Maths A - Chapter 12

12syllabussyllabusrrefefererenceence
Strand:
Statistics and probability
Core topic:
Exploring and understanding
data
In thisIn this chachapterpter
12A Informal description of
chance
12B Sample space
12C Tree diagrams
12D Equally likely outcomes
12E Using the fundamental
counting principle
12F Relative frequency
12G Single event probability
12H Writing probabilities as
decimals and percentages
12I Range of probabilities
12J Complementary events
Introduction
to probability
MQ Maths A Yr 11 - 12 Page 487 Thursday, July 5, 2001 11:45 AM

488 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Introduction
Sam’s dream is to win lotto and be instantly swept into the millionaire set. Each week
he submits his six numbers to Q-lotto (an imaginary system) and anxiously awaits the
results of the draw. Many Australians subscribe each week to the various systems, some
using the same numbers, week after week, year after year, hoping one day that their
numbers will ‘come up’; but what chance does each entry have of winning? How many
numbers, and what combinations of numbers are necessary to win any prize? Will Sam
win a million dollars, even if his six numbers do come up? Are there some numbers
that appear to be chosen more frequently than others? Lotto systems are complex. We
will endeavour to understand a simplified version of them.
In a Q-lotto draw, balls numbered 1 to 45 are placed in a barrel and agitated in order
to mix them thoroughly. Six balls are selected at random, in succession. It is these six
numbers that entitle an entrant to first prize (the jackpot prize), or a share in first prize
if there is more than one entry with these six correct numbers.
Another two balls are then randomly selected. These two numbers are called bonus
or supplementary numbers. Either of these two supplementary numbers may be used
with three, four or five of the previous six numbers to give lower level prizes (called
division prizes). Visit a lotto web site, which will display a table similar to the
following one, which advises requirements for the various prize divisions.
Combinations that win
There are five prize divisions in Q-lotto. This table explains what you need to win.
Any potential prize obviously depends on the jackpot size (the total amount of
money available to be distributed to all first-prize winners), and the number of winners
in each of the divisions. So what is the chance that Sam will win money in any of the
divisions? A web site will display a table similar to this, showing the chance of winning
in any prize division. It appears that Sam’s chance of becoming a millionaire is quite
remote (1 in 8.145 million, in fact)!
Chances of winning
Even if Sam happens to pick the correct six numbers, he will not necessarily win a
million dollars. Examine the following Q-lotto draws for three consecutive weeks. Note
Division Numbers required to win
Division 1
Division 2
Division 3
Division 4
Division 5
All 6 winning numbers
Any 5 winning numbers plus either supplementary
Any 5 winning numbers
Any 4 winning numbers
Any 3 winning numbers plus either supplementary
Division Chance based on four games
Division 1
Division 2
Division 3
Division 4
Division 5
1 in 8.145 million
1 in 679 000
1 in 37 000
1 in 733
1 in 211

C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 489
that in Week 1, when the jackpot prize was over three million dollars, there were only
three winners, so each received over one million dollars. In Week 2, the jackpot was
twenty-four million dollars. There were 30 winners, so each winner’s share was less than
one million dollars. The four winners in Week 3 received the same as each of the 30
winners in Week 2.
To gain a better understanding of systems such as this, we need to understand the
theory of chance and probability. After considering several simpler examples, we shall
return to continue our investigation of Sam’s chance of fulﬁlling his dream.
1 Classify the chance of the following events
occurring on a scale of impossible to certain,
then position each on the scale at right.
a Drawing a black card from a deck of cards
b Winning the lotto
c The sun rising tomorrow
d A 6 will turn up on one roll of a 6-sided die
e A Head will result if a coin is tossed
f A total of 15 will result if two 6-sided dice are rolled.
2 Consider rolling a normal 6-sided die.
a How many faces are on the die?
b List the numbers on the die.
c What would be the chance of rolling a 6?
d How many even numbers are on the die?
e What would be the chance of rolling an even number?
f How many of the numbers are prime?
g What would be the chance of rolling a number which was not prime?
Data Draw 1 Draw 2 Draw 3
Date Week 1 Week 2 Week 3
Numbers 20, 30, 38, 40, 41,
43
5, 11, 22, 33, 40,
45
4, 5, 8, 15, 22, 30
Bonus nos. 1, 29 7, 12 23, 35
No. of winners 3 30 4
Jackpot size $3 600 000 $24 000 000 $3 200 000
6 out of 6 paid: $1 200 000 $800 000 $800 000
5 out of 6 + bonus
nos. paid:
$14 000 $10 000 $12 000
5 out of 6 paid: $1300 $1000 $1100
4 out of 6 paid: $40 $350 $30
3 out of 6 + either
bonus no. paid:
$25 $20 $20
Impossible Certain

3 In a family there are 3 children.
a Copy and complete the following table to show all possibilities of the order of birth
of the children.
b How many different outcomes are possible?
c In how many cases are all 3 of the children of the same sex?
d If there are both girls and boys in the family, how many situations occur where girls
outnumber boys?
4 Convert the following fractions to decimals (to 3 decimal places, if necessary).
a b c d e
5 Convert the fractions in question 4 to percentages (to 1 decimal place).
6 Calculate the following, giving your answer in both fraction and decimal form.
a × × b ( )2
c 1 −
d 1 − ( )2
e 1 − ( )3
f ( )2
× ( )3
Child 1 Child 2 Child 3
Boy Girl Girl
Boy Boy Girl
Terms used in chance
Resources: Pen, paper, newspapers, other print material, other forms of media,
World Wide Web or Internet, library.
This activity would be best performed in groups.
We frequently hear words such as:
‘There is absolutely no chance . . .’
‘I am certain . . .’
‘The chance of . . . is very slim.’
‘. . . a ﬁfty-ﬁfty chance.’
These terms represent an expression of the chance that a particular event might
occur.
Task 1
1 As a group, make a list of as many terms of chance as you can.
2 Rank these terms in order of the likelihood of their occurring (some terms you
may consider to be of equivalent chance of occurrence to each other).
3 Draw a line graph and place your terms in an appropriate position on this line.
4 Compare your graph with those compiled by other groups.
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Informal description of chance
You have booked a ski holiday to Thredbo for the middle of July. What is the chance
that there will be enough snow on the ground for you to ski? There is no exact answer
to this question, but by looking at the amount of snow in Thredbo during July over past
years, we know that there is a very good chance that there will be enough snow to ski
again this year.
We can say that it is very likely that we will be able to ski during July at Thredbo.
Terms such as very likely, almost certain, unlikely and fifty-fifty are used in everyday
language to describe the chance of an event occurring. For the purposes of probability,
an event is the outcome of an experiment that we are interested in. We can describe an
outcome as a possible result to the probability experiment.
Imagine you are playing a board game and it is your turn to roll the die. To win the
game you need to roll a number less than 7. If you roll one die you must get a number
less than 7. We would describe the chance of this event occurring as certain.
When an event is certain to occur, the probability of that event occurring is 1.
Now let’s consider an impossible situation.
In a board game you have one last throw of the die. To win you must roll a 7. We
know that this cannot be done. We would say that this is impossible.
When an event is impossible, the probability of the event is 0.
Task 2
The media (printed material, radio, television) often exaggerate events (positively
or negatively) in their reports of incidents.
1 Search for and collect statements from a range of media where terms of chance
have been used. Comment on the appropriateness of their use.
Task 3
The English language has many colourful expressions to describe the chance of an
event occurring.
1 Consider the following expressions, and research them through the Word Wide
Web and your library, then answer the questions that follow.
A ‘That will happen once in a blue moon.’
a What is a blue moon?
b How often does a blue moon occur?
B ‘There is Buckley’s chance of that happening.’
a Who was Buckley?
b How did this saying originate?
2 Investigate to find any similar expressions. What are their origins?

The chance of any event occurring will often be somewhere
between being certain and impossible and we use a variety of
terms to describe where the chance lies in this range as shown in
the figure at right.
We use these terms based on our general knowledge of the
world, the total possible outcomes and how often an event occurs.
You will need to use terms of chance such as those used above to describe events that
are more likely to occur than others.
The term frequency refers to how often an event occurs. We use our knowledge about
possible outcomes to order outcomes from the most frequent to the least frequent.
Very unlikely
Impossible
Unlikely
Fifty-fifty
Probable
Certain
Almost certain
1–
2
1
0
SkillS
HEET 12.1
Describe the chance of each of the following events occurring.
a Tossing a coin and its landing Heads.
b Rolling a 6 with one die.
c Winning the lottery.
d Selecting a numbered card from a standard deck of playing cards.
THINK WRITE
a There is an equal chance of the coin
landing Heads and Tails.
a The chance of tossing a Head is fifty-fifty.
b There is only one chance in six of
rolling a 6.
b It is unlikely that you will roll a 6.
c There is only a very small chance of
winning the lottery.
c It is very unlikely that you will win the
lottery.
d There are more numbered cards than
picture cards in a deck.
d It is probable that you will select a numbered
card.
1WORKEDExample
Mrs Graham is expecting her baby to be born between July 20 and 26. Is it more likely
that her baby will be born on a weekday or a weekend?
THINK WRITE
There are 5 chances that the baby will be
born on a weekday and 2 chances that it
will be born on a weekend.
It is more likely that Mrs Graham’s baby will
be born on a weekday.
2WORKEDExample

In the above examples, we have been able to calculate which event is more likely by
counting the number of ways an event may occur. This is not always possible. In some
cases we need to use general knowledge to describe the chance of an event occurring.
Consider the following probability problems.
‘The letters of the alphabet are written on cards and one card is selected at random.
Which letter has the greatest chance of being chosen, E or Q?’
Each letter has an equal chance of being chosen because there is one chance that E
will be chosen and one chance that Q will be chosen.
‘Stacey sticks a pin into a page of a book and she writes down the letter nearest to
the pin. Which letter has the greater chance of being chosen, E or Q?’
This question is more difficult to answer because each letter does not occur with
equal frequency in written text. However, we know from our experience with the
English language that Q will occur much less often than most other letters. We can
therefore say that E has a greater chance than Q of being chosen.
This is an example of using your knowledge of the world to make predictions about
which event is more likely to occur. In this way, we make predictions about everyday
things such as the weather and which football team will win on the weekend.
A card is chosen from a standard deck. List the following outcomes in order from least
likely to most likely: selecting a picture card, selecting an Ace, selecting a diamond,
selecting a black card.
THINK WRITE
There are 12 picture cards in the deck.
There are 4 Aces in the deck.
There are 13 diamonds in the deck.
There are 26 black cards in the deck. The order of events in ascending order of likeli-
hood:
selecting an Ace
selecting a picture card
selecting a diamond
selecting a black card.
1
2
3
4
3WORKEDExample
During the 1999 NRL season, the Sydney Roosters won 10 of their first 12 games. In
Round 13 they played the Northern Eagles who had won 3 of their first 12 games. Who
would be more likely to win?
THINK WRITE
Sydney Roosters have won more
games than the Eagles.
Sydney Roosters would be more likely to win, based
on their previous results. (Footynote: The Eagles won
the game 36–14. Sydney was more likely to win the
game but nothing in football is certain.)
4WORKEDExample

This is one example of past results being used to predict future happenings. There are
many other such examples.
Informal description of
chance
1 Describe the chance of each of the following events occurring, using an appropriate
probability term.
a Selecting a ball with a double-digit number from a bag with balls numbered 1 to
40
b Selecting a female student from a class with 23 boys and 7 girls
c Selecting a green marble from a barrel with 40 blue marbles and 30 red marbles
d Choosing an odd number from the numbers 1 to 100
2 For each of the events below, describe the chance of it occurring as impossible,
unlikely, even chance (fifty-fifty), probable or certain.
a Rolling a die and getting a negative number
b Rolling a die and getting a positive number
c Rolling a die and getting an even number
d Selecting a card from a standard deck and getting a red card
e Selecting a card from a standard deck and getting a numbered card
f Selecting a card from a standard deck and getting an Ace
g Reaching into a moneybox and selecting a 30c piece
h Selecting a blue marble from a bag containing 3 red, 3 green and 6 blue marbles
Weather records show that it has rained on Christmas Day 12 times in the last 80 years.
Describe the chance that it will rain on Christmas Day this year.
THINK WRITE
It has rained only 12 times on the last 80
Christmas Days. This is much less than half
of all Christmas Days.
It is unlikely that it will rain on Christmas Day
this year.
5WORKEDExample
remember
1. The chance of an event occurring ranges from being certain to impossible.
2. (a) An event that is certain has a probability of 1.
(b) An event that is impossible has a probability of 0.
3. There are many terms that we use to describe the chance of an event occurring,
such as improbable, unlikely, fifty-fifty, likely and probable.
4. Sometimes we can describe the chance of an event occurring by counting the
possible outcomes, while other times we need to rely on our general knowledge
to make such a description.
remember
12A
WORKED
Example
1

3 Give an example of an event which has a probability that could be described as:
4 Is it more likely that a person’s birthday will occur during a school term or during the
school holidays?
5 For each event on the left, state whether it is more likely, less likely or equally likely
to occur than the event on the right.
a Fine weather Christmas Day Wet weather Christmas Day
b A coin landing Heads A coin landing Tails
c Rolling a total of 3 with two dice Rolling a total of 7 with two dice
d Winning a raffle made up of 50 tickets Winning a raffle made up of 200 tickets
e Winning a prize in the Lotto draw Not winning a prize in the Lotto draw
6 A die is thrown and the number rolled is noted. List the following events in order
from least likely to most likely.
Rolling an even number
Rolling a number less than 3
Rolling a 6
Rolling a number greater than 2
7 Write the following events in order from least to most likely.
Winning a raffle with 5 tickets out of 30
Rolling a die and getting a number less than 3
Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marbles
Selecting a court card (Ace, King, Queen, Jack) from a standard deck
Tossing a coin and having it land Heads
8 Before meeting in the cricket World Cup in 1999, Australia had beaten Zimbabwe in
10 of the last 11 matches. Who would be more likely to win on this occasion?
9 Which of the following two runners would be expected to win the final of the 100
metres at the Olympic Games?
Carl Bailey — best time 9.92 s and won his semi-final
Ben Christie — best time 10.06 s and 3rd in his semi-final
Give an explanation for your answer.
10
A stack of 26 cards has the letters of the alphabet written on them. Vesna draws a card
from that stack. The probability of selecting a card that has a vowel written on it could
best be described as:
11
Which of the following events is the most likely to occur?
A Selecting the first number drawn from a barrel containing 20 numbered marbles
B Selecting a diamond from a standard deck of cards
C Winning the lottery with one ticket out of 150 000
D Drawing the inside lane in the Olympic 100-metre final with eight runners
a certain b probable c even chance
d unlikely e impossible.
A unlikely B even chance C probable D almost certain
WORKED
Example
2
WORKED
Example
3
WORKED
Example
4
mmultiple choiceultiple choice

12
The ski season opens on the first weekend of June. At a particular ski resort there has
been sufficient snow for skiing on that weekend on 32 of the last 40 years. Which of
the following statements is true?
A It is unlikely to snow at the opening of the ski season this year.
B There is a fifty-fifty chance that it will snow at the opening of the ski season this year.
C It is probable that it will snow at the opening of the ski season this year.
D It is certain to snow at the opening of the ski season this year.
13 On a production line, light globes are tested to see how long they will last. After
testing 1000 light globes it is found that 960 will burn for more than 1500 hours.
Wendy purchases a light globe. Describe the chance of the light globe burning for
more than 1500 hours.
14 Of 12 000 new cars sold last year, 1500 had a major mechanical problem during the
first year. Edwin purchased a new car. Describe the chance of Edwin’s car having a
major mechanical problem in the first year.
15 During an election campaign, 2000 people were asked for their voting preferences.
One thousand said that they would vote for the government. If one person is chosen at
random, describe the chance that he or she would vote for the government.
Sample space
At some time in our lives, most of us have tossed or will toss a coin. Many sports begin
with the toss of a coin.
What is the chance that the coin will land showing Heads? Most people would cor-
rectly say fifty-fifty. We need to develop a method of accurately describing the prob-
ability of an event.
Before we can calculate probability, we need to be able to list all possible outcomes
in a situation. This is called listing the sample space. When tossing a coin, the sample
space has two elements: Heads and Tails. To calculate a probability, we need to know
the number of elements in the sample space and what the elements are.
WORKED
Example
5

In many cases, several elements of the sample space may be the same. In such cases,
we can distinguish between the number of elements in the sample space and the
number of distinct (different) elements.
In some situations there may be more than one element in the sample space that gives
us the desired outcome. Favourable outcomes are the elements from the sample space
that will meet the requirements for an event to occur.
List the sample space for rolling a die.
THINK WRITE
The sample space is the numbers 1 to 6. S = {1, 2, 3, 4, 5, 6}
6WORKEDExample
In a barrel there are 4 red marbles, 5 green marbles and 3 yellow marbles. One marble is
drawn from the barrel.
a List the sample space.
b How many elements are in the sample space?
c How many distinct elements are in the sample space?
THINK WRITE
a List each marble in the barrel. a S = {red, red, red, red, green, green, green,
S = green, green, yellow, yellow, yellow}
b Count the number of elements in the
sample space.
b The sample space has 12 elements.
c Count the number of different elements
in the sample space.
c The sample space has 3 distinct elements.
7WORKEDExample
Tegan is playing a board game. To win the game, Tegan must roll a number greater than 2
with one die.
a List the sample space. b List the favourable outcomes.
THINK WRITE
a List all possible outcomes for one roll of
a die.
a S = {1, 2, 3, 4, 5, 6}
b List all elements of the sample space
which are greater than 2.
b E = {3, 4, 5, 6}
8WORKEDExample

Sample space
1 The numbers 1 to 10 are written on cards that are turned face down. The cards are
shuffled and one is chosen. List the sample space.
2 For each of the following probability experiments, state the sample space.
a Tossing a coin
b Rolling a die
c The total when rolling two dice
d Choosing a letter of the alphabet
e The day of the week on which a baby could be born
f The month in which a person’s birthday falls
3 For each of the following probability experiments, state the number of elements in the
sample space.
a Choosing a card from a standard deck
b Selecting the winner of a 15-horse race
c Selecting the first ball drawn in a Lotto draw (The balls are numbered 1–45.)
d Drawing a raffle ticket from tickets numbered 1 to 1500
e Selecting a number in the range 100 to 1000, inclusive
f Drawing a ball from a bag containing 3 yellow, 4 red and 4 blue balls
4 The letters of the word MISSISSIPPI are written on cards and turned face down. A
card is then selected at random.
a List the sample space.
b How many elements has the sample space?
c How many distinct elements in the sample space?
5 A card is to be selected from a standard deck.
a How many elements has the sample space?
b How many different elements has the sample space if we are interested in:
i the suit of the card?
ii the colour of the card?
iii the face value of the card?
remember
1. The sample space is the list of all possible outcomes in a probability
experiment.
2. The number of elements in a sample space is the total number of possible
outcomes.
3. In the sample space, there are sometimes several elements that are the same.
We may be asked to count the number of distinct (different) elements in the
sample space.
4. Favourable outcomes are the elements from the sample space which meet the
requirements for a certain event to occur.
remember
12B
WORKED
Example
6
WORKED
Example
7

6 Jane is playing a game of snakes and ladders. It is her turn to roll the die and to win
she needs a number greater than 4.
a List the sample space for this roll of the die.
b List the favourable outcomes for this roll of the die.
7 A bag holds 60 black marbles and 40 white marbles. Tony is to choose one of these
marbles from the bag. Tony wants to select a white marble.
a How many elements are in the sample space?
b How many favourable outcomes are contained in the sample space?
8
A bag contains 5 blue discs, 9 red discs and 6 yellow discs. To win a game, Jenny
needs to draw a yellow disc from the bag. How many elements are in the sample
space?
9
To win a game Jenny needs to draw a yellow disc from the bag in question 8. How
many favourable outcomes are there?
10
A raffle has 100 tickets. Chris buys 5 tickets in the raffle. Which of the following
statements is correct?
A There are two elements in the sample space.
B There are five favourable outcomes.
C There are 100 elements in the sample space.
D Both B and C
11 New South Wales are playing Queensland in a State of Origin match.
a List the sample space for the possible outcomes of the match.
c Is each element of the sample space equally likely to occur? Explain your answer.
12 A bag contains five 20c pieces, four 10c pieces and one 5c piece. A coin is selected at
random from the bag. Without replacing the first coin, a second coin is then selected.
a List the sample space for the first coin selected. How many elements has the
sample space?
b Assume that the first coin chosen was a 20c piece. List the sample space for the
second coin chosen.
c Assume that the first coin chosen was a 10c piece. List the sample space for the
second coin chosen.
d Assume that the first coin chosen was the 5c piece. List the sample space for the
second coin chosen.
13 Write down an example of an event that has 4 elements in the sample space.
14 Write down an example of an event that has 10 elements in the sample space but only
4 distinct elements.
A 3 B 6 C 14 D 20
A 3 B 6 C 14 D 20
WORKED
Example
8

1 A die is rolled. Describe the chance that the uppermost face is 4.
2 A card is drawn from a standard pack. Describe the chance of selecting a black card.
3 A bag contains four $1 coins and seven $2 coins. Describe
the chance that a coin drawn at random from the
bag will be a $2 coin.
4 A barrel containing balls
numbered 1 to 100 has one
ball selected at random from
it. How many elements has the
sample space?
5 Five history books, three
reference books and ten sporting
books are arranged on a shelf. A
book is chosen at random from
the shelf. How many elements are
in the sample space?
6 For the example in question 5, how
many distinct elements has the
sample space?
7 For the example in question 5, if
you want a sporting book, how
many favourable outcomes are
there?
8 Copy and complete. If an event is
certain, then the probability of it
occurring is .
9 Copy and complete. If an event is
impossible, then the probability of it
occurring is .
10 If Jane needs to select an Ace from a
standard deck to win the game, how
many favourable outcomes are there?
1

Matching actual and expected results
Resources: Coin, six-sided die, calculator.
This investigation is best performed in small groups.
Task 1
Consider tossing a coin once.
1 List the sample space.
2 Theoretically, if you tossed the coin 60 times, how many Heads and how many
Tails would you expect to obtain?
3 In practice, these results may vary. Set up an experiment to determine the
outcomes. Within your group, toss a coin 60 times and record the outcomes by
copying and completing the table below:
4 How closely do your experimental results match the theoretical ones?
5 Combine your results with those of another group. These ﬁgures then represent
the tossing of a coin 120 times. How do they compare with what you would
expect in theory?
6 Collate and combine the results of all of the groups in your class. Compare
these ﬁgures with what you would expect in theory. Have you reached any
conclusions?
Task 2
Repeat the processes in Task 1 using a six-sided die in place of the coin. Tabulate
your results, comparing the theoretical and experimental outcomes. What
conclusions can you draw?
Task 3
Your calculator can be set to generate random integers. If you are unsure how to do
this, your teacher will advise you.
1 Tossing a coin can be simulated on a calculator by the random generation of a
1 (representing, say a Tail) or a 2 (representing a Head). Using your calculator,
repeat Task 1, simulating tossing a coin 60 times.
2 Set your calculator to generate the random integers 1, 2, 3, 4, 5 or 6. This can
be used to simulate the rolling of a die. Repeat Task 2.
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Outcome Tally
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Random
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Dice 1
GCpr
ogram
Coin flip

Computer simulation:Tossing a coin
and rolling a die
Resources: Computer spreadsheet.
Task 1
This task uses a computer to simulate tossing a coin 60 times. The following
instructions refer to the Excel spreadsheet. If you use an alternative spreadsheet
your teacher will advise you of the equivalent instructions and formulas.
1 In the spreadsheet, type the entries shown in cells A1, A3, D3, G3, D4, G4,
D5 and G5. The ﬁgures in H4 and H5 are generated by the results of the
experiment. Leave these two cells blank at this stage.
2 Enter the expected results of the experiment in cells E4 and E5.
3 Type in the heading shown in cell A7.
4 The 60 cells in the range A9 to J14 represent the results of the simulation of
the 60 coin tosses. In these cells we are going to generate integers 1 or 2
randomly. If a 2 results, we will let that represent a Head and if a 1 results, we
will say that represents a Tail. The formula to generate these Heads or Tails
randomly is:
=IF(INT(RAND()*2+1)=2,”Head”,”Tail”)
Type this formula in cell A9 and copy it to all cells in the range A9 to J14.
This will randomly generate a Head or a Tail in each of these cells every time
the program is run.
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L Spreadshe
et
Random
number
generator
EXCE
L Spreadshe
et
Simulating
coin
tosses DIY
EXCE
L Spreadshe
et
Simulating
coin
tosses

5 The number of Heads and Tails generated are to be counted and the results
displayed in cells H4 and H5.
In cell H4 enter the formula
=COUNTIF(A9:J14,”Head”)
In cell H5 enter the formula
=COUNTIF(A9:J14,”Tail”)
6 The function key F9 causes the computer to simulate the 60 tosses of the coin.
Press the key and note the count of the resulting number of Heads and Tails in
cells H4 and H5 respectively. Press the F9 key again. Record the results over a
number of simulations. How do these ﬁgures compare with the theoretical
expected results?
Task 2
Repeat the previous experiment, simulating the rolling of a die 60 times.
1 Enter the expected numbers of 1s, 2s, etc in cells E4–E9.
2 To generate the integers 1, 2, 3, 4, 5 or 6 randomly in cells A13 to J18,
a enter, in cell A13, the formula:
=INT(RAND()*6+1)
b Copy the formula to the other cells.
3 To count the numbers of 1s, 2s, etc
in cell H4, enter the formula =COUNTIF(A13:J18,1)
in cell H5, enter the formula =COUNTIF(A13:J18,2)
. . . and so on.
4 Run the simulation a number of times (by pressing F9). Record and accumulate
the results. Comment on the outcomes.
Task 3
Formally record the results of this investigation in report form.
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Tree diagrams
A multi-stage event is an event where there is more than one part to the probability
experiment. Tree diagrams are used to find the elements in the sample space in a multi-
stage probability experiment. Consider the case of tossing two coins. How many
elements are there in the sample space? We draw a tree diagram to develop a system
that will list the sample space for us.
The tree diagram branches out once, for every stage of the probability experiment.
At the end of each branch, one element of the sample space is found by following the
branches that lead to that point. All of these elements together give us the outcomes of
the experiment.
Therefore, when two coins are tossed, the sample space can be written:
S = {Head–Head, Head–Tail, Tail–Head, Tail–Tail}
There are four elements in the sample space: Head–Tail and Tail–Head are distinct
elements of the sample space.
In many cases, the second branch of the tree diagram will be different from the first
branch. This occurs in situations such as those outlined in the following worked
examples, where the first event has an influence on the second event. The card chosen
first can then not be chosen in the second event.
Head
Head
Tail
Head
Tail
Tail
1st coin 2nd coin
Head–Head
Head–Tail
Tail–Head
Tail–Tail
Outcomes
A coin is tossed and a die is rolled. List all elements of the sample space.
THINK WRITE
Draw the branches for the coin toss.
From each branch for the coin toss,
draw the branches for the die roll.
List the sample space by following the
path to the end of each branch.
S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4,
S = T5, T6}
1
2
Head
Tail
Coin toss Die roll
1
2
3
4
5
6
1
2
3
4
5
6
Outcomes
H1
H2
H3
H4
H5
H6
T1
T2
T3
T4
T5
T6
3
9WORKEDExample

Each question must be read carefully, to see if repetition is possible or not. In the above
example, the numbers cannot be repeated because we are drawing two cards without
replacing the first card. In examples such as tossing two coins, it is possible for the
same outcome on both coins.
When drawing a tree diagram, the tree needs to
branch once for every stage of the experiment. When
we roll two dice, there are two levels to the tree
diagram. If we were to toss three coins, there would
be three levels to the diagram, as shown at right.
The numbers 2, 4, 7 and 8 are written on cards and are chosen to form a two-digit
number. List the sample space.
THINK WRITE
Draw the first branch of the tree
diagram to show each possible
first digit.
Draw the second branch of the
tree diagram to show each
possible second digit. When
drawing the second branch, the
digit from which the tree branches
can’t be repeated.
List the sample space by following
the tree to the end of each branch.
S = {24, 27, 28, 42, 47, 48, 72, 74, 78, 82, 84, 87}
Note that in this case, 24 is not the same as 42.
1
2
4
7
8
1st digit 2nd digit
4
7
8
2
7
8
2
4
8
2
4
7
2
3
4
10WORKEDExample
Head
Head
Head
Tail
Head
Tail
Tail
Head
Tail
Tail
Head
Tail
Head
Tail
1st coin 2nd coin 3rd coin
A coin showing ‘Heads’

For a family of four children:
a Draw a tree diagram to list all possible combinations of boys and girls.
c How many elements of the sample contain 3 boys and a girl?
THINK WRITE
a Draw the tree diagram. a
b List the sample space by following the
paths to the end of each branch.
b S = {BBBB, BBBG, BBGB, BBGG, BGBB,
BGBG, BGGB, BGGG, GBBB, GBBG,
GBGB, GBGG, GGBB, GGBG, GGGB,
GGGG}
There are 16 elements in the sample space.
c Count the number of elements that
contain 3 boys and 1 girl.
c There are four elements of the sample space
which contain 3 boys and 1 girl.
Boy
Boy
Boy
Boy
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Girl
Girl
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Boy
Girl
2nd child1st child 3rd child 4th child
11WORKEDExample
remember
1. A tree diagram is necessary in any example where there is more than one stage
to the probability experiment.
2. The tree diagram must branch out once for every stage of the probability
experiment.
3. Once the tree is drawn, the sample space is found by following the branches to
each end.
remember

Tree diagrams
1 Two coins are tossed. Use a tree diagram to list the sample space.
2 On three red cards, the numbers 1, 2 and 3 are written. On three blue cards, the same
numbers are written. A red card and a blue card are then chosen to form a two-digit
number. Draw a tree diagram to list the sample space.
3 A family consists of 3 children. Use a tree diagram to list all possible combinations of
boys and girls.
4 A coin is tossed and then a die is rolled.
a How many elements are in the sample space?
b Does it make any difference to the sample space if the die is rolled first and then
the coin is tossed?
5 The digits 1, 3, 4 and 8 are written on cards. Two cards are then chosen to form a two-
digit number. List the sample space.
6 Darren, Zeng, Melina, Kate and Susan are on a committee. From among themselves,
they must select a chairman and a secretary. The same person cannot hold both
positions. Use a tree diagram to list the sample space for the different ways the two
positions can be filled.
7 A tennis team consists of six players, three males and three females. The three males
are Andre, Pat and Yevgeny. The three females are Monica, Steffi and Lindsay. A male
and a female must be chosen for a mixed doubles match. Use a tree diagram to list the
sample space.
8 Chris, Aminta, Rohin, Levi and Kiri are on a Landcare group. Two of them are to
represent the group on a field trip. Use a tree diagram to list all the different pairs that
could be chosen. [Hint: In this example, a pairing of Chris and Aminta is the same as
a pairing of Aminta and Chris.]
9 Four coins are tossed into the air.
a Draw a tree diagram for this experiment.
b Use your tree to list the sample space.
c How many elements have an equal number of Heads and Tails?
10
Three coins are tossed into the air. The number of elements in the sample space is:
11
A two-digit number is formed using the digits 4, 6 and 9. If the same number can be
repeated, the number of elements in the sample space is:
12
A two-digit number is formed using the digits 4, 6 and 9. If the same number cannot
be used twice, the number of elements in the sample space is:
A 3 B 6 C 8 D 9
A 3 B 6 C 8 D 9
A 3 B 6 C 8 D 9
12C
WORKED
Example
9
WORKED
Example
10
WORKED
Example
11

13 When two coins are tossed there are three elements in the sample space, 2 Heads,
2 Tails or 1 Head and 1 Tail. Is this statement correct? Explain why or why not.
14 A two-digit number is to be formed using the digits 2, 5, 7 and 8.
a If the same number can be used twice, list the sample space.
b If the same number cannot be repeated, list the sample space.
15 The numbers 1, 2, 5 and 8 are written on cards and placed face down.
a If two cards are chosen and used to form a two-digit number, how many elements
are in the sample space?
b If three cards are chosen and used to form a three-digit number, how many
elements are in the sample space?
c How many four-digit numbers can be formed using these digits?
16 A school captain and vice-captain need to be elected. There are five candidates. The
three female candidates are Tracey, Jenny and Svetlana and the male candidates are
Richard and Mushtaq.
a Draw a tree diagram to find all possible combinations of captain and vice-captain.
b How many elements has the sample space?
c If boys are filling both positions, how many elements are there?
d If girls are filling both positions, how many elements are there?
e If students of the opposite sex fill the positions, how many elements are there?
17 Two dice are rolled.
a Use a tree diagram to calculate the number of elements in the sample space.
b Steve is interested in the number of elements for each total. Copy and complete the
table below.
c How many elements of the sample space have a two-digit number?
Equally likely outcomes
Below is the field for the 1999 Melbourne Cup.
Source: Courier-Mail 2 November 1999
Total 2 3 4 5 6 7 8 9 10 11 12
No. of elements
MELBOURNE CUP ODDS
Horse
Latest
odds Horse
Latest
odds
1 Tie The Knot
2 Central Park
3 Sky Heights
4 Maridpour
5 Arena
6 Yavana’s Pace
7 The Hind
8 Travelmate
9 Skybeau
10 The Message
11 Streak
12 Lahar
10-1
40-1
13-4
80-1
15-1
33-1
10-1
15-2
66-1
125-1
16-1
66-1
13 Second Coming
14 Able Master
15 Bohemiath
16 Figurehead
17 Rogan Josh
18 Laebeel
19 Brew
20 Lady Elsie
21 Rebbor
22 The Warrior
23 Zabuan
24 Zazabelle
66-1
30-1
25-1
100-1
5-1
7-1
33-1
60-1
50-1
50-1
125-1
80-1

There are 24 horses in the race. The sample space therefore has 24 elements. How-
ever, in this case, each outcome is not equally likely. This is because each horse in
the race is not of equal ability. Some horses have a greater chance of winning than
others. It is true in many practical situations that each outcome is not equally likely
to occur.
The weather on any day could be wet or ﬁne. Each outcome is not equally likely as
there are many factors to consider, such as the time of year and the current weather
patterns.
In each probability example, it is important to consider whether or not each outcome
is equally likely to occur. In general, when the selection is made randomly then equally
likely outcomes will result.
In some cases we need to use tree diagrams to calculate if each outcome is equally
likely. A statement may seem logical, but unless further analysis is conducted, we
cannot be sure.
In a football match between Brisbane and Parramatta there are three possible outcomes:
Brisbane win, Parramatta win and a draw. Is each outcome equally likely? Explain your
answer.
THINK WRITE
Each team may not be of equal ability and
draws occur less often than one of the
teams winning.
Each outcome is not equally likely as the teams
may not be of equal ability and draws are fairly
uncommon in football.
12WORKEDExample

Two-stage experiments
(tossing two coins)
Resources: Pen, paper, two coins, calculator, computer spreadsheet.
This investigation would be best performed in pairs.
Task 1
Consider the theoretical tossing of two coins.
1 Draw a tree diagram to show all possible outcomes.
2 List the elements of the sample space. Consider outcomes of Head–Tail and
Tail–Head both to be equivalent to an outcome of 1 Head and 1 Tail. Is each
outcome equally likely?
3 Consider tossing a pair of coins 36 times. In theory, how many of each outcome
should result?
Task 2
Consider an experiment consisting of tossing two coins 36 times.
1 Each pair of participants should have two coins. Draw up the following table to
record the tossing of the pair of coins 36 times.
When two coins are tossed there are three possible outcomes, 2 Heads, 2 Tails and one of
each. Is each outcome equally likely?
THINK WRITE
There is more than one coin being
tossed and so a tree diagram must be
drawn.
There are actually four outcomes, two
of which involve 1 Head and 1 Tail.
Therefore each of the outcomes
mentioned is not equally likely to
occur.
Each outcome is not equally likely. There are
two chances of getting one Head and one Tail.
There is only one chance of getting 2 Heads
and one chance of getting 2 Tails.
1
Head
Head
Tail
Head
Tail
Tail
1st coin 2nd coin
2
13WORKEDExampleinv
estigat
ioninv
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ion
Outcome Tally
HH
HT/TH
TT

2 Compare your tallies with the theoretical results you predicted in Task 1.
3 Compare your results with those obtained by other pairs in the class.
Task 3
Set your calculator to generate random integers 1 or 2. Let a 1 represent a Tail and
a 2 represent a Head.
1 With each person in the pair generating these random integers, simulate tossing
two coins 36 times. Record your results in a table as you did in Task 2.
2 Compare these results with those obtained in Task 1 and Task 2.
3 Compare your answer with those from other pairs in the class.
Task 4
This task uses a spreadsheet to simulate the tossing of the coins. Instructions and
formulas relate to the Excel spreadsheet. If you are using a different one, your
teacher will advise you of variations.
1 In the spreadsheet, type the entries shown in cells A1, E1, A3, D3, G3, E4,
H4, E5, H5, E6, H6 and A8. Leave the cells I4, I5 and I6 blank at this stage.
The results generated by the computer will be displayed there.
2 Enter the expected tally for each outcome in cells F4, F5 and F6.
3 The computer can randomly generate a 1 to represent a Tail or a 2 to represent
a Head. To simulate the tossing of 2 coins, in the cell A10 enter the formula:
=IF(INT(RAND()*2+1)+INT(RAND()*2+1)=4,”HH”,IF(INT(RAND()*2+1)
+INT)RAND()*2+1)=2,”TT”,”HT/TH”))
Copy this formula to the 36 cells in the range A10:I13.

4 To count the number of HH outcomes which result, in cell I4 enter the formula
=COUNTIF(A10:I13,”HH”)
For the HT/TH count, in the cell I5 enter the formula
=COUNTIF(A10:I13,”HT/TH”)
For the TT tally, in the cell I6, enter the formula
=COUNTIF(A10:I13,”TT”)
5 Simulate the 36 tosses by pressing the F9 key. Note the results. Repeat the
simulation until you have accumulated the tallies of 10 experiments. Comment
on the results.
Rolling two 6-sided dice
Resources: Pen, paper, two 6-sided dice, calculator, computer spreadsheet.
This activity is best performed in pairs and is designed as a follow-on from the
investigation of tossing two coins. It relies on the skills developed previously, with
only minimal guidance provided. (If you experience difﬁculties, consult the
previous investigations where detailed information is supplied.) Approach this
activity as you would an alternative assessment item, recording results in an
orderly manner, culminating in the production of a report detailing your results.
Task 1
Consider rolling two 6-sided dice 36 times. Explain the theoretical outcomes you
would expect.
Task 2
As a pair, roll two 6-sided dice 36 times. Record your results.
Task 3
Use a calculator each to simulate the rolling of the two dice 36 times. Record your
results.
Task 4
Simulate the rolling of the two dice 36 times with the aid of a spreadsheet. Note
your results.
Reporting
Write up the results of this investigation in the form of a report which must indicate
detailed understanding of:
a the theoretically expected results
b the technique involved in undertaking the practical experiment and the results
obtained
c formulas used in the calculator simulation and the tallies resulting
d formulas and results for the computer simulation — include at least two
printouts showing the diversity of results
e comparison of results from all tasks
f summary and conclusion.
inv
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ogram
Dice 2
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Equally likely outcomes
1 A tennis match is to be held between Lindsay and Anna. There are two possible out-
comes, Lindsay to win and Anna to win. Is each outcome equally likely? Explain your
answer.
2 There are 80 runners in the Olympic Games marathon. The sample space for the
winner of the race therefore has 80 elements. Is each outcome equally likely? Explain
your answer.
3 The numbers 1 to 40 are written on 40 marbles. The marbles are then placed in a bag
and one is chosen from the bag. There are 40 elements to the sample space. Is each
outcome equally likely? Explain your answer.
4 For each of the following, state whether each element of the sample space is equally
likely to occur.
a A card is chosen from a standard deck.
b The result of a volleyball game between two teams.
c It will either rain or be dry on a summer’s day.
d A raffle with 100 tickets has one ticket drawn to
win first prize.
5 For each of the following, state whether the
statement made is true or false. Give a reason for
your answer.
a Twenty-six cards each have one letter of the
alphabet written on them. One card is then
chosen at random. Each letter of the alphabet has
an equal chance of being selected.
b A book is opened on any page and a pin is stuck in
the page. The letter closest to the pin is then noted. Each
letter of the alphabet has an equal chance of being selected.
6
In which of the following is each member of the sample space equally likely to occur?
A Kylie’s softball team is playing a match that they could win, lose or draw.
B A bag contains 4 red counters and 2 blue counters. One counter is selected from
the bag.
C The temperature on a January day will be between 20°C and 42°C.
D A rose is planted in the garden that may bloom to be red, yellow or white.
remember
1. Each element of the sample space will not always be equally likely.
2. Outcomes will be equally likely if a selection is random. When other factors
influence the selection, each outcome is not equally likely.
3. When there is more than one event involved, examine the tree diagram to
determine if events described are equally likely.
remember
12D
WORKED
Example
12

7 A couple have two children. They could have two boys, two girls or one of each. The
sample space therefore has three elements that are all equally likely. Is this statement
correct? Explain your answer.
8 In a game, two dice are rolled and the total of the two dice is the player’s score.
a What is the sample space for the totals of two dice?
b Is each element of the sample space equally likely to occur?
9 A restaurant offers a three course
meal. The menu is shown below.
a A diner selects one plate from
each course. Draw a tree
diagram to determine the
number of elements in the sample space.
b Is each element of the sample space equally likely to occur?
10 There are 10 horses in a race. Ken hopes to select the winner of the race.
a How many elements in the sample space?
b Is each element of the sample space equally likely to occur? Explain your answer.
c Loretta selects her horse by drawing the names out of a hat. In this case, is the
sample space the same? Is each element of the sample space equally likely to
occur? Explain your answer.
1 Describe the chance of selecting an Ace from a standard deck of cards.
For questions 2–5. A bag contains 3 black marbles, 4 white marbles and a red marble.
2 How many elements in the sample space?
3 How many distinct elements in the sample space?
4 If Julie needs to draw a red marble from the bag, how many favourable outcomes are
there?
5 Is each element of the sample space equally likely to occur?
6 A pair of twins is born. Draw a tree diagram and then list the sample space for all
possible combinations of boys and girls.
7 Amy and Samantha are in Year 11, while Luke, Matthew and John are in Year 12. One
Year 11 student and one Year 12 student are to represent the school at a conference.
List the sample space for all pairs that could be chosen.
8 A two-digit number is formed using the digits 1, 2, and 3. How many elements has the
sample space if the same digit can be used twice?
9 A two-digit number is formed using the digits 5, 6, and 7. How many elements in the
sample space if the same digit cannot be used twice?
10 A student has an exam in mathematics that she could either pass or fail. Is each
element of the sample space equally likely to occur?
WORKED
Example
13
Entree Main course
Prawn cocktail
Oysters
Soup
Seafood platter
Chicken supreme
Roast beef
Vegetarian quiche
Dessert
Pavlova
Ice-cream
2

Using the fundamental
counting principle
A three-course meal is to be served at a 21st birthday party.
Guests choose one plate from each course, as shown in the
menu below.
In how many different ways can the three courses for
the meal be chosen?
There are two possible choices of entree,
four choices for main course and three
dessert choices. To ﬁnd the sample space
for all possible outcomes, we draw a tree
diagram.
By following the path to the end of
each branch we can see that there are 24
elements in the sample space. If we simply
need to know the number of elements in
the sample space, we multiply the number
of possible choices at each level.
Number of elements = 2 × 4 × 3
= 24
There are 24 ways in which the three-course
meal can be chosen.
This multiplication principle is called the
fundamental counting principle.
The total number of ways that a
succession of choices can be made is
found by multiplying the number of ways
each single choice could be made.
The fundamental counting
principle is used when each
choice is made independently of
every other choice. That is, when
one selection is made it has no
bearing on the next selection. In
the case above, the entree that is
chosen has no bearing on what
main course or dessert is chosen.
Entree Main course
Beef broth
Calamari
Spaghetti
Roast chicken
Pasta salad
Grilled fish
Dessert
Ice-cream
Banana split
Strawberries
Spaghetti
Beef broth
Roast
chicken
Pasta salad
Grilled fish
Main courseEntree Dessert
Ice-cream
Banana split
Strawberries
Ice-cream
Banana split
Strawberries
Ice-cream
Banana split
Strawberries
Ice-cream
Banana split
Strawberries
Spaghetti
Calamari
Roast
chicken
Pasta salad
Grilled fish
Ice-cream
Banana split
Strawberries
Ice-cream
Banana split
Strawberries
Ice-cream
Banana split
Strawberries
Ice-cream
Banana split
Strawberries

A poker machine has three wheels. There are 20 symbols on each wheel. In how many
different ways can the wheels of the poker machine finish, once they have been spun?
THINK WRITE
There are 20 possibilities for how the
first wheel can finish, 20 for the second
wheel and 20 for the third wheel.
Multiply each of these possibilities
together.
Total outcomes = 20 × 20 × 20
= 8000
Give a written answer. There are 8000 different ways in which the
wheels of the poker machine can land.
1
2
14WORKEDExample
In Year 11 at Blackhurst High School, there are four classes with 20, 22, 18 and 25
students in them respectively. A committee of four people is to be chosen, one from each
class to represent Year 11 on the SRC. In how many ways can this group of four people be
chosen?
THINK WRITE
There are 20 possible choices from the
first class, 22 from the second, 18 from
the third and 25 from the fourth class.
Multiply these possibilities together.
Total possible outcomes = 20 × 22 × 18 × 25
= 198 000
Give a written answer. The committee of four people can be chosen in
198 000 different ways.
1
2
15WORKEDExample

Sometimes we need to reconsider examples that have some type of restriction placed
on the possible selections.
If number plates consist of
3 letters and 3 digits, how
many are possible if the
first letter must be T, U
or V, and the first digit
cannot be 0 or 1?
THINK WRITE
There are 3 possible first letters.
There are 26 possible second and
third letters.
There are 8 possible first digits.
There are 10 possible second and
third digits.
Multiply all these possibilities
together.
Total number plates = 3 × 26 × 26 × 8 × 10 × 10
= 1 622 400
Give a written answer. There are 1 622 400 possible number plates under
this system.
1
2
3
4
5
6
16WORKEDExample
remember
1. The fundamental counting technique allows us to calculate the number of
different ways that separate events can occur.
2. This method can be used only when each selection is made independently of
the others. To use this method, we multiply the number of ways that each
selection can be made.
remember

Using the fundamental
counting principle
1 A poker machine has four reels, with 15 symbols on each wheel. If the wheels are
spun, in how many ways can they finish?
2 Consider each of the following events.
a A 10c coin and a 20c coin are tossed.
In how many ways can they land?
b A red die and blue die are cast.
How many ways can the two dice land?
c A coin is tossed and a die is rolled.
How many possible outcomes are there?
3 A briefcase combination lock has a combination
of three dials, each with 10 digits. How many
possible combinations to the lock are there?
4 In the game of Yahtzee, five dice are rolled. In how
many different ways can they land?
5 Some number plates have two letters followed by
4 numbers. How many of this style of plate are
possible?
6
Personalised number plates that have six symbols can be any combination of letters or
digits. How many of these are possible?
7
A restaurant menu offers a choice of four entrees, six main courses and three desserts.
If one extra choice is offered in each of the three courses, how many more combi-
nations of meal are possible?
8 There are 86 students in Year 11 at Narratime High School. Of these, 47 are boys and
39 are girls. One boy and one girl are to be chosen as school captains. In how many
different ways can the boy and girl school captain be chosen?
9 A travel agency offers Queensland holiday packages flying from Sydney with
QANTAS and Ansett; travelling in First, Business and Economy class to Brisbane, the
Gold Coast, the Great Barrier Reef and Cairns for periods of 7, 10 and 14 days.
How many holiday packages does the traveller have to choose from?
A 1 000 000 B 17 576 000 C 308 915 776 D 2 176 782 336
A 3 B 68 C 72 D 140
12E
WORKED
Example
14
WORKED
Example
15

10 A punter at the racetrack tries to pick the Daily Double. This requires her to pick the
winner of race 6 and race 7. How many selections of two horses can she make if there
are:
a 8 horses in each race?
b 12 horses in each race?
c 14 horses in race 6 and 12 in race 7?
d 16 horses in race 6 and seven in race 7?
e 24 horses in race 6 and 16 horses in race 7?
11 A poker machine has five wheels and 20 symbols on each wheel.
a In how many ways can the wheels of the poker machine finish when spun?
b There are 4 Aces on the first wheel, 5 on the second wheel, 2 on the third wheel, 6
on the fourth wheel and 1 on the fifth. In how many ways can five Aces be spun on
this machine?
12 Radio stations on the AM band have a call sign of a digit from 2 to 9, followed by two
letters.
a How many radio stations could there be under this system?
b In Qld all stations begin with a 4. How many stations are possible in Qld?
13 At a shoe store a certain pair of shoes can be bought in black, brown or grey; lace up
or buckle up and in six different sizes. How many different pairs of shoes are poss-
ible?
14 Home telephone numbers in Australia have eight digits.
a How many possible home telephone numbers are there?
b If a telephone number can’t begin with either a 0 or 1, how many are possible?
c Freecall 1800 numbers begin with 1800 and then six more digits. How many of
these are possible?
d A certain mobile network has numbers beginning with 015 or 018 followed by six
digits. How many numbers can this network have?
15 Madako can’t remember his PIN number for his bank account. He knows that it has
four digits, does not begin with nine, is an odd number and that all digits are greater
than five. How many possible PIN numbers could he try?
16
Postcodes in Australia begin with either 2, 3, 4, 5, 6, 7 or 8 followed by three more
digits. How many of these postcodes can there be?
17 Nadia goes to a restaurant that has a choice of 8 entrees, 15 main courses and 10 des-
serts.
a How many combinations of entree, main course and dessert are possible?
b Nadia is allergic to garlic. When she examines the menu she finds that three
entrees and four main courses are seasoned with garlic. How many possible
choices can she make without choosing a garlic dish?
18 Bill is trying to remember Tom’s telephone number. It has eight digits and Bill can
remember that it starts with 963 and finishes with either a 4 or a 6. How many poss-
ible telephone numbers could Tom try?
19 A representative from each of six classes must be chosen to go on a committee. There
are four classes of 28 students, a class of 25 students and a class of 20 students. How
many committees are possible?
A 70 B 1000 C 7000 D 10 000
WORKED
EExample
16
Work
SHEET 12.1

Q-lotto: Frequencies
Now that we have some understanding of the theory of chance, let us continue to
investigate Sam’s chance of winning Q-lotto.
Each Q-lotto entry card has spaces to complete
12 games. Each game consists of 45 boxes. Six of
these boxes are selected and marked with a cross.
Consider the following questions:
1 Describe, in words, Sam’s chance of selecting
one of the 6 correct numbers from the 45 available.
2 Describe Sam’s chance of selecting the correct 6 numbers from the 45 available.
3 Is he more or less likely to select the 6 correct than he is to choose 1 correct
number?
4 Is it possible to select the same number more than once in each game?
5 List the sample space for each game. How many elements are in this sample
space?
6 Considering the Q-lotto draw for Week 3 (see beginning of chapter), list the
favourable outcomes.
7 How many elements are there in the set of favourable outcomes? Does this
number change from week to week?
8 Of these elements, are they all required to win a prize? If not, how many are
required?
9 Which ones are required to win the Division 1 prize?
10 Does each of the 45 numbers have an equal chance of being selected? Why/
why not?
Let us consider the following statistics of frequencies of draws of the 45 balls in
Q-lotto. (B1 = Ball no. 1; B2 = Ball no. 2 etc.; shaded rows show the number of
times the ball number has been drawn.)
investigat
ioninv
estigat
ion
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
41 42 43 44 45 Game 1
Number of times drawn
B1 B2 B3 B4 B5 B6 B7 B8 B9
60 54 54 65 48 43 60 55 70
B10 B11 B12 B13 B14 B15 B16 B17 B18
56 70 51 60 49 62 59 64 64
B19 B20 B21 B22 B23 B24 B25 B26 B27
55 62 60 52 52 59 50 58 71
B28 B29 B30 B31 B32 B33 B34 B35 B36
76 71 62 59 52 49 65 69 57
B37 B38 B39 B40 B41 B42 B43 B44 B45
56 70 65 60 55 59 70 50 55

Consider the following:
11 Do these tables reﬂect the numbers chosen by entrants?
12 How many times has the number 1 been drawn as one of the lucky
numbers?
13 How many weeks is it since the number 1 was drawn?
14 What number has been drawn most frequently? How many times has it been
drawn?
15 What number has been drawn least frequently? How many times has it been
drawn?
16 Which numbers were drawn last week? You should have obtained eight
numbers. Why are there 8 numbers and not only 6?
17 If you were to put in a Q-lotto entry next week, basing your numbers on those
which have been chosen more than others, which numbers might you include
in your 6?
18 Basing your Q-lotto entry next week on the fact that those numbers which
haven’t turned up for a while might turn up next week, which numbers might
you choose?
We’re well on the way to answering the lotto questions posed at the beginning of
this chapter. We’ll resume our investigation after we consider relative frequency. At
that stage, we should be able to verify Sam’s chance of picking the 6 winning
numbers as quoted earlier in the chapter.
Number of weeks since drawn
B1 B2 B3 B4 B5 B6 B7 B8 B9
3 13 3 5 13 4 6 10 3
B10 B11 B12 B13 B14 B15 B16 B17 B18
0 4 18 1 1 13 8 1 6
B19 B20 B21 B22 B23 B24 B25 B26 B27
1 0 0 3 1 9 0 6 1
B28 B29 B30 B31 B32 B33 B34 B35 B36
1 8 9 1 6 8 2 2 5
B37 B38 B39 B40 B41 B42 B43 B44 B45
10 0 2 0 0 3 0 6 6

Relative frequency
You are planning to go skiing on the first weekend in July. The trip is costing you a lot
of money and you don’t want your money wasted on a weekend without snow. So what
is the chance of it snowing on that weekend? We can use past records only to estimate
that chance.
If we know that it has snowed on the first weekend of July for 54 of the last 60 years,
we could say that the chance of snow this year is very high. To measure that chance, we
calculate the relative frequency of snow on that weekend. We do this by dividing the
number of times it has snowed by the number of years we have examined. In this case,
we can say the relative frequency of snow on the first weekend in July is 54 ÷ 60 = 0.9.
The relative frequency is usually expressed as a decimal and is calculated using the
formula:
Relative frequency =
In this formula, a trial is the number of times the probability experiment has been conducted.
The relative frequency is used to assess the quality of products. This is done by finding
the relative frequency of defective products.
number of times an event has occurred
number of trials
---------------------------------------------------------------------------------------------
The weather has been fine on Christmas Day in Sydney for 32 of the past 40 Christmas
Days. Calculate the relative frequency of fine weather on Christmas Day.
THINK WRITE
Write the formula. Relative frequency =
Substitute the number of fine Christmas Days
(32) and the number of trials (40).
Calculate the relative frequency as a decimal. = 0.8
1
number of trials
---------------------------------------------------------------------------------------------
2
32
40
------
3
17WORKEDExample
A tyre company tests its tyres and finds that 144 out of a batch of 150 tyres will withstand
20 000 km of normal wear. Find the relative frequency of tyres that will last 20 000 km.
Give the answer as a percentage.
THINK WRITE
Substitute 144 (the number of times the event
occurred) and 150 (number of trials).
Calculate the relative frequency. = 0.96
Convert the relative frequency to a percentage. = 96%
1
number of trials
---------------------------------------------------------------------------------------------
2
144
150
---------
3
4
18WORKEDExample

Relative frequencies can be used to solve many practical problems.
A batch of 200 light globes was tested. The batch is considered unsatisfactory if more than
15% of globes burn for less than 1000 hours. The results of the test are in the table below.
Determine if the batch is unsatisfactory.
No. of hours No. of globes
Less than 500 4
500 to less than 750 12
1000 to less than 1250 102
1500 or more 35
THINK WRITE
Count the number of light globes that burn
for less than 1000 hours.
31 light globes burn for less than 1000 hours.
Substitute 31 (number of times the event
occurs) and 200 (number of trials).
Calculate the relative frequency. = 0.155
Convert the relative frequency to a
percentage.
= 15.5%
Make a conclusion about the quality of the
batch of light globes.
More than 15% of the light globes burn for
less than 1000 hours and so the batch is
unsatisfactory.
1
2
number of trials
---------------------------------------------------------------------------------------------
3
31
200
---------
4
5
6
19WORKEDExample
remember
1. The relative frequency is used to estimate the probability of an event.
2. The relative frequency, usually expressed as a decimal, is a ﬁgure which
represents how often an event has occurred.
3. The relative frequency is calculated using the formula:
Relative frequency = .
4. The relative frequency can also be written as a percentage and is used to solve
practical problems.
number of trials
---------------------------------------------------------------------------------------------
remember

Relative frequency
1 At the opening of the ski season, there has been sufficient snow for skiing for 37 out
of the past 50 years. Calculate the relative frequency of sufficient snow at the begin-
ning of the ski season.
2 A biased coin has been tossed 100 times with the result of 79 Heads. Calculate the
relative frequency of the coin landing Heads.
3 Of eight maths tests done by a class during a year, Peter has topped the class three
times. Calculate the relative frequency of Peter topping the class.
4 Farmer Jones has planted a wheat crop. For the wheat crop to be successful farmer
Jones needs 500 mm of rain to fall over the spring months. Past weather records show
that this has occurred on 27 of the past 60 years. Find the relative frequency of:
a sufficient rainfall
b insufficient rainfall.
5 Of 300 cars coming off an assembly line, 12 are found to have defective brakes.
Calculate the relative frequency of a car having defective brakes. Give the answer as a
percentage.
6 A survey of 25 000 new car buyers found that 750 cars had a major mechanical
problem in the first year of operation. Calculate the relative frequency of the car:
a having mechanical problems in the first year
b not having mechanical problems in the first year.
7 On a production line, light globes are tested to see how long they will last. After
testing 1000 light globes, it is found that 960 will burn for more than 1500 hours.
Wendy purchases a light globe. What is the relative frequency that the light globe
will:
a burn for more than 1500 hours?
b burn for no more than 1500 hours?
8
A study of cricket players found that of 150 players, 36 batted left handed. What is
the relative frequency of left-handed batsmen?
9
Four surveys were conducted and the following results were obtained. Which result
has the highest relative frequency?
A Of 1500 P-plate drivers, 75 had been involved in an accident.
B Of 1200 patients examined by a doctor, 48 had to be hospitalised.
C Of 20 000 people at a football match, 950 were attending their first match.
D Of 50 trucks inspected, 2 were found to be unroadworthy.
A 0.24 B 0.36 C 0.64 D 0.76
12F
WORKED
Example
17
WORKED
Example
18

10 During an election campaign 2000 people were asked for their voting preferences.
One thousand and fifty said that they would vote for the government, 875 said they
would vote for the opposition and the remainder were undecided. What is the relative
frequency of:
a government voters?
b opposition voters?
c undecided voters?
11 Research over the past 25 years shows that each November there is an average of two
wet days on Sunnybank Island. Travelaround Tours offer one-day tours to Sunnybank
Island at a cost of $150 each, with a money back guarantee against rain.
a What is the relative frequency of wet November days as a percentage?
b If Travelaround Tours take 1200 bookings for tours in November, how many
refunds would they expect to give?
12 An average of 200 robberies takes place each year in the town of Amiak. There are
10 000 homes in this town.
a What is the relative frequency of robberies in Amiak?
b Each robbery results in an average insurance claim of $20 000. What would be the
minimum premium that the insurance company would need to charge to cover
these claims?
13 A car maker recorded the first time that its cars came in for mechanical repairs. The
results are in the table below.
The assembly line will need to be upgraded if the relative frequency of cars needing
mechanical repair in the first year is greater than 25%. Determine if this will be neces-
sary.
14 For the table in question 13 determine, as a percentage, the relative frequency of:
a a car needing mechanical repair in the first 3 months
b a car needing mechanical repair in the first 2 years
c a car not needing mechanical repair in the first 3 years.
Time taken No. of cars
0 to <3 months 5
3 to <6 months 12
6 to <12 months 37
1 to <2 years 49
2 to <3 years 62
3 or more years 35
WORKED
Example
19

15 A manufacturer of shock absorbers measures the distance that its shock absorbers can
travel before they must be replaced. The results are in the table below.
The relative frequency of the shock absorber lasting is 0.985 for a certain guaranteed
distance. What is the maximum distance the manufacturer will guarantee?
16 A soccer team plays 40 matches over a season and the results (wins, losses and draws)
are shown below.
W W W D L L L D W L W D L D
W W L L L D W W D L L W W
W L D L D D L W W W D D L
a Put this information into a table showing the number of wins, losses and draws.
b Calculate the relative frequency of each result over a season.
No. of kilometres No. of shock absorbers
0 to <20 000 1
20 000 to <40 000 2
40 000 to <60 000 46
60 000 to <80 000 61
80 000 to <100 000 90
Relative frequencies
Resources: Calculator, World Wide Web.
Task 1
1 Set your calculator to generate random integers in the range 1 to 10 inclusive.
2 Draw up the table below to record the result of generating 100 random
numbers:
3 Calculate the relative frequency of each number. Is it what you would have
expected?
4 Repeat the experiment once more. Comment on any differences.
inv
estigat
ioninv
estigat
ion
Integer Tally Frequency
Relative
frequency
1
2
3
4
5
6
7
8
9
10

Single event probability
Previously we discussed the chances of certain events occurring. In doing so, we used
informal terms such as probable and unlikely. However, while these terms give us an
idea of whether something is likely to occur or not, they do not tell us how likely they
are. To do this, we need an accurate way of stating the probability.
We stated earlier that the chance of any event occurring was somewhere between
impossible and certain. We also said that:
1. if an event is impossible the probability was 0
2, if an event is certain the probability was 1.
It therefore follows that the probability of any event must lie between 0 and 1 inclusive.
A probability is a number that describes the chance of an event occurring. All prob-
abilities are calculated as fractions but can be written as fractions, decimals or percent-
ages. Probability is calculated using the formula:
The total number of favourable outcomes is the number of different ways the event can
occur, while the total number of outcomes is the number of elements in the sample
space.
Task 2
The Bureau of Meteorology has a web site detailing the temperature, rainfall,
cloudy days etc. for a large number of towns throughout Queensland. This is found
at: http://www.bom.gov.au/climate
5 Visit the web site and search to locate the town closest to where you live.
Choose some aspect of the climate and compile a relative frequency table
displaying the occurrence and variation of the weather in your area. Write a
conclusion highlighting your ﬁndings. Is it consistent with your experiences?
Task 3
The World Wide Web records an abundance of statistical information — for
example, results of sporting teams, stock market movements, Melbourne Cup
winners, world leaders, movie attendance.
6 Choose a topic which interests you and research its statistics through the Web.
Draw up a relative frequency table and summarise its results.
P event( )
number of favourable outcomes
total number of outcomes
----------------------------------------------------------------------------=
Zoran is rolling a die. To win a game, he must roll a number greater than 2. List the
sample space and state the number of favourable outcomes.
THINK WRITE
There are 6 possible outcomes. S = {1, 2, 3, 4, 5, 6}
The favourable outcomes are to roll a 3,
4, 5 or 6.
There are 4 favourable outcomes.
1
2
20WORKEDExample

Consider the case of tossing a coin. If we are calculating the probability that it will land
Heads, there is 1 favourable outcome out of a total of 2 possible outcomes. Hence we
can then write P(Heads) = . This method is used to calculate the probability of any
single event.
In the above example the fraction could be simpliﬁed to .
Some questions have more than one favourable outcome. In these cases, we need to add
together each of these outcomes to calculate the number of outcomes that are favourable.
1
2
---
Andrea selects a card from a standard deck. Find the probability that she selects an Ace.
THINK WRITE
There are 52 cards in the deck (total
number of outcomes).
There are 4 Aces (number of favourable
outcomes).
Write the probability. P(Ace) =
1
2
3
4
52
------
21WORKEDExample
4
52
------
1
13
------
In a barrel there are 6 red marbles, 2 green marbles and 4 yellow marbles. One marble is
drawn at random from the barrel. Calculate the probability that the marble drawn is red.
THINK WRITE
There are 12 marbles in the barrel (total
There are 6 red marbles in the barrel
(number of favourable outcomes).
Write the probability. P(red) = =
1
2
3
6
12
------
1
2
---
22WORKEDExample
On a bookshelf there are 4 history books, 7 novels, 2 dictionaries and 5 sporting books. If
I select one at random, what is the probability that the one chosen is not a novel?
THINK WRITE
There are 20 books on the shelf (total
Seven of these books are novels,
meaning that 13 of them are not novels
(number of favourable outcomes).
Write the probability. P(not a novel) =
1
2
3
13
20
------
23WORKEDExample

Some questions do not require us to calculate the entire sample space, only the sample
space for a small part of the experiment.
The digits 1, 3, 4, 5 are written on cards and these cards are then used to form a four-digit
number. Calculate the probability that the number formed is:
a even
b greater than 3000.
THINK WRITE
a If the number is even the last digit
must be even.
a
There are four cards that could go in
the final place (total number of
outcomes).
Only one of these cards (the 4) is
even (number of favourable
outcomes).
Write the probability. P(even) =
b If the number is greater than 3000,
then the first digit must be a 3 or
greater.
b
There are four cards that could go in
the first place.
Three of these cards are a 3 or
greater.
Write the probability. P(greater than 3000) =
1
2
3
4
1
4
---
1
2
3
4
3
4
---
24WORKEDExample
remember
1. The sample space is the list of all possible outcomes in a probability
experiment.
2. The probability of an event is calculated using the formula:
P(event) =
number of favourable outcomes
total number of outcomes
----------------------------------------------------------------------------
remember

Single event probability
1 A coin is tossed at the start of a cricket match. Manuel calls Heads. List the sample
space and the number of favourable outcomes.
2 For each of the following
probability experiments,
list the sample space and
state the number of
favourable outcomes.
a Rolling a die and
needing a 6
b Rolling two dice and
needing a total greater
than 9
c Choosing a letter of the
alphabet and it being
a vowel
d The chance a baby will
be born on the weekend
e The chance that a person’s
birthday will fall in summer
3 For each of the following probability experiments, state the number of favourable
outcomes and the total number of outcomes.
a Choosing a red card from a standard deck
b Selecting the winner of a 15-horse race
c Selecting the ﬁrst ball drawn in a lotto draw (The balls are numbered 1 to 45.)
d Winning a rafﬂe with 5 tickets out of 1500
e Selecting a yellow ball from a bag containing 3 yellow, 4 red and 4 blue balls
4 A coin is tossed. Find the probability that the coin will show Tails.
5 A regular die is rolled. Calculate the probability that the uppermost face is:
6 A barrel contains marbles with the numbers 1 to 45 written on them. One marble is
drawn at random from the bag. Find the probability that the marble drawn is:
a 6 b 1 c an even number
d a prime number e less than 5 f at least 5.
a 23 b 7 c an even number
d an odd number e a multiple of 5 f a multiple of 3
g a number less than 20 h a number greater than 35 i a square number.
12G
WORKED
Example
20
WORKED
Example
21

7 Many probability questions are asked about decks of cards. You should know the
cards making up a standard deck.
A card is chosen from a standard deck. Find the probability that the card chosen is:
8 A bag contains 12 counters: 7 are orange, 4 are red and 1 is yellow. One counter is
selected at random from the bag. Find the probability that the counter chosen is:
9 The digits 2, 3, 5 and 9 are written on cards. One card is then chosen at random. Find
the probability that the card chosen is:
10 In a bag of fruit there are 4 apples, 6 oranges and 2 pears. Larry chooses a piece of
fruit from the bag at random but he does not like pears. Find the probability that Larry
does not select a pear.
11 The digits 2, 3, 5 and 9 are written on cards. They are then used to form a four-digit
number. Find the probability that the number formed is:
12
A die is rolled. The probability that the number on the uppermost face is less than 4 is:
a the Ace of diamonds b a King c a club
d red e a picture card f a court card.
a yellow b red c orange.
a the number 2 b the number 5 c even
d odd e divisible by 3 f a prime number.
a even b odd c divisible by 5
d less than 3000 e greater than 5000.
A B C D
K
Q
J 10 9 8 7 6 5 4 3 2
2A
K
Q
J 10 9 8 7 6 5 4 3 2
A
2
K
Q
J 10 9 8 7 6 5 4 3 2
A
2
K
Q
J 10 9 8 7 6 5 4 3 2
A
2
WORKED
Example
22
WORKED
Example
23
WORKED
Example
24
1
6
---
1
3
---
1
2
---
2
3
---
MQ Maths A Yr 11 - 12 Page 531 Thursday, July 5, 2001 12:09 PM

13
When a die is rolled, which of the following outcomes does not have a probability
equal to ?
A The number on the uppermost face is greater than 3.
B The number on the uppermost face is even.
C The number on the uppermost face is at least a 3.
D The number on the uppermost face is a prime number.
14
A card is chosen from a standard deck. The probability that the card chosen is a court
card is:
15
When a card is chosen from a standard deck, which of the following events is most
likely to occur?
16 One thousand tickets are sold in a raffle. Craig buys five tickets.
a One ticket is drawn at random. The holder of that ticket wins first prize. Find the
probability of Craig winning first prize.
b After the first prize has been drawn, a second prize is drawn. If Craig won first
prize, what is the probability that he now also wins second prize?
17 A lottery has 160 000 tickets. Janice buys one ticket. There are 3384 cash prizes in the
lottery.
a What is the probability of Janice winning a cash prize?
b If there are 6768 consolation prizes of a free ticket for being one number off a
cash prize, what is the probability that Janice wins a consolation prize?
c What is the probability that Janice wins either a cash prize or a consolation prize?
18 A number is formed using all five of the digits 1, 3, 5, 7 and 8. What is the probability
that the number formed:
19 Write down an example of an event which has a probability of:
20 A three-digit number is formed using the digits 2, 4 and 7.
a Explain why it is more likely that an even number will be formed than an odd
number.
b Which is more likely to be formed: a number less than 400 or a number greater
than 400?
A B C D
A choosing a seven B choosing a club
C choosing a picture card D choosing a black card
a begins with the digit 3 b is even c is odd
d is divisible by 5 e is greater than 30 000 f is less than 20 000.
a b c .
1
2
---
1
52
------
1
13
------
3
13
------
4
13
------
1
2
---
1
4
---
2
5
---

1 A die is rolled. Find the probability that the uppermost face is 4.
2 A card is drawn from a standard pack. Find the probability of selecting a Jack.
3 A bag contains four $1 coins and seven $2 coins. Find the probability that a coin
drawn at random from the bag will be a $2 coin.
4 A barrel, containing balls numbered 1 to 100, has one ball selected at random from
it. Find the probability that the ball selected is a multiple of 3.
5 Five history books, 3 reference books and 10 sporting books are arranged on a
shelf. What is the probability of a sporting book being on the left-hand end of the
bookshelf?
6 A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The relative
frequency of this coin landing Heads is 0.7; true or false?
7 A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The probability of
this coin landing Tails is ; true or false?
8 In 60 rolls of a die, there have been 12 sixes. What is the relative frequency of
rolling a six?
9 During a football season a team has won 15 matches and lost 5. Calculate the
relative frequency of the team winning.
10 A car assembly line finds that five in every 1000 cars have faulty paintwork. If I
purchase one of these cars find, as a percentage, the relative frequency that the
paintwork is faulty.
3
7
10
------
MQ Maths A Yr 11 - 12 Page 533 Thursday, July 5, 2001 12:09 PM

Comparing probabilities with actual
results
In this activity, we compare the probability of certain events to practical results.
You may be able to do a simulation of these activities on a spreadsheet.
Tossing a coin
If we toss a coin, P(Heads) = .
1 If you toss a coin, how many Heads would you expect in:
a 4 tosses b 10 tosses c 50 tosses d 100 tosses.
2 Toss a coin 100 times and record the number of Heads after:
a 4 tosses b 10 tosses c 50 tosses d 100 tosses.
3 Combine your results with those of the rest of the class. How close to 50% is
the total number of Heads thrown by the class?
Rolling a die
1 When you roll a die, what is the probability of rolling a 1? (In fact, the
probability for each number on the die is the same.)
2 Roll a die 120 times and record each result in the table below.
How close are the results to the results that were expected?
Rolling two dice
1 Roll two dice and record the total on the faces of the two dice. Repeat this 100
times and complete the table below.
2 Do you notice anything different about the results of this activity, compared to
the others?
investigat
ioninv
estigat
ion
1
2
---
Number Occurrences Percentage of throws
1
2
3
4
5
6
Number Occurrences Percentage of throws
2
3
4
5
6
7
8
9
10
11
12

Writing probabilities as decimals and
percentages
In our exercises so far, we have been writing probabilities as fractions. This is the way
that most mathematicians like to express chance. However, in day-to-day language,
decimals and percentages are also used. Therefore, we need to be able to write prob-
abilities as both decimals and percentages.
When writing a probability as a decimal, we use the same formula and divide the
numerator by the denominator to convert to a decimal.
The chance of an event occurring is commonly expressed as a percentage. This is the
percentage chance of an event occurring. When writing a probability as a percentage,
we take the fractional answer and multiply by 100% to convert to a percentage.
If I select a card from a standard deck, what is the probability of selecting a heart,
expressed as a decimal?
THINK WRITE
There are a total of 52 cards in the deck
(elements of the sample space).
There are 13 hearts in the deck
(elements of the event space).
Write the probability. P(heart) =
Convert to a decimal. = 0.25
1
2
3
13
52
------
4
25WORKEDExample
In a bag there are 20 counters: 7 are green, 4 are blue and the rest are yellow. If I select
one at random, ﬁnd the probability (as a percentage) that the counter is yellow.
THINK WRITE
There are 20 counters in the bag
(elements of the sample space).
There are 9 yellow counters in the bag
(elements of the event space).
Write the probability. P(yellow counter) = × 100%
Convert to a percentage. = 45%
1
2
3
9
20
------
4
26WORKEDExample
remember
1. Sometimes it is necessary to write a probability as a decimal or a percentage.
2. To write a probability as a decimal, we calculate the probability as a fraction,
then divide the numerator by the denominator to convert to a decimal.
3. To write a probability as a percentage, we calculate the probability as a
fraction, then multiply by 100% to convert to a percentage.
remember

Writing probabilities as
decimals and percentages
1 A die is rolled. What is the probability of rolling an even number, expressed as a
decimal?
2 A barrel contains 40 marbles. There are
10 blue marbles, 15 red marbles and
15 white marbles. A marble is selected
at random from the barrel. Calculate as a
percentage the probability of selecting a
red marble.
3 Write down the probability that a tossed
coin will land Tails:
a as a decimal
b as a percentage.
4 A student is rolling a die. Write down
each of the following probabilities as
decimals, correct to 2 decimal places.
a Getting a 1
b Getting an odd number
c Getting a number greater than 4
5 For rolling a die, write down the following probabilities as percentages. Give your
answers correct to 1 decimal place.
a Getting a 3
b Getting an even number
c Getting a number less than 6
6 From a standard deck of cards, one is selected at random. Write down the probability
of each of the following as a decimal (correct to 2 decimal places where necessary).
a Selecting the King of hearts
b Selecting a spade
c Selecting any 5
d Selecting a red card
e Selecting a court card (any King, Queen or Jack — the Jack is also called a
Knave)
7 When selecting a card from a standard deck, what would be the probabilities of the
following, written as percentages? Give your answers correct to 1 decimal place.
a Selecting a Jack of clubs
b Selecting a diamond
c Selecting any 2
d Selecting a black card
e Selecting a court card
12H
SkillS
HEET 12.2 WORKED
Example
25
SkillS
HEET 12.3
WORKED
Example
26

8
A raffle has 400 tickets. Sonya has bought 8 tickets. The probability that Sonya wins
first prize in the raffle is:
9
In a class of 25 students, there are 15 boys and 10 girls. If a student is chosen at
random from the class, the probability that the student is a boy is:
10
Which of the following does not describe the chance of selecting a diamond from a
standard deck of cards?
11 The diagram on the right shows a spinner that can be used in a
board game. When the player spins the spinner, what is the
probability of getting the following results (expressed as a decimal)?
a A 5
b An even number
c An odd number
d A number greater than one
12 The board game in question 11 has the following rules. A player spinnning a 2 or a 5
is out of the game. A player spinning a 3 collects a treasure and automatically wins
the game. Write down the probability, as a percentage, that with the next spin a
player:
a wins the game
b is out of the game
c neither wins nor is out of the game.
A 0.02 B 0.08 C 0.2 D 0.8
A 10% B 15% C 40% D 60%
A B 0.13 C 0.25 D 25%
13
52
------
1
2
3
4
5

Range of probabilities
Consider the following problem:
A die is rolled. Calculate the probability that the uppermost face is a number less than 7.
We know this is certain to occur but we will look at the solution using the probability
formula. There are 6 elements in the sample space and 6 elements that are favourable.
Therefore:
P(no. less than 7) =
= 1
When the probability of an event is 1, the event is certain to occur.
Now let’s consider an impossible situation:
A die is rolled. Calculate the probability that the uppermost face is a number greater
than 7.
There are 6 elements in the sample space and there are 0 elements that are favour-
able. Therefore:
P(no. greater than 7) =
= 0
When the probability of an event is 0, the event is impossible.
All probabilities therefore lie in the range 0 to 1. An event with a
probability of has an even chance of occurring or not occurring.
The range of probabilities can be seen in the figure at right.
This figure allows us to make a connection between the formal
probabilities that we calculated in the previous exercise, and the
informal terms we used earlier in the chapter.
The closer a probability is to 0, the less likely it is to occur.
The closer the probability is to 1, the more likely it is to occur.
0 ≤ P(E) ≤ 1
We read this as: ‘The probability of an event is greater than or equal to zero, and less
than or equal to 1’.
6
6
---
0
6
---
Very unlikely
Impossible
Unlikely
Fifty-fifty
Probable
Certain
Almost certain
1–
2
1
0
1
2
---
For the following probabilities, describe whether the event would be certain, probable,
fifty-fifty, unlikely or impossible.
a b 0 c
THINK WRITE
a is less than and is therefore unlikely
to occur.
a The event is unlikely as it has a probability
of less than .
b A probability of 0 means the event is
impossible.
b The event is impossible as it has a
probability of 0.
c = . Therefore, the event has an even
chance of occurring.
c The event has an even chance of occurring as
the probability = .
4
9
---
18
36
------
4
9
---
1
2
---
1
2
---
18
36
------
1
2
---
1
2
---
27WORKEDExample

You need to be able to recognise when you can and cannot measure the probability. You
cannot measure probability when each outcome is not equally likely.
In a batch of 400 televisions, 20 are defective. If one television is chosen, find the
probability of its not being defective and describe this chance in words.
THINK WRITE
There are 400 televisions (elements of
the sample space).
There are 380 televisions which are not
defective (number of favourable
outcomes).
Write the probability. P(not defective) =
=
Since the probability is much greater
than and very close to 1, it is very
probable that it will not be defective.
It is very probable that the television chosen
will not be defective.
1
2
3
380
400
---------
19
20
------
4
1
2
---
28WORKEDExample
State whether the following statements are true or false, and give a reason for your
answer.
a The probability of correctly selecting a number between 1 and 10 drawn out of a barrel
is .
b The weather tomorrow could be fine or rainy, therefore the probability of rain is .
THINK WRITE
a Each outcome is equally likely. a True, because each number is equally likely
to be selected.
b Each outcome is not equally likely. b False, because there is not an equal chance
of the weather being fine or rainy.
1
10
------
1
2
---
29WORKEDExample
remember
1. Probabilities range from 0 to 1. A probability of 0 means that the event is
impossible, while a probability of 1 means the event is certain.
2. By calculating the probability, we are able to make a connection with the more
informal descriptions of chance.
3. The rules of probability can be applied only when each outcome is equally
likely to occur.
remember

Maths A - Chapter 12

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