn diagram
Imogen
✓
i i a two-way table.
Jen
✓
✓
b If one student is selected at
Luke
✓
✓
random, find:
Pam
✓
i P(H)
Petra
i i P(H ∪ C)
Roger
✓
i ii P(H ∩ C)
Seedevi
✓
iv
P(student likes classical or hip
Tomas
hop but not both).
SA
M
PL
E
EV
AL
U
AT
IO
N
O
10
12
Place the elements of the following sets of numbers in their correct position in a single
Venn diagram.
A = {prime numbers from 1 to 20}
B = {even numbers from 1 to 20}
C = {multiples of 5 from 1 to 20}
ξ = {numbers between 1 and 20 inclusive}
492 Maths Quest 9
c13Probability.indd 492
18/05/16 4:29 PM
STATistics and probability
REASONING
One hundred Year 9 Maths students were asked to indicate their favourite topic in
mathematics. Sixty chose Probability, 50 chose Measurement and 43 chose Algebra.
Some students chose two topics: 15 chose Probability and Algebra, 18 chose
Measurement and Algebra, and 25 chose Probability and Measurement. Five students
chose all three topics.
ξ
a Copy and complete the Venn diagram at right.
b How many students chose Probability only?
Probability 20 Measurement
c How many students chose Algebra only?
d How many students chose Measurement only?
5
e How many students chose any two of the three
topics?
A student is selected at random from this group. Find
Algebra
the probability that this student has chosen:
f Probability
g Algebra
h Algebra and Measurement
i Algebra and Measurement but not Probability
j all of the topics.
14 Create a Venn diagram using two circles to accurately describe the relationships
between the following quadrilaterals: rectangle, square and rhombus.
15 Use the Venn diagram at right to write the numbers
8
ξ
A
B
of the correct regions for each of the following
4
problems.
1
5
a A′ ∪ (B′ ∩ C)
b A ∩ (B ∩ C′)
3
c A′ ∩ (B′ ∪ C′)
d (A ∪ B ∪ C)′
2
6
16 A recent survey taken at a cinema asked 90 teenagers
what they thought about three different movies. In
7
C
total, 47 liked ‘Hairy Potter’, 25 liked ‘Stuporman’
and 52 liked ‘There’s Something About Fred’.
16 liked ‘Hairy Potter’ only.
4 liked ‘Stuporman’ only.
27 liked ‘There’s Something About Fred’ only.
There were 11 who liked all three films and 10 who liked none of them.
a Construct a Venn diagram showing the results of the survey.
b What is the probability that a teenager chosen at random liked ‘Hairy Potter’ and
‘Stuporman’ but not ‘There’s Something About Fred’?
SA
M
PL
E
EV
AL
U
AT
IO
N
O
N
LY
13
PROBLEM SOLVING
17
120 children attended a school holiday
program during September. They were
asked to select their favourite board game
from Cluedo, Monopoly and Scrabble.
They all selected at least one game, and
4 children chose all three games.
In total, 70 chose Monopoly and
55 chose Scrabble.
Topic 13 • Probability 493
c13Probability.indd 493
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STATistics and probability
EV
AL
U
AT
IO
N
O
N
LY
Some children selected exactly two games — 12 chose Cluedo and Scrabble, 15 chose
Monopoly and Scrabble, and 20 chose Cluedo and Monopoly.
a Draw a Venn diagram to represent the children’s selections.
b What is the probability that a child selected at random did not choose Cluedo as a
favourite game?
18 Valleyview High School offers three sports at Year 9: baseball, volleyball and soccer.
There are 65 students in Year 9.
2 have been given permission not to play sport due to injuries and medical conditions.
30 students play soccer.
9 students play both soccer and volleyball but not baseball.
9 students play both baseball and soccer (including those who do and don’t play
volleyball).
4 students play all three sports.
12 students play both baseball and volleyball (including those who do and don’t
play soccer).
The total number of players who play baseball is 1 more than the total of students who
play volleyball.
a Determine the number of students who play volleyball.
b If a student was selected at random, what is the probability that this student plays
soccer and baseball only?
19 A Venn diagram consists of overlapping ovals which are used to show the relationships
between sets.
Consider the numbers 156 and 520. Show how a Venn diagram could be used to
determine their:
a HCF b
LCM.
20 A group of 200 shoppers was asked which type of fruit they had bought in the last
week. The results are shown in the table.
SA
M
PL
E
Fruit
Apples (A) only
Bananas (B)
only
Cherries (C)
only
A and B
A and C
B and C
A and B and C
Number of shoppers
45
34
12
32
15
26
11
Display this information in a Venn diagram.
Calculate n(A ∩ B′ ∩ C).
c How many shoppers purchased apples and bananas but not cherries?
d Calculate the relative frequency of shoppers who purchased:
i apples
ii bananas or cherries.
e Estimate the probability that a shopper purchased cherries only.
a
b
494 Maths Quest 9
c13Probability.indd 494
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
cHAllEnGE 13.1
N
LY
13.5 Two-step experiments
The sample space
int-2772
Bag 1
AL
U
AT
IO
N
O
• Imagine two bags (that are not transparent) that contain coloured counters. The first
bag has a mixture of black and white counters, and the second bag holds red, green and
yellow counters. In a probability experiment, one counter is to be selected at random
from each bag and its colour noted.
• The sample space for this experiment can be found using a table called an array that
systematically displays all the outcomes.
Bag 2
Bag 2
G
Y
B
BR
BG
BY
W
WR
WG
WY
EV
R
E
Bag 1
SA
M
PL
The sample space, ξ = {BR, BG, BY, WR, WG, WY}.
• The sample space can also be found using a tree diagram.
First
selection
B
W
Second
selection
Sample
space
R
BR
G
BG
Y
BY
R
WR
G
WG
Y
WY
Topic 13 • Probability 495
c13Probability.indd 495
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
WorKEd EXAmplE 11
WriTE/drAW
Draw an array (a table) showing all the possible
outcomes.
2
1, 2
2, 2
3, 2
4, 2
5, 2
6, 2
O
1
2
3
4
5
6
1
1, 1
2, 1
3, 1
4, 1
5, 1
6, 1
AT
IO
First die
Second die
3
1, 3
2, 3
3, 3
4, 3
5, 3
6, 3
N
THinK
N
LY
Two dice are rolled and the numbers uppermost are noted.
List the sample space in an array.
a How many outcomes are there?
b How many outcomes contain at least one 5?
c What is P(at least one 5)?
4
1, 4
2, 4
3, 4
4, 4
5, 4
6, 4
a
The table shows 36 outcomes.
b
Count the outcomes that contain 5. The cells are
shaded in the table.
b
Eleven outcomes include 5.
c
There are 11 favourable outcomes and 36 in
total.
c
P(at least one 5) = 11
36
There are 36 outcomes.
EV
E
M
PL
WorKEd EXAmplE 12
6
1, 6
2, 6
3, 6
4, 6
5, 6
6, 6
AL
U
a
5
1, 5
2, 5
3, 5
4, 5
5, 5
6, 5
SA
Two coins are tossed and the outcomes are noted. Show the
sample space on a tree diagram.
a How many outcomes are there?
b Find the probability of tossing at least one Head.
THinK
1
Draw a tree representing the outcomes for the
toss of the first coin
WriTE/drAW
First coin
H
T
496
Maths Quest 9
c13Probability.indd 496
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
2
For the second coin the tree looks like this:
First coin
Second coin
H
Second coin
Sample space
H
HH
T
HT
H
TH
H
T
T
List the outcomes.
TT
Count the outcomes in the sample space.
a
There are 4 outcomes (HH, HT, TH, TT).
b
Three outcomes have at least one Head.
b
P(at least one Head) = 34
N
O
a
Two-step experiments
• When a coin is tossed, P(H) = 12, and when a die
is rolled, P(3) = 16.
AL
U
If a coin is tossed and a die is rolled, what is the
probability of getting a Head and a 3?
• Consider the sample space.
AT
IO
3
N
LY
T
Add this tree to both ends of the first tree.
2
3
4
5
6
H
H, 1
H, 2
H, 3
H, 4
H, 5
H, 6
T
T, 1
T, 2
T, 3
T, 4
T, 5
T, 6
EV
1
1
.
There are 12 outcomes, and P(Head and 3) = 12
E
1
= 12 × 16.
• In this case, P(Head and 3) = P(H) × P(3); that is, 12
SA
M
PL
• In general, if A is the outcome of one event and B is the outcome of a separate event,
then
P(A ∩ B) = P(A) × P(B).
WorKEd EXAmplE 13
In one cupboard Joe has 2 black t-shirts and 1 yellow one. In his drawer there are 3 pairs of white
socks and 1 black pair. If he selects his clothes at random, what is the probability that his socks and
t-shirt will be the same colour?
THinK
WriTE
If they are the same colour then they must be black.
P(Bt ∩ Bs) = P(Bt) × P(Bs)
P(black t-shirt) = P(Bt) = 23
= 23 ×
P(black socks) = P(Bs) = 14
= 16
1
4
Topic 13 • Probability 497
c13Probability.indd 497
18/05/16 4:29 PM
STATISTICS AND PROBABILITY
Choosing with replacement
• Consider what happens when replacement is allowed in an experiment. Worked example 14
illustrates this situation.
WORKED EXAMPLE 14
1
Draw a tree for the first trial. Write the
probability on the branch.
Note: The probabilities should sum to 1.
R
3
5
O
a
WRITE/DRAW
N
THINK
AT
IO
2
5
For the second trial the tree is the same.
Add this tree to both ends of the first tree.
3
5
2
5
For both draws P(R) = 35 and P(B) = 25.
Use the rule P(A ∩ B) = P(A) × P(B)
to determine the probabilities.
SA
M
PL
E
b
List the outcomes.
EV
3
b
B
3
5
R
AL
U
2
N
LY
A bag contains 3 red and 2 blue counters. A counter is taken at random from the bag,
its colour is noted, then it is returned to the bag and a second counter is chosen.
a Show the outcomes on a tree diagram.
b Find the probability of each outcome.
c Find the sum of the probabilities.
B
R
2
3 5 B
5
R
RR
B
BB
2
5
RB
BR
P(R ∩ R) = P(R) × P(R)
= 35 × 35
9
= 25
P(R ∩ B) = P(R) × P(B)
= 35 × 25
6
= 25
P(B ∩ R) = P(B) × P(R)
= 25 × 35
6
= 25
P(B ∩ B) = P(B) × P(B)
= 25 × 25
4
= 25
c
498
Add the probabilities.
9
c 25
6
6
4
+ 25
+ 25
+ 25
=1
Maths Quest 9
c13Probability.indd 498
18/05/16 4:34 PM
STATISTICS AND PROBABILITY
• In Worked example 14, P(R) = 35 and P(B) = 25 for both trials.
This would not be so if a counter is selected but not replaced.
Choosing without replacement
If the first counter randomly selected
is red, then the sample space for the
second draw looks like this:
N
LY
• Let us consider again the situation described in Worked example 14, and
consider what happens if the first marble is not replaced.
• Initially the bag contains 3 red and 2 blue counters, and either a red
counter or a blue counter will be chosen.
P(R) = 35 and P(B) = 25.
• If the counter is not replaced, then the sample space is affected as
follows:
WORKED EXAMPLE 15
So P(R) = 34 and P(B) = 14.
EV
So P(R) = 24 and P(B) = 24.
AL
U
AT
IO
N
O
If the first counter randomly selected is
blue, then the sample space for the second
draw looks like this:
SA
THINK
M
PL
E
A bag contains 3 red and 2 blue counters. A counter is taken at
random from the bag and its colour is noted, then a second
counter is drawn, without replacing the first one.
a Show the outcomes on a tree diagram.
b Find the probability of each outcome.
c Find the sum of the probabilities.
a
Draw a tree diagram, listing the
probabilities.
WRITE/DRAW
2
4
a
3
5
RR
B
RB
R
BR
B
BB
R
2
4
2
5
R
3
4
B
1
4
Topic 13 • Probability 499
c13Probability.indd 499
18/05/16 4:34 PM
STATiSTicS And probAbiliTy
b
Use the rule P(A ∩ B) = P(A) × P(B) to
determine the probabilities.
b
P(R ∩ R) = P(R) × P(R)
= 35 ×
2
4
6
= 20
3
= 10
P(R ∩ B) = P(R) × P(B)
6
= 20
3
= 10
2
4
N
LY
= 35 ×
P(B ∩ R) = P(B) × P(R)
3
4
O
= 25 ×
AT
IO
N
6
= 20
3
= 10
Add the probabilities.
3
c 10
= 25 ×
1
4
2
= 20
1
= 10
3
3
1
+ 10
+ 10
+ 10
=1
EV
c
AL
U
P(B ∩ B) = P(B) × P(B)
E
Exercise 13.5 Two-step experiments
individUAl pATHWAyS
500
⬛
prAcTiSE
M
PL
SA
rEFlEcTion
How does replacement affect
the probability of an event
occurring?
Questions:
1–10, 12
⬛
conSolidATE
⬛
Questions:
1–15
⬛ ⬛ ⬛ Individual pathway interactivity
mASTEr
Questions:
1–17
int-4537
FlUEncy
1
In her cupboard Rosa has 3 scarves (red,
blue and pink) and 2 beanies (brown and
purple). If she randomly chooses 1 scarf
and 1 beanie, show the sample space in
an array.
Maths Quest 9
c13Probability.indd 500
18/05/16 4:29 PM
STATistics and probability
2
If two dice are rolled and their sum is noted, complete the array below to show
the sample space.
WE11 Die 1
1
3
4
6
2
2
7
3
4
O
Die 2
5
N
LY
1
2
5
6
AT
IO
N
9
What is P(rolling a total of 5)?
b What is P(rolling a total of 1)?
c What is the most probable outcome?
3 One box contains red and blue pencils, and a second box contains red, blue and green
pencils. If one pencil is chosen at random from each box and the colours are noted, draw
a tree diagram to show the sample space.
A bag contains 3 discs labelled 1, 3 and 5, and another bag contains two discs,
labelled 2 and 4, as shown below. A disc is taken from each bag and the larger number is
recorded.
SA
4
M
PL
E
EV
AL
U
a
WE12 5
1
3
2
4
Topic 13 • Probability 501
c13Probability.indd 501
18/05/16 4:29 PM
STATistics and probability
a
Complete the tree diagram below to list the sample space.
2
2
4
4
2
3
1
N
LY
3
What is:
i P(2)
ii P(1)
iii P(odd number)?
5Two dice are rolled and the difference between the two numbers is found.
a Use an array to find all the outcomes.
b Find:
i P(odd number)
ii P(0)
iii P(a number more than 2)
iv P(a number no more than 2).
6 WE13 A die is rolled twice. What is the probability of rolling:
a a 6 on the first roll b a double 6
c an even number on both dice d a total of 12?
7A coin is tossed twice.
a Show the outcomes on a tree diagram.
b What is:
i P(2 Tails)
ii P(at least 1 Tail)?
8 WE14 A bag contains 3 red counters and 1 blue counter. A counter is chosen at
random. A second counter is drawn with replacement.
a Show the outcomes and probabilities on a tree diagram.
b Find the probability of choosing:
i a red counter then a blue counter
ii two blue counters.
9 WE15 A bag contains 3 black balls and 2 red balls. If two balls are selected, randomly,
without replacement:
a show the outcomes and their probabilities on a tree diagram
b find P(2 red balls).
SA
M
PL
E
EV
AL
U
AT
IO
N
O
b
Understanding
The kings and queens from a deck of cards
are shuffled, then 2 cards are chosen. Find
the probability that 2 kings are chosen:
a if the first card is replaced
b if the first card is not replaced.
11 Each week John and Paul play 2 sets
of tennis against each other. They each
have an equal chance of winning the first
set. When John wins the first set, his
10
502 Maths Quest 9
c13Probability.indd 502
18/05/16 4:29 PM
STATiSTicS And probAbiliTy
probability of winning the second set rises to 0.6, but if he loses the first set, he has
only a 0.3 chance of winning the second set.
a Show the possible outcomes on a tree diagram.
b What is:
i P(John wins both sets)
ii P(Paul wins both sets)
iii P(they win 1 set each)?
N
LY
rEASoninG
A bag contains 4 red and 6 yellow balls. If the first ball drawn is yellow, explain the
difference in the probability of drawing the second ball if the first ball was replaced
compared to not being replaced.
13 Three dice are tossed and the total is recorded.
a What are the smallest and largest possible totals?
b Calculate the probabilities for all possible totals.
N
O
12
AT
IO
problEm SolvinG
You draw two cards, one after the other without replacement, from a deck of 52 cards.
a What is the probability of drawing two aces?
b What is the probability of drawing two face cards (J, Q, K)?
c What is the probability of getting a ‘pair’? (22, 33, 44 … QQ, KK, AA)?
15 A chance experiment involves flipping a coin and rolling two dice. Determine the
probability of obtaining Tails and two numbers whose sum is greater than 4.
16 In a jar there are 10 red balls and 6 green balls. Jacob takes out two balls, one at a
time, without replacing them. What is the probability that both balls are the same
colour?
17 In the game of ‘Texas Hold’Em’ poker, 5 cards are
progressively placed face up in the centre of the table
for all players to use. At one point in the game there are
3 face-up cards (two hearts and one diamond). You have
2 diamonds in your hand for a total of 3 diamonds. Five
diamonds make a flush. Given that there are 47 cards left,
what is the probability that the next two face-up cards are
both diamonds?
M
PL
E
EV
AL
U
14
SA
cHAllEnGE 13.2
doc-6314
Topic 13 • Probability 503
c13Probability.indd 503
18/05/16 4:30 PM
STATiSTicS And probAbiliTy
13.6 Mutually exclusive and independent
events
Mutually exclusive events
AT
IO
N
O
N
LY
• If two events cannot both occur at the same time then it is said the two events are
mutually exclusive.
For example, when rolling a die, the events ‘getting a 1’ and ‘getting a 5’ are mutually
exclusive.
• If two sets are disjoint (have no elements in common), then the sets are mutually
exclusive.
For example, if A = {prime numbers > 10} and B = {even numbers}, then A and B are
mutually exclusive.
• If A and B are two mutually exclusive events (or sets),
ξ
then P(A ∩ B) = ø.
A
B
• Consider the Venn diagram shown. Since A and
2
4
B are disjoint, then A and B are mutually
1
6
5
exclusive sets.
3
• If two events A and B are mutually exclusive, then
P(A or B) = P(A ∪ B) = P(A) + P(B).
Examples of mutually exclusive events
AL
U
• Draw a card from a standard deck: the drawn card is a heart or a club.
– Reason: it is impossible to get both a heart and a club at the same time.
• Record the time of arrival of overseas flights: a flight is late, on time or it is early.
– Reason: it is impossible for the flight to arrive late, on time or early all at the
same time.
EV
Examples of non-mutually exclusive events
M
PL
E
• Draw a card from a standard deck: the drawn card is a heart or a king.
– Reason: it is possible to draw the king of hearts.
• Record the mode of transport of school students: count students walking or going
by bus.
– Reason: a student can walk (to the bus stop) and take a bus.
WorKEd EXAmplE 16
SA
A card is drawn from a pack of 52 cards. What is the probability that the card is a diamond
or a spade?
THinK
WriTE
1
The events are mutually exclusive because diamonds
and spades cannot be drawn at the same time.
The two events are mutually exclusive as
P(A ∩ B) = ∅.
2
Determine the probability of drawing a diamond and
the probability of drawing a spade.
Number of diamonds, n(E1) = 13
Number of spades, n(E2) = 13
Number of cards, n(S) = 52
P(spade) = 13
P(diamond) = 13
52
52
= 14
504
= 14
Maths Quest 9
c13Probability.indd 504
18/05/16 4:30 PM
STATiSTicS And probAbiliTy
3
Write the probability.
4
Evaluate and simplify.
P(A ∪ B) = P(A) + P(B)
P(diamond or spade)
= P(diamond) + P(spade)
= 14 + 14
= 12
N
LY
Independent events
O
• Two events are considered independent if the outcome of one event is not dependent on
the outcome of the other event.
• For example, if E1 = {rolling a 4 on a first die} and E2 = {rolling a 2 on a second die},
the outcome of event E1 is not influenced by the outcome of event E2, so the events are
independent.
N
WorKEd EXAmplE 17
THinK
Link each outcome of the first flip with
the outcomes of the second part of the
experiment (flipping the second coin).
Link each outcome from the second
flip with the outcomes of the third part
of the experiment (flipping the third
coin).
SA
M
PL
E
3
Use branches to show the individual
outcomes for the first part of the
experiment (flipping the first coin).
AL
U
2
WriTE/drAW
EV
1
AT
IO
Three coins are flipped simultaneously. Draw a tree diagram for the experiment. Calculate the
following probabilities.
a P(3 Heads)
b P(2 Heads)
c P(at least 1 Head)
First coin
1–
2
1–
2
H
T
First coin
1–
2
1–
2
Second coin
H
T
First coin
1–
2
H
1–
2
T
1–
2
H
1–
2
T
Second coin
Third coin
1–
2
1–
2
1–
2
H
T
1–
2
H
1–
2
T
1–
2
1–
2
1–
2
1–
2
H
1–
2
T
1–
2
H
T
1–
2
H
T
1–
2
H
T
1–
2
H
T
Topic 13 • Probability 505
c13Probability.indd 505
18/05/16 4:30 PM
STATistics and probability
Determine the probability of each
outcome. Note: The probability of each
result is found by multiplying along
the branches and in each case this will
be 12 × 12 × 12 = 18.
1
1–
2
1–
2
H
T
2
1–
2
H
1–
2
T
1–
2
1–
2
1–
2
1–
2
1–
2
H
1–
2
T
1–
2
1–
2
1–
2
1–
2
2
Write your answer.
1
At least 1 Head means
any outcome that contains
one or more Head. This
is every outcome except
three Tails. That is, it is the
complementary event to
obtaining 3 Tails.
1–
2
×
1–
2
=
H
HTH
×
H
THH
T
THT
H
TTH
1–
2
1–
2
1–
2
1–
2
1–
2
1–
2
=
HTT
1–
2
1–
2
1–
2
1–
2
1–
2
1–
2
×
T
1–
2
1–
2
1–
2
1–
2
1–
2
1–
2
T
×
×
×
×
×
×
×
×
TTT
P(2 Heads)
= P(H, H, T) + P(H, T, H) + P(T, H, H)
= 18 + 18 + 18
×
=
=
=
=
—
1
O
b
×
=
1–
8
1–
8
1–
8
1–
8
1–
8
1–
8
The probability of obtaining exactly
2 Heads is 38.
E
Write your answer.
×
P(3 Heads) = 18.
= 38
M
PL
2
1–
2
N
{2 Heads} has 3 satisfactory
outcomes: (H, H, T),
(H, T, H) and (T, H, H),
which are mutually exclusive.
HHT
a
c
EV
c
1
T
1–
8
1–
8
AT
IO
b
The probability of three heads is
P(H, H, H)
Outcomes Probability
1–
1–
1–
HHH
2 × 2 × 2 =
AL
U
a
3
H
N
LY
4
P(at least 1 Head)
= 1 − P(T, T, T)
= 1 − 18
= 78
The probability of obtaining at least
1 Head is 78.
SA
Note: The probabilities of all outcomes add to 1.
Dependent events
•• Many real-life events have some dependence upon each other, and their probabilities are
likewise affected.
Examples include:
–– the chance of rain today and the chance of a person taking an umbrella to work
–– the chance of growing healthy vegetables and the availability of good soil
–– the chance of Victory Soccer Club winning this week and winning next week
–– drawing a card at random, not replacing it, and drawing another card.
•• It is important to be able to recognise the difference between dependent events and
independent events.
506 Maths Quest 9
c13Probability.indd 506
18/05/16 4:30 PM
STATiSTicS And probAbiliTy
WorKEd EXAmplE 18
A jar contains three black marbles, five red marbles, and two white marbles. Find the probability of
choosing a black marble (with replacement), then choosing another black marble.
THinK
The events, draw 1 and draw 2, are independent
E1 and E2 are independent events.
because the result of the first draw is not dependent
3
––
B
10
on the result of the second draw.
5
––
10
B
2
3
––
10
Demonstrate using a tree diagram.
2
––
10
3
––
10
5
––
10
W
5
––
10
B
R
O
R
R
N
LY
1
WriTE/drAW
3
AT
IO
2
––
10
3
––
10
W
W
5
––
10
2
––
10
B
R
W
P(E1 and E2) = P(E1) × P(E2)
P(black and black) = P(black) × P(black)
Determine the probability.
3
P(black and black) = 10
×
3
10
9
= 100
EV
AL
U
Evaluate and simplify.
N
2
––
10
E
• If the first marble had not been replaced in the previous worked example, the second
draw would be dependent on the outcome of the first draw, and so it follows that the
sample space for the second draw is different from that for the first draw.
M
PL
WorKEd EXAmplE 19
Repeat Worked example 15 without replacing the first marble before the second
one is drawn.
THinK
The words ‘without replacing’ indicate that the two
events are dependent.
Write the sample space and state the probability of
choosing a black marble on the first selection.
There are 10 marbles and 3 of these are black.
The sample space is
{B, B, B, R, R, R, R, R, W, W}.
Assume that a black marble was chosen in the first
selection. Determine how many black ones remain,
and the total number of remaining marbles.
Write the sample space and state the probability of
choosing a black marble on the second selection.
A black one was chosen, leaving 2 black ones
and a total of 9 marbles.
The sample space is
{B, B, R, R, R, R, R, W, W}.
SA
1
2
WriTE/drAW
3
P(E1) = P(black) = 10
P(E2) = P(black) = 29
Topic 13 • Probability 507
c13Probability.indd 507
18/05/16 4:30 PM
STATiSTicS And probAbiliTy
Demonstrate using a tree diagram.
2–
9
B
3
––
10
2–
9
3–
9
5
––
10
R
5–
9
3–
9
W
4–
9
B
R
W
5–
9
1–
9
B
R
O
W
P(E1 and E2) = P(E1) × P(E2)
3
× 29
= 10
Multiply the probabilities.
N
4
R
W
2–
9
2
––
10
B
N
LY
3
5
AT
IO
1
= 15
Answer the question.
The probability of choosing two black marbles
1
.
without replacing the first marble is 15
AL
U
Exercise 13.6 Mutually exclusive
and independent events
⬛
rEFlEcTion
What is the difference between
independent events and
mutually exclusive events?
prAcTiSE
EV
individUAl pATHWAyS
M
PL
E
Questions:
1–3, 5, 7, 9, 11, 13–16, 18, 20,
22, 24, 26, 29
⬛
conSolidATE
⬛
Questions:
1, 3–6, 8, 10, 12–15, 18, 20,
24–26, 28, 29
⬛ ⬛ ⬛ Individual pathway interactivity
mASTEr
Questions:
1, 5–8, 11, 13, 14, 16, 17, 19–23,
25, 27, 28, 30
int-4538
FlUEncy
SA
1
508
If a card is drawn from a pack of 52 cards, what is the probability that the card is
not a queen?
MC
A
4
52
b
4
48
c
13
12
d
48
52
Which events are not mutually exclusive?
A Drawing a queen and drawing a jack from 52 playing cards
b Drawing a red card and drawing a black card from 52 playing cards
c Drawing a vowel and drawing a consonant from cards representing the 26 letters of the
alphabet
d Obtaining a total of 8 and rolling doubles (when rolling two dice)
1
3 When a six-sided die is rolled 3 times, the probability of getting 3 sixes is
. What is
216
the probability of not getting 3 sixes?
2
MC
Maths Quest 9
c13Probability.indd 508
18/05/16 4:30 PM
STATistics and probability
Eight athletes compete
in a 100-m race. The
probability that the athlete
in lane 1 will win is 15. What
is the probability that one of
the other athletes wins?
(Assume that there are no
dead heats.)
4 MC 1
5
B
5
8
8
4
C D
5
5
N
LY
A
pencil case has 4 red pens,
3 blue pens and 5 black pens. If a pen is randomly drawn from the pencil case, find:
a P(drawing a blue pen)
b P(not drawing a blue pen).
6Seventy Year 9 students were surveyed. Their ages ranged from 13 years to 15 years, as
shown in the table below.
13
10
 7
17
14
20
15
35
15
 9
 9
18
AT
IO
Age
Boys
Girls
Total
N
O
5A
Total
39
31
70
SA
M
PL
E
EV
AL
U
A student from the group is selected at random. Find:
a P(selecting a student of the age of 13 years)
b P(not selecting a student of the age of 13 years)
c P(selecting a 15-year-old boy)
d P(not selecting a 15-year-old boy).
7 WE16 A card is drawn from a pack of 52 cards.
What is the probability that the card is a king or
an ace?
8 MC A die is rolled. Find the probability of
getting an even number or a 3.
3
4
1
5
4
1
A B
C D
6
6
6
6
9If you spin the following spinner, what is the probability of
obtaining:
3
2
a a 1 or a 3
b an even number or an odd number?
10 The probabilities of Dale placing 1st, 2nd, 3rd or 4th in the
local surf competition are:
7
.
1st = 162nd = 153rd = 254th = 30
Find the probability that Dale places:
a 1st or 2nd b 3rd or 4th c 1st, 2nd or 3rd d not 1st.
11 WE17 A circular spinner that is divided into two equal halves, coloured red and blue, is
spun 3 times.
a Draw a tree diagram for the experiment.
b Calculate the following probabilities.
i P(3 red sectors)
ii P(2 red sectors)
iii P(1 red sector)
iv P(0 red sectors)
v P(at least 1 red sector)
vi P(at least 2 red sectors)
Topic 13 • Probability 509
c13Probability.indd 509
18/05/16 4:30 PM
STATistics and probability
13
14
15
16
There are two yellow tickets, three green
tickets, and four black tickets in a jar. Choose one
ticket, replace it, then choose another ticket. Find
the probability that a yellow ticket is drawn first,
then a black ticket.
WE19 Repeat question 12 with the first ticket not
being replaced before the second ticket is drawn.
A coin is tossed two times. Determine P(a Head and a Tail in any order).
A coin is tossed three times. Determine P(H, H, T) (in that order).
A coin and a die are tossed. What is the probability of a Heads–2 outcome?
WE18 N
LY
12
UNDERSTANDING
M
PL
E
21
AL
U
20
EV
19
AT
IO
N
18
Holty is tossing two coins. He claims that flipping two Heads and flipping zero Heads
are complementary events. Is he correct? Explain your answer.
Each of the numbers 1, 2, 3, . . . 20 is written on a card and placed in a bag. If a card is
drawn from the bag, find:
a P(drawing a multiple of 3 or a multiple of 10)
b P(drawing an odd number or a multiple of 4)
c P(drawing a card with a 5 or a 7)
d P(drawing a card with a number less than 5 or more than 16).
From a shuffled pack of 52 cards, a card is drawn. Find:
a P(hearts or the jack of spades)
b P(a queen or a jack)
c P(a 7, a queen or an ace)
d P(neither a club nor the king of spades).
MC Which are not mutually exclusive?
A Obtaining an odd number on a die and obtaining a 4 on a die B Obtaining a Head on a coin and obtaining a Tail on a coin
C Obtaining a red card and obtaining a black card from a pack of 52 playing cards
D Obtaining a diamond and obtaining a king from a pack of 52 playing cards
Greg has a 30% chance of scoring an A on an exam, Carly has 70% chance of scoring
an A on the exam, and Chilee has a 90% chance of scoring an A on the exam. What is
the probability that all three can score an A on the exam?
From a deck of playing cards, a card is drawn at random, noted, replaced and another
card is drawn at random. Find the probability that:
a both cards are spades b neither card is a spade
c both cards are aces d both cards are the ace of spades
e neither card is the ace of spades.
Repeat question 22 with the first drawn card not being replaced before the second card
is drawn.
Assuming that it is equally likely that a boy or a girl will be born, answer the
following.
a Show the gender possibilities of a 3-child family on a tree diagram.
b In how many ways is it possible to have exactly 2 boys in the family?
c What is the probability of getting exactly 2 boys in the family?
d Which is more likely, 3 boys or 3 girls in the family?
e What is the probability of having at least 1 girl in the family?
O
17
SA
22
23
24
510 Maths Quest 9
c13Probability.indd 510
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STATistics and probability
REASONING
Give an example of mutually exclusive events that are not complementary events
using:
a sets b a Venn diagram.
26 Explain why all complementary events are mutually exclusive but not all mutually
exclusive events are complementary.
27 A married couple plans to have four children.
a List the possible outcomes in terms of boys and girls.
b What is the probability of them having exactly two boys?
c Another couple plans to have two children. What is the probability that they have
exactly one boy?
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PROBLEM SOLVING
A bag contains 6 marbles, 2 of which are red, 1 is green
and 3 are blue. A marble is drawn, the colour is noted,
the marble is replaced and another marble is drawn.
a Show the possible outcomes on a tree diagram.
b List the outcomes of the event ‘the first marble
is red’.
c Calculate P(the first marble is red).
d Calculate P(2 marbles of the same colour are drawn).
29 A tetrahedron (prism with 4 identical triangular faces) is numbered 1, 1, 2, 3 on its
4 faces. It is rolled twice. The outcome is the number facing downwards.
a Show the results on a tree diagram.
b Are the outcomes 1, 2 and 3 equally likely?
c Find the following probabilities:
i P(1, 1)
ii P(1 is first number)
iii P(both numbers the same)
iv P(both numbers are odd).
30 Robyn is planning to watch 3 footy games
on one weekend. She has a choice of two
games on Friday night: (A) Carlton vs West
Coast and (B) Collingwood vs Adelaide. On
Saturday, she can watch one of three games:
(C) Geelong vs Brisbane, (D) Melbourne
vs Fremantle and (E) North Melbourne vs
Western Bulldogs. On Sunday, she also has
a choice of three games: (F) St Kilda vs
Sydney, (G) Essendon vs Port Adelaide and
(H) Richmond vs Hawthorn. She plans to
watch one game each day and will choose a
game at random.
a To determine the different combinations of games Robyn can watch, she draws a tree
diagram using codes A, B, . . . H. List the sample space for Robyn’s selections.
b Robyn’s favourite team is Carlton. What is the probability that one of the games
Robyn watches involves Carlton?
c Robyn has a good friend who plays for St Kilda. What is the probability that Robyn
watches both the matches involving Carlton and St Kilda?
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Topic 13 • Probability 511
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STATiSTicS And probAbiliTy
13.7 Conditional probability
• The probability that an event occurs given that another event has already occurred is
called conditional probability.
• The probability that event B occurs, given that event A has already occurred is denoted
by P(B | A). The symbol ‘|’ stands for ‘given’.
• The formula for conditional probability is:
P(A ∩ B)
, P(A) ≠ 0.
P(A)
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P(B | A) =
WorKEd EXAmplE 20
12
ξ
B
7
10
N
A
O
This Venn diagram below shows the results of a survey where students were
asked to indicate whether they liked apples or bananas.
AT
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4
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If one student is selected at random:
a What is the probability that the student likes bananas?
b What is the probability that the student likes bananas, given that they also like
apples?
c Comment on any differences between the answers for parts a and b.
THinK
Find the total number of students.
2
Find the total number of students
who like bananas.
3
Determine the probability using
the correct formula.
a
EV
1
SA
b
E
4
Write the answer.
1
Determine the number of students
who like apples.
2
Find the probability that a student
likes apples.
Total number of students in survey
= 12 + 7 + 10 + 4
= 33
Total number who like bananas
= 7 + 10
= 17
P(bananas) = P(B)
Total number who like bananas
=
Total number of students
M
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a
WriTE
= 17
33
The probability that a student likes
.
bananas is 17
33
b
Number of students who like apples
= 12 + 7
= 19
P(apples) = P(A)
=
Number of students who like apples
Total number of students
= 19
33
512
Maths Quest 9
c13Probability.indd 512
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STATiSTicS And probAbiliTy
3
Note the number liking both
apples and bananas. This is the
overlapping region of the two sets.
Number who like both apples and bananas
= n(A ∩ B)
=7
4
Determine the probability a student
likes both apples and bananas.
7
P(A ∩ B) = 33
5
Apply the formula to determine
the conditional probability.
P(B | A) =
=
The probability that a student likes
bananas, given that they also like apples
7
is 19
.
This answer is also supported
by the figures in the Venn diagram.
O
Write the answer.
Why aren’t the answers for parts
a and b both the same?
c
The answer for part a determines the
proportion of students who like bananas
out of the whole group of students. The
part b answer gives the proportion of
students who like bananas out of those
who like apples.
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6
7
33
19
33
7
19
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=
P(A ∩ B)
P(A)
EV
Note: These probabilities could also be expressed as decimals or percentages.
• It is possible to transpose the conditional probability formula to determine P(A ∩ B).
P(A ∩ B) = P(A) × P(B | A)
E
WorKEd EXAmplE 21
SA
THinK
M
PL
In a student survey, the probability that a student likes apples is 19
. The
33
7
.
probability that a student likes bananas, given that they also like apples, is 19
What is the probability that a student selected at random likes both apples and
bananas?
1
2
3
Write the given information.
WriTE
P(A) = 19
33
7
P(B | A) = 19
Apply the rearranged
conditional probability
formula.
P(A ∩ B) = P(A) × P(B | A)
Answer the question.
The probability that a student selected at 7
.
random likes both apples and bananas is 33
×
= 19
33
7
19
7
= 33
Topic 13 • Probability 513
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STATiSTicS And probAbiliTy
• Conditional probability can also be determined by examining outcomes from
a tree diagram.
WorKEd EXAmplE 22
Draw a tree diagram to
display the flipping of three
coins. Write the individual
outcomes.
a
1
2
H
H
T
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N
a
WriTE/drAW
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THinK
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Three coins are flipped simultaneously.
a Display the outcomes as a tree diagram.
b Determine the probability that a Head will result from the third coin, given that
the first two coins resulted in a Head (H) and a Tail (T).
T
SA
T
Outcomes
H
HHH
T
HHT
H
HTH
T
HTT
H
THH
T
THT
H
TTH
T
TTT
1
Look for the outcomes
where the first two flips
resulted in a Head and
a Tail.
2
How many of these
outcomes have a Head for
the third flip?
There are two of these outcomes where the third
flip resulted in a Head — HTH and THH.
3
Calculate the probability.
From four possible outcomes, two satisfy
the conditions.
P(H on third flip | H and T on first two flips)
= 24
b
There are four outcomes where the first two flips
are a Head and a Tail — HTH, HTT, THH and
THT.
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b
H
3
= 12
• A two-way table can also be used to determine conditional probability.
WorKEd EXAmplE 23
Two dice are rolled and the numbers are added.
a Show the results in a two-way table.
b Determine the probability that the sum of the two dice is 7, given that their total
is greater than 6.
514
Maths Quest 9
c13Probability.indd 514
18/05/16 4:30 PM
STATiSTicS And probAbiliTy
THinK
WriTE/diSplAy
Show the results of rolling
two dice in a two-way table.
a
a
Die 2
2
1
1
3
4
5
6
(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)
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)
1
Which outcomes have a
total greater than 6?
2
Which of these outcomes
have a total equal to 7?
There are 6 of these outcomes that have a total of
7 − (6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6).
3
Write the probability.
6
= 27
P(total of 7 | total greater than 6) = 21
b
There are 36 outcomes. 21 of these have a total
greater than 6 (6, 1), (5, 2), (4, 3), (3, 4), (2, 5),
(1, 6), (6, 2) . . . etc.
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b
O
6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
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Die 1
3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
Exercise 13.7 Conditional probability
individUAl pATHWAyS
prAcTiSE
⬛
conSolidATE
EV
⬛
Questions:
1–3, 5, 7, 9–11
M
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E
int-4539
A group of motocross racers was asked to comment on which of two tracks, A or B,
they used. The results were recorded in the Venn diagram below.
WE21
SA
1
rEFlEcTion
How can you determine when
a probability question is a
conditional one?
mASTEr
Questions:
1–15
⬛ ⬛ ⬛ Individual pathway interactivity
FlUEncy
⬛
Questions:
1–4, 6–13
ξ
A
23
B
16
15
6
How many motocross racers were surveyed?
b Calculate P(A ∩ B).
c Calculate:
i P(A)
ii P(B | A).
d Calculate:
i P(B)
ii P(A | B).
a
Topic 13 • Probability 515
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STATistics and probability
Consider your answers to question 1.
a Use your answers from part c to determine P(A ∩ B).
b Use your answers from part d to determine P(A ∩ B).
c Comment on your answers to parts a and b in this question.
3 A survey was conducted to determine whether a
group of students preferred drink A or drink B.
The results of the survey produced the following
probabilities.
7
and P(B | A) = 37.
P(A) = 10
Determine P(A ∩ B).
4 WE22 Two fair coins are tossed.
a Display the outcomes as a tree diagram.
b Determine the probability that a Head results
on the second coin, given that the first coin also
resulted in a Head.
5 WE23 Two standard dice are rolled and the numbers are added together.
a Show the results in a two-way table.
b Determine the probability that the sum of the two dice is even, given that their total is
greater than 7.
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2
UNDERSTANDING
A group of 40 people was surveyed regarding the types of movies, comedy or drama,
that they enjoyed. The results are shown below.
28 enjoyed comedy.
17 enjoyed drama.
11 liked both comedy and drama.
6 did not like either type.
a Draw a Venn diagram to display the
results of the survey.
b Determine the probability that a
person selected at random:
i likes comedy
ii likes drama
iii likes both comedy and drama
iv likes drama, given that they also like comedy
v likes comedy, given that they also like drama.
c Arrange the probabilities in part b in order from least probable to most probable.
7 A teacher gave her class two tests. Only 25% of the class passed both tests, but 40%
of the class passed the first test. What percentage of those who passed the first test also
passed the second test?
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6
REASONING
If P(A) = 0.3, P(B) = 0.5 and P(A ∪ B) = 0.6, calculate:
a P(A ∩ B)
b P(B | A)
c P(A | B).
9 A group of 80 boys is auditioning for the school musical. They are all able to either
sing, play a musical instrument, or both. Of the group, 54 can play a musical instrument
and 35 are singers. What is the chance that if a randomly selected student is a singer
he can also play a musical instrument?
8
516 Maths Quest 9
c13Probability.indd 516
18/05/16 4:30 PM
STATiSTicS And probAbiliTy
A white die and a black die are rolled. The dice are 6-sided and unbiased. Consider the
following events.
Event A: the white die shows a 6.
Event B: the black die shows a 2.
Event C: the sum of the two dice is 4.
Determine the following probabilities.
a P(A | B)
b P(B | A)
c P(C | A)
d P(C | B)
11
A die is rolled and the probability of rolling a 6 is 16. However, with the condition
that the number rolled was an even number, its probability is 13. Explain why the
probabilities are different, using conditional probability.
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10
problEm SolvinG
H
35
J
12
29
N
ξ
O
A group of students was asked to nominate their favourite form of dance, hip hop (H)
or jazz (J ). The results are illustrated in the Venn diagram. Use the Venn diagram given
to calculate the following probabilities relating to a student’s favourite form of dance.
AT
IO
12
14
What is the probability that a randomly selected student likes jazz?
b What is the probability that a randomly selected student enjoys hip hop, given that
they like jazz?
13 At a school classified as a ‘Music school for
excellence’, the probability that a student
elects to study Music and Physics is 0.2. The
probability that a student takes Music is 0.92.
What is the probability, correct to 2 decimal
places, that a student takes Physics, given
that the student is taking Music?
14 A medical degree requires applicants to
participate in two tests, an aptitude test and
an emotional maturity test. This year, 52%
passed the aptitude test and 30% passed
both tests. Use the conditional probability
formula to calculate the probability, correct
to 2 decimal places, that a student who
passed the aptitude test also passed the
emotional maturity test.
15 The probability that a student is well and
misses a work shift the night before an exam
is 0.045, while the probability that a student misses a work shift is 0.05. What is the
probability that a student is well, given they miss a work shift the night before an
exam?
SA
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a
doc-6315
Topic 13 • Probability 517
c13Probability.indd 517
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c13Probability.indd 518
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STATiSTicS And probAbiliTy
13.8 Review
www.jacplus.com.au
The Maths Quest Review is available in a customisable format
for students to demonstrate their knowledge of this topic.
AT
IO
Language
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int-2711
E
EV
array
certain
chance
chance experiment
complementary
conditional
dependent
equally likely
event
M
PL
int-3212
SA
Link to assessON for
questions to test your
readiness For learning,
your progress AS you learn and your
levels oF achievement.
assessON provides sets of questions
for every topic in your course, as well
as giving instant feedback and worked
solutions to help improve your mathematical
skills.
www.assesson.com.au
Download the Review
questions document
from the links found in
your eBookPLUS.
N
A summary of the key points covered and a concept
map summary of this topic are available as digital
documents.
int-2712
Review
questions
O
The Review contains:
• Fluency questions — allowing students to demonstrate the
skills they have developed to efficiently answer questions
using the most appropriate methods
• problem Solving questions — allowing students to
demonstrate their ability to make smart choices, to model
and investigate problems, and to communicate solutions
effectively.
N
LY
ONLINE ONLY
experiment
experimental probability
favourable outcome
impossible
independent
intersection
likely
mutually exclusive
outcome
probability
random
sample space
scale
theoretical probability
tree diagram
trial
two-way table
Venn diagram
The story of mathematics
is an exclusive Jacaranda
video series that explores the
history of mathematics and
how it helped shape the world
we live in today.
What are the chances? (eles-1700) takes a look at
the history of mathematical probability, then goes
on to see how probability plays a crucial role in the
modern world to the extent that it saves thousands
of lives every year.
Topic 13 • Probability 519
c13Probability.indd 519
18/05/16 4:30 PM
<invESTiGATion> For ricH TASK or <STATiSTicS And probAbiliTy> For pUZZlE
invESTiGATion
ricH TASK
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Sand-rings
B
A
C
520
Maths Quest 9
c13Probability.indd 520
18/05/16 4:30 PM
STATiSTicS And probAbiliTy
0
C
AT
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N
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The first sand-rings puzzle requires 8 shells to be
arranged inside the circles, so that 4 shells appear
inside circle A, 5 shells appear inside circle B and
6 shells appear inside circle C. The overlapping of
the circles shows that the shells can be counted in
2 or 3 circles. One possible arrangement is shown
below. Use this diagram to answer questions 1 to 4.
1 How many shells
ξ
appear inside
A
B
circle A, but not
circle B?
1
1
0
2 How many shells
1
appear in circles
3
2
B and C, but not
circle A?
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A shell is selected at random from the sand.
3 What is the probability it came from circle A?
4 What is the probability it was not in circle C?
5 The class was challenged to find the rest of the arrangements of the 8 shells. (Remember: 4 shells need to appear in circle A, 5 in circle B and 6 in circle C.)
SA
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E
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After completing the first puzzle, the students are given new rules. The number of shells to be
arranged in the circles is reduced from 8 to 6. However, the number of shells to be in each circle
remains the same; that is, 4 shells in circle A, 5 shells in circle B and 6 shells in circle C.
6 Using 6 shells, in how many ways can the shells be
arranged so that there are 4, 5 and 6 shells in the
three circles?
7 Explain the system or method you used to
determine your answer to question 6 above.
Draw diagrams in the space provided to show the
different arrangements.
8 Using 7 shells, in how many ways can the shells be
arranged so that there are 4, 5 and 6 shells in the
three circles?
9 Again, explain the system or method you
used to determine your answer to question 8.
Draw diagrams in the space provided to show the
different arrangements.
10 What would be the minimum number of shells required to play sand-rings, so that there are 4, 5 and
6 shells in the three circles?
11 Modify the rules of this game so that different totals are required for the 3 circles. Challenge your
classmates to find all possible solutions to your modified game.
Topic 13 • Probability 521
c13Probability.indd 521
18/05/16 4:30 PM
<invESTiGATion>
STATiSTicS
And probAbiliTy
For ricH TASK or <STATiSTicS And probAbiliTy> For pUZZlE
codE pUZZlE
N
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Roy G. Biv is a mnemonic
(memory aid) for what purpose?
The number of elements in the regions of the Venn diagrams gives
the puzzle’s answer code. The numbers shown indicate the number
of elements in the region.
n (ε ) = 20
P
ε
Y
O
ε
n (P ) = 16
X
Q
20 7
15
8
5
Z
Q )´
=2
a = n (Q ) =
g = n (Y ) =
Q)=
⊂
h = n (Z ) =
ε
k = n (ε ) =
n (ε ) = 50
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n (R ) = 38
S
C
8
n (S ) = 21
⊂
⊂
E
⊂
H
SA
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n (ε ) = 45
r = n (B
[A C ]´ ) =
s = n (ε ) =
K
t = n (K ) =
11
v = n (H
⊂
K´ ) =
3
z = n (H ) =
7
40
9
522
[A B ]´ ) =
⊂
ε
4
q = n (C
⊂
S )´ =
n (B ) = 17
B´ ) =
o = n (A C
⊂
n = n (R
B
n (A ) = 19
R´ ) =
6
2
4
A
S´ ) =
m = n (S
3
n (C ) = 14
⊂
EV
18
l = n (R
ε
⊂
d = n (P
⊂
P´ ) =
⊂
b = n (Q
R
Y´ ) =
e = n (X
AT
IO
⊂
= 5, n (P
⊂
Q)
⊂
n (P
N
n (X ) = 31
18
9
18
5
11
7
20
18 20 18 20
40
42
18
11
7
5
15
18
5
31
15
2
23
11
9
66
3
8
9
7
15
15
9
4
5
8
27
Maths Quest 9
c13Probability.indd 522
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STATiSTicS And probAbiliTy
Activities
13.3 Experimental probability
digital doc
• WorkSHEET 13.1 (doc-6313): Experimental probability
interactivity
• IP interactivity 13.3 (int-4535): Experimental probability
N
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O
13.7 conditional probability
digital doc
• WorkSHEET 13.3 (doc-6315): Probability III
interactivity
• IP interactivity 13.7 (int-4539): Conditional probability
13.8 review
interactivities
• Word search (int-2711)
• Crossword (int-2712)
• Sudoku (int-3212)
digital docs
• Topic summary (doc-13667)
• Concept map (doc-13668)
E
EV
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13.4 venn diagrams and two-way tables
digital docs
• SkillSHEET (doc-6311): Determining
complementary events
• SkillSHEET (doc-6312): Calculating the
probability of a complementary event
interactivity
• IP interactivity 13.4 (int-4536): Venn
diagrams and two-way tables
13.6 mutually exclusive and independent events
interactivity
• IP interactivity 13.6 (int-4538): Mutually
exclusive and independent events
N
13.2 Theoretical probability
digital docs
• SkillSHEET (doc-6307): Probability scale
• SkillSHEET (doc-6308): Understanding
a deck of playing cards
• SkillSHEET (doc-6309): Listing the sample space
• SkillSHEET (doc-6310): Theoretical probability
interactivities
• IP interactivity 13.2 (int-4534): Theoretical probability
• Random number generator (int-0089)
13.5 Two-step experiments
interactivities
• Two-step chance (int-2772)
• IP interactivity 13.5 (int-4537): Two-step experiments
digital doc
• WorkSHEET 13.2 (doc-6314): Probability II
AT
IO
13.1 overview
video
• The story of mathematics: What are
the chances? (eles-1700)
www.jacplus.com.au
SA
M
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To access ebookplUS activities, log on to
Topic 13 • Probability 523
c13Probability.indd 523
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STATistics and probability
Answers
topic 13 Probability
0.5
1
2 a
d
g
3 a
c
e
4 a
Certain
b Certain
c Unlikely
Likely
e Certain
f Likely
Likely
h Likely
{1, 2, 3, 4, 5, 6}
b {H, T}
{defective, not defective}
d {red, black}
{hearts, clubs, diamonds, spades}
{1, 2, 3}
b {1, 3, 5}
c {1, 2, 4, 6}
d {3, 4, 5, 6}e
{1, 2, 3, 4}f
{5, 6}
5 a 3, 5
b 1, 2c
6
d 1, 2, 5, 6e
1, 2, 3, 5
f no favourable outcomes
g3
6 a {hearts, clubs, diamonds, spades}
b i {clubs, spades}
ii {hearts, diamonds}
iii {clubs, diamonds, spades}
iv {hearts, clubs, diamonds, spades}
7 a 6
b 2c
52d
4e
5
1
1
1
1
8 a b
c
d
13
13
52
4
1
1
1
1
1
b
c
d
e
52
2
26
4
2
1
4
2
12
f g
h
i
0
j
13
52
13
13
2
9
4
7
10 a b
c
d
11
11
11
11
1
1
5
3
11 a b
c
d
0e
4
8
16
4
2
1
12 ai ii
3
3
EV
b Answers will vary.
13, 14 Check with your teacher.
15 Probabilities must be between 0 and 1, so −0.5 and 1.05 can’t be
E
probabilities.
16 The coloured portions outside the arc of the spinner shown are of
M
PL
no consequence. The four colours within the arc of the spinner
are of equal area (each 14 circle), so there is equal chance of
falling on each of the colours.
17
conditions.
e
Not reasonable; monthly rainfall in Sydney is not consistent
throughout the year.
5
59
7 a b
24
72
8 A
9 200
10 20
11 B
12 a
11
3
17
3
b
c
d
20
10
20
20
33
40
3
b
= 25 c100
100
100
13 a
14 a i 6ii
{356, 365, 536, 563, 635, 653}
2
3
1
3
1
3
b c
d
5 D
1
16 Svetlana 6, Sarah 4, Leonie 3, Trang 2
17 C
18 a 27
b {111, 112, 115, 121, 122, 125, 151, 152, 155, 211, 212, 215,
AL
U
9 a
N
LY
0
e
b
a
2 a i r.f.(H) = 0.44, r.f.(T) = 0.56ii
1
b i r.f.(H) = 0.62, r.f.(T) = 0.38ii
1
3 Each answer will be different.
4 a 0.365b
0.33c
0.25d
0.875
5 a 0.275b
0.64c
0.365d
0.635
6 a Not reasonable; not enough trials were held.
b Not reasonable; the conditions are different under each trial.
c Not reasonable; there are seasonal influences.
d
Reasonable; enough trials were performed under the same
O
h
g
f
d
N
1
AT
IO
Exercise 13.2 Theoretical probability
2
3
18 There are 36 outcomes, 15 under 7, 6 equal to 7 and 15 over 7.
SA
So, you would have a greater chance of winning if you chose
‘under 7’ or ‘over 7’ rather than ‘equal to 7’.
9 0.09
1
Exercise 13.3 Experimental probability
1 a i 16ii
4
iii Outcome
1
2
3
4
Relative
0.19 0.06 0.31 0.13
frequency
iv 1
b i 270ii
40
iii Outcome
1
Relative
0.19
frequency
iv 1
5
0.25
6
221, 222, 225, 251, 252, 255, 511, 512, 515, 521, 522, 525,
551, 552, 555}
1
2
3
3
1
1
b
12
20
1
3
c d
e
19 a
c No, because John has a higher probability of winning.
20 a
7
10
b, c Yes, far too many mistakes
21 32
22 1000 balls
23 Red = 10, yellow = 50, blue = 40
7
2
24a b
20
5
25a
30
241
b
91
241
c
59
241
Exercise 13.4 Venn diagrams and two-way tables
1 a A′ = {11, 13, 14, 16, 17, 19, 20}b
B′ = {20}
c C′ = {12, 14, 15, 16, 18, 20}d
D′ = {12, 14, 16}
2 a 27b
11c
13d
6
e 6f
7g
9
3
T
6
7
S
S′
0.06
4 ξ
T′
5
9
S
2
3
4
5
6
V
0.14
0.17
0.18
0.15
0.17
7
21
2
10
524 Maths Quest 9
c13Probability.indd 524
18/05/16 4:31 PM
STATistics and probability
5 a i W′
b i A ∩ B
iiA ∩ B′
c i A′ ∩ B′ ii
A′ ∩ B
6 a
14
Left-handed
Right-handed
Male
15
40
Female
12
33
Rectangle
L
33
16 a
73
2
3
c i ii
iii
100
5
25
8
17
1
3
7 a b
c
d
25
25
5
25
1
1
31
8 a b
c
2
10
40
27
9
d e
40
40
5
3
5
7
9 a 16
bi ii
iii
iv
16
16
8
16
3
H
H′
C
3
2
C′
4
3
N
8
90
b
19
1
13
ξ
156
SA
520
2
2
13
ξ
12
12
b4
c21
81
d i 200
13
Algebra
15
e
3
5
b 25c
15d
12e
43f
43
9
13
1
g h
i
j
100
50
100
20
34
11 15
C
58
Measurement
5
10
B
21
4
Favourite topic
25
5
A
45
20
2
a HCF = 2 × 2 × 13 = 52
b LCM = 3 × 2 × 2 × 13 × 2 × 5 = 1560
0a ξ
2
15
Probability
15
7
12
3
C
13 a
4
S 24
E
4 6 8
B 12
2
14
16
10
18
20
M
31
20
18 a 31 students
M
PL
1 73
A
9 11
13
17
19 5
C
14
12
3
1
1
7
b i ii
iii
iv
12
4
4
2
12
10
4
= 45
ξ
b
EV
ii 2
27
Fred
AL
U
2
4
11
12
17 a
4
3
S
8
16
b
H
C
HP
AT
IO
0 a Children who are not swimmers
1
b Children who are swimmers and runners
c Children who neither swim nor run
d Children who swim or run or both
i ξ
N
LY
15
40
11 a
Rhombus
5 a 2, 5, 6, 7, 8
1
b 4
c 5, 7, 8
d 8
12
M
Square
O
b ξ
Quadrilaterals
ξ
ii
97
200
3
50
Challenge 13.1
120
Topic 13 • Probability 525
c13Probability.indd 525
18/05/16 4:31 PM
STATistics and probability
Bl
Pi
1
4
d
Br
Br, R
Br, Bl
Br, Pi
Pu
Pu, R
Pu, Bl
Pu, Pi
Die 2
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
7
8
9
10
11
7 a
b i 1
2
1
2
H
1
2
T
B
RB
G
RG
R
BR
B
BB
G
BG
8 a
12
3
4
R
1
4
B
2
3
4
4
EV
4
E
5
4
5
1
2
b i ii
0iii
SA
3
5
2
5
b
1
10
10 a
1
4
b
3
14
5
M
PL
2
Die 2
TH
1
2
T
TT
3
4
R
1
4
3
4
B
RR
RB
R
BR
1
4
B
BB
2
4
B
BB
3
10
2
4
3
4
R
BR
3
10
B
RB
3
10
1
4
R
RR
1
10
0.6
J
JJ
0.4
P
JP
0.3
J
PJ
0.7
P
PP
3
16
9 a
2
3
1
2
H
1
16
4
5 a
HT
ii 1
1
6
T
3
4
b i 2
4 a
1
2
1
2
AT
IO
B
RR
HH
ii AL
U
R
R
H
1
4
1
a b
0c
A total of 7
9
3
1
36
c Die 1
6
1
36
N
LY
R
2
b
N
Scarves
Beanies
1
6
6 a 1
O
Exercise 13.5 Two-step experiments
11 a
0.5
Die 1
B
R
J
1
2
3
4
5
6
1
0
1
2
3
4
5
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
b i 0.3
ii 0.35
iii 0.35
5
4
3
2
1
0
1
12 If the first ball is replaced, the probability of drawing a yellow
6
5
4
3
2
1
0
1
6
1
3
2
3
b i ii
iii
iv
0.5
P
ball stays the same on the second draw, i.e. (35).
If the first ball isn’t replaced, the probability of drawing a yellow
ball on the second draw decreases, i.e. (59).
526 Maths Quest 9
c13Probability.indd 526
18/05/16 4:31 PM
STATistics and probability
3 a Smallest total: 3, largest total: 18
1
b
15
17 No, because there is also the possibility of 1 Head (HT or TH).
18 a
1
216
4
3
216
5
6
216
6
10
216
7
15
216
8
21
216
23 a
9
25
216
d 0e
10
27
216
24 a
11
27
216
14
15
216
15
10
216
16
6
216
17
3
216
18
1
216
1
16 2
1–
2
9 a
11 a
1–
2
1–
2
R
B
c
EV
2
1–
2
R
1–
2
B
1–
2
1–
2
R
B
G
G
1–
2
1–
2
B
1–
2
G
1–
2
1–
2
1–
2
1–
2
1–
2
B
G
BGB
BGG
B
G
GBB
GBG
B
G
GGB
GGG
1–
8
1–
8
1–
8
1–
8
1–
8
1–
8
—
1
3
8
7
8
d They are equally likely.
e
b
3
1
1–
2
c
A
1–
2
1–
2
1–
2
1–
2
1–
2
1–
2
1–
2
1–
2
3
R
B
B
2
1
R
B
BRR
BRB
R
B
BBR
BBB
8
7
26 If two events are complementary, they cannot occur at the same
time, thus their intersection is ø, the same as mutually exclusive
sets. However, if events are mutually exclusive, they do not need
to have a sum equal to 1.
7 a {BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB,
2
BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG,
GGGB, GGGG}
3
8
1
2
b c
1
1–
3
R
1–
3
Outcomes Probability
1–
RRR
8
1–
RRB
8
RBR
RBB
ξ
4
6
28 a
R
B
5
3
1
b
1
2
11
19
23
5
b
c
d
30
30
30
6
SA
10 a
1–
2
B
Outcomes Probability
1–
BBB
8
1–
BBG
8
3
B
G
1–
2
b 3
1
17
E
8 B
2
25 a S = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, B = {4, 5, 6}
215
4 D
216
1
3
b
4
4
17
53
9
61
b
c
d
70
70
70
70
2
13
B
1–
2
M
PL
7
1
1–
2
Exercise 13.6 Mutually exclusive and independent events
6 a
1
19
1
b
c
17
34
221
25
26
Advantaged. The chance of getting a total of 7 would be 13.
2 A
or 0.189
1
9
1
22 a
b
c
16
16
169
1
2601
d
e
2704
2704
11
b
221
45
17
1081
189
1000
O
21
216
21
N
13
20 D
AT
IO
12
25
216
19 a
Challenge 13.2
5 a
8
15
4
8
b
c
d
20
20
20
20
14
3
38
2
b
c
d
52
13
52
13
3
5
12
1 D
1
12
16
AL
U
14a
1
221
Probability
1
8
N
LY
Total
15
1–
8
1–
8
1–
8
1–
8
1–
8
1–
8
—
1
1
3
3
8
8
8
1
7
1
iv v
vi
8
8
2
8
1
1
12 13 14
81
9
2
b i ii
iii
1–
6
1–
2
1–
3
G
1–
2
1–
6
1–
6
1–
2
1–
3
B
1–
2
1–
6
2 Outcomes Probability
1–
RR
9
R
G
RG
B
RB
R
GR
G
GG
B
GB
R
BR
G
BG
B
BB
1
—
18
1–
6
1
—
18
1
—
36
1
—
12
1
—
6
1
—
12
1–
4
—
1
b {(R, R), (R, G), (R, B)}
c
1
3
d
7
18
Topic 13 • Probability 527
c13Probability.indd 527
18/05/16 4:31 PM
STATistics and probability
1–
2
1–
4
1
1–
2
1–
4
1–
2
1–
4
1–
4
2
1–
4
1–
4
1–
2
1–
4
3
1–
4
2 Outcomes Probability
1–
11
4
2
12
3
13
1
21
2
22
3
23
1
31
2
32
3
33
Die 2
5 a
1
1
1–
8
1–
8
1
1–
8
1
—
16
1
—
16
1–
8
1
—
16
1
—
16
—
1
4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
1–
3
B
1–
3
1–
3
1–
3
D
E
1–
3
1–
3
ADF
ADG
ADH
F
G
AEF
AEG
AEH
BCF
H
F
–1
3 G
1–
3
1–
3
BCG
BCH
H
F
1–
3 G
BDF
BDG
BDH
BEF
H
1–
3
1–
3
1–
3
F
G
H
1–
3
C
1–
3
F
G
H
BEG
BEH
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
1
—
18
C
17
—
1
1
6
c
E
1
2
Exercise 13.7 Conditional probability
4
15
13
20
16
39
ci ii
M
PL
1 a 60b
31
16
d i ii
60
31
4
15
4
15
b
SA
11
6
6
7
17
10
40
11
11
iv v
28
17
11
40
c Order from least to most probable: iii, iv, ii, v, i.
7 62.5%
2
2
8 a 0.2b
c
9
3
5
9
35
10 a
1
1
1
b
c
0d
6
6
6
11 Conditional probability reduces the sample space that the
probability is calculated from. In this instance the sample
space is reduced from 6 numbers (1, 2, 3, 4, 5, 6) to
3 numbers (2, 4, 6).
41
12aP(J) =
90
b P(H | J) =
130.22
140.58
150.9
12
41
Investigation — Rich task
c They are the same, and equal to the probability calculated in
question 1 part b.
Coin 1
D
b i ii
iii
Sample space = {ACF, ACG, . . ., BEG, BEH}
b O
ACG
ACH
ξ
N
1–
3
D
E
ACF
F
G
H
1–
3
1–
3
1–
3
1–
2
1–
3
3
5
AT
IO
1–
3
1–
3
1–
2
4 a
3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
6 a
3 Outcomes Probability
C
A
3
10
6
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
AL
U
2
b
EV
1
1–
3
3
5
6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
1–
3
2 a
4
5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
1
1
c i ii
4
2
3
9
iii iv
8
16
3
2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
b No
30 a
2
N
LY
1
Die 1
29 a
Coin 2 Outcomes
HH
H
H
1 3
24
1
1
3 4
2
4
5 18
6 2
7 Answers will vary.8
8
9 Answers will vary.10
6
11 Answers will vary.
Code puzzle
T
HT
H
TH
T
TT
To remember colours of the rainbow
T
b
1
2
528 Maths Quest 9
c13Probability.indd 528
18/05/16 4:31 PM
N
LY
O
N
AT
IO
AL
U
EV
E
M
PL
SA
c13Probability.indd 529
18/05/16 4:31 PM