Download Year 11 General Mathematics

Area of Study:
Probability
Multistage events and
applications of probability
Topic:
Brief Summary from Syllabus:
The focus of this topic is on counting the number of outcomes for an experiment, or the number of ways in which an
event can occur, and the calculation of outcomes expected from simple experiments and comparing them with
experimental results. The probability of particular outcomes can then be established.
Syllabus pages: 96-97
Outcomes addressed:
MG2H-1, MG2H-2, MG2H-8, MG2H-9, MG2H-10
Estimated number of lessons:
Syllabus Content, Applications, Considerations, Links to Resources,
Teaching strategies, Learning Experiences etc
Notes
Note: The symbol # denotes that this is non-compulsory extension / enrichment material
Formulae involving factorial notation are not required in the Preliminary Mathematics General
course, the HSC Mathematics General 2 course or the HSC Mathematics General 1 course.
Tex
t
Ref.

Students will have different levels of familiarity with probability contexts such as those involving
dice and card games.
While games of chance are to be investigated, the small chance of winning large prizes in popular
lotteries and number draws should be emphasised.
‘Probability tree diagrams’ can be developed as a shorthand for tree diagrams in which
every branch represents an equally likely event. Using the multiplication principle and a tree
diagram in which every branch represents an equally likely event establishes the conceptual
background for multiplying along branches of a probability tree diagram.
Technology
Students should use probability simulations for large numbers of trials, eg tossing a coin, or rolling
a die, two hundred times. The graphing facility of a spreadsheet could be used to investigate
results.
Students could use appropriate technology to develop simulations of experiments such as those
listed above
Content

multiply the number of choices at each stage to determine the
number of outcomes for a multistage event
7B

Content

establish that the number of ways in which n different items can be
arranged in a line is n n 1n  2  … 1 , eg the number of
7C

arrangements of four different items is 4  3 2 1  24 ; the number
of arrangements of three different items is 3 2 1  6
Resource
PB_Resource_Library\3_Counting_Techniques\10_GenMathsCountingTechniquesProb
ability.doc
PB_Resource_Library\3_Counting_Techniques\20_Counting_Techniques.flp
PB_Resource_Library\3_Counting_Techniques\30_Probability_cards.pdf
Application

Content

construct and use tree diagrams to establish the outcomes for a
simple multistage event
Consideration

Establishing the number of arrangements (ordered and unordered) that can be obtained
should include the use of tree diagrams and making lists
Content

establish the number of ordered selections that can be made from
a group of different items (small numbers only), eg if selecting two
particular positions (such as captain and vice-captain) from a team
of five people, the number of selections is 5  4  20
7D

Content

establish the number of unordered selections that can be made
from a group of different items (small numbers only), eg if selecting
a pair of people to represent a team of five, the number of
selections is half of the number of ordered selections
7E

In how many ways can the names of three candidates be listed on a ballot paper?
What is the probability that a particular candidate’s name will be at the top of the
paper? Check by listing.
PB3 – Multi Stage Events
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7D, 
E, F
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Content

use the formula for the probability of an event to calculate the
probability that a particular selection will occur
7F

Content

use probability tree diagrams to solve problems involving two-stage
events
7G

Consideration

Content

7H

Resource
PB_Resource_Library\2_Financial_Expectation\Financial_Expectation.d
oc
Content

7I

Applications

7K

Class discussion should include whether a particular event is obviously dependent or
independent, eg a set of free throws in basketball. Some people are of the view that the
success of each shot is independent of the result of the last shot. Others suggest that there
is a dependent psychological impact of success or failure based on the result of the last shot.
Despite many statistical studies of basketball free throws, neither view has been established
as correct. There may be similar implications for the number of faults served in a tennis
match.
calculate the expected number of times a particular event would
occur, given the number of trials of a simple experiment, by
establishing the theoretical probability of the event and multiplying
by the number of trials
compare the result in the previous dot point with an experimental
result
Forty people are given Drug X for the treatment of a disease. Drug X has a success rate of
80%. How many patients can be expected to be treated successfully?
Solution
Number of patients expected to be treated successfully

 40  0.80  32 .
Data collected in a certain town suggests that the probabilities of there being 0, 1, 2, 3, 4
or 5 car thefts in one day are 0.10, 0.35, 0.30, 0.08, 0.15 and 0.02, respectively. What is the
expected number of car thefts occurring on any particular day?
Solution
Expected number of car thefts
 0  0.10 1 0.35  2  0.30  3 0.08  4  0.15  5  0.02
 1.89
The expected number of car thefts is 1.89 cars per day.

Given the following options:
(A)
a sure gain of $300
(B)
a 30% chance of gaining $1000 and a 70% chance of gaining nothing,
calculate the financial expectation of option A and of option B. In this example, the financial
expectations are equal. Given a choice, many people would select option A because it is
certain. This example illustrates that many decisions are not made based on financial
expectation alone.

Consideratio
n
Paul plays a game involving the tossing of two coins. He gains $5 if they both show heads
and $1 if they show a head and a tail, but loses $6 if they both show tails. What is his
financial expectation for the game?

Expected value includes financial expectation calculations.

A financial loss is regarded as a negative.

When playing games of chance, any entry fee into a game is
considered a financial loss.
Teacher’s evaluation of the topic and
suggestions for changes to the program:
PB3 – Multi Stage Events
Assessment strategies used:
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Additional resources used:
Date of completion:
PB3 – Multi Stage Events
Signature of teacher:
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Year 12 General Mathematics
HSC Course - Lessons
Topic – Multi Stage Events (PB2)
Lesson 1 – Listing the Sample Space
What is “Sample Space”?
In short, the sample space refers to all of the possible outcomes of an event.
For example, For flipping a coin the sample space is Heads or Tails… SS = {H,T}
For rolling a die… SS = {1,2,3,4,5,6}
What is a multi-stage event?
A multi-stage event is a larger event that is made up of smaller simple events. To see the sample
space for a multi-stage event it is useful to look at a tree diagram.
Example 1
List the sample space for the event of a coin flipped and a die rolled.
(i)
(ii)
Draw a tree diagram…
List the sample space… SS = {H1, H2, etc… }
Example 2
List the sample space for when a couple has three children.
(i)
(ii)
Draw a tree diagram…
List the sample space… SS = {BBB, BBG, BGB, … … }
Class Work
Exercise 7A (p221)
Q1-10
Lesson 2 – Fundamental Counting Theorem
What is the Fundamental Counting Theorem?
In the example of a coin tossed and a die rolled there where 12 outcomes. When we looked at the
amount of possibilities for a couple having three children, there were eight possibilities.
How do we work these out?
The Fundamental Counting Theorem is a fancy name for a law that says; “If there is more than one
event, then you can calculate the number of total possibilities by multiplying the number of choices
of each event.” See p212, Blue Box for official definition!
Example 1 On a restaurant menu, there are 5 entrees, 8 mains and 6 desserts. How many different
combinations of meals are there?
Example 2 Number plates can be made up of two letters, followed by four digits. How many
different number plates can be made?
Class Work
Exercise 7B (p222) Q1-12
Lesson 3 – Ordered Arrangements of n items
PB3 – Multi Stage Events
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Example 1
There are three letters on cards; A, B, C.
In how many ways can the cards be placed in a line? List the possibilities.
Example 2
Group work!
In a group of four:
 Stand in a line of four people – how many different possibilities are there?
 How did you work this out?
 How many times does a person stand in 1st place; 2nd place; 3rd place; 4th place?
Class Work
Exercise 7C (p224)
Q1-13
Lesson 4 – Ordered Selections
Example 1
Group work!
In a group of four:
 Take turns at standing in a line of two people from your group of four – how many different
possibilities are there?
 How did you work this out?
 How many times does a person stand in 1st place; 2nd place; not in the queue?
Example 2
How many three letter combinations can be made from the letters in {B A T H}?
(Draw a tree diagram to verify the result)
Class Work
Exercise 7D (p225) Q1-7
PB3 – Multi Stage Events
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Lesson 5 – Unordered Selections
What is the difference between an ordered selection and an unordered selection?
In an ordered selection the order is important (eg. Standing in a line; filling the first 3 places of a race
etc) while in an unordered selection the order doesn’t matter (eg. Choosing members of a
team/committee)?
Example 1
Five people (A, B, C, D, E) are nominated for a committee. In how many ways can two people be
selected for the committee?
Solution: At first it appears as though the answer is 5  4 = 20. But AB is the same as BA; and BC
is the same as CB etc… So we need to divide by the number of ways that two people can be selected
5 4
(2  1). Hence the answer is:
= 10 ways
2 1
Example 2
In how many ways can two people be selected to play tennis from four people called W, X, Y, Z?
Solution: At first it appears as though the answer is 4  3 = 12. But WX is the same as XW; and XY
is the same as YX etc… So we need to divide by the number of ways that two people can be
43
rearranged amongst themselves (2  1). Hence the answer is:
= 6 ways
2 1
Example 3
In how many ways can a team of five basketball players be selected from ten people?
Sol:
The number of ways 5 people can be chosen from 10 people = 10  9  8  7  6 = ???
The number of ways 5 people can be rearranged amongst themselves = 5  4  3  2  1 = ?
Sol = (10  9  8  7  6)  ( 5  4  3  2  1) =
The quicker and easier way to do all of this???
Use the nCr key on the calculator. Eg 5 choose 2 = 5C2 = 60…
Class Work
Exercise 7E (p227) Q1-12
PB3 – Multi Stage Events
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Lesson 6 – Probability
If all the possible outcomes for an event are equally likely then:
The theoretical probability of an event, E, happening is:
number of desired outcomes
P(E) =
total number of possible outcomes
Example 1
A family decides to have four children. Assuming that boys and girls are equally likely;
(a)
How many outcomes are possible, if order of birth is important?
(b)
List all the possible outcomes
(c)
Find the probability that the family has:
(i)
all boys.
(ii)
two boys then two girls, where order is important.
(iii) two boys then two girls, where order is not important.
Class Work
Exercise 7F (p230)
Q1-17 (odd questions)
Lesson 7 – Harder Probability Trees
Example 1
One bag contains one red ball and one blue ball. Another bag contains one red, blue and yellow ball.
One marble is drawn from each bag. Find the probability of picking two blue balls?
Example 2
One bag contains two red balls and one blue ball. Another bag contains two red, one blue and one
yellow ball. One marble is drawn from each bag.
(a)
Draw a tree diagram of the sample space (all possibilities)
(b)
Find the probability of picking two red balls?
Rule – Finding Probabilities from Tree Diagrams…
To find the probability of a tow stage events, find the product (multiply) of the probabilities
along the branches leading to the desired event.
Example 3
The probability of Mr Kastelan hitting the dart board is 0.7.
If he has two throws, find the probability that he
hits the target:
(a)
both times
(b)
no times
(c)
one time
Note: What occurs when you add up all of the probabilities?
Class Work
Exercise 7G (p235) Q1,3,5,7,9,10
PB3 – Multi Stage Events
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Lesson 8 – Expected Frequency of an Event
What is “Expected Frequency”?
Expected Frequency = probability of an event happening  number of trials
Example 1
The probability of a Mr Kastelan hitting the dart board if 08. How many times would you expect
him to hit the dart board if he had 70 trials?
Example 2
In a single roll of a die, list the probability of rolling a: (a) 6, (b) odd number, (c) number less than 3.
If a die is rolled 600 times, how many times would you expect to roll a: : (a) 6, (b) odd number, (c)
number less than 3.
Class Work
Exercise 7H (p237) Q1,3,5,7,9,,11,13
Lesson 9 – Expected Value
Class Work
Exercise 7J (p242) Q1, 3, 5, 6
PB3 – Multi Stage Events
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Lesson 10 – Financial Expectation
What is “Financial Expectation”?
The financial expectation of an event is the theoretical expected return for the event. As a rule,
Financial Expectation = (financial outcome  probability of an outcome)
Example 1
A game is played where a fair coin is tossed. If it’s heads, you win $10. If it’s tails, you lose $9.
Calculate the financial expectation of the game.
Outcome
Financial Outcome
Probability of an outcome
H
$10
0.50
T
-$9
0.50
Financial Expectation = $10  0.50 + $9  0.50 = $0.50
i.e. If you were to play this game, over the long term you would expect to win $0.50 per game
played. Eg. If you played 100 times, you’d expect 100  $0.50 = $50.
Example 2
A business has a financial expectation where there is a 20% chance of making a $2000 profit, but a
10% chance of losing $3000. What is the Financial Expectation for this business?
Outcome
Profit
Loss
Financial Outcome
$2000
$3000
Probability of an outcome
0.20
0.10
F.E. = 0.20  $2000 + 0.10  $3000 = +$100.
Example 3
In a game, two coins are tossed. If you toss two heads you get $6. If you toss one head, you get $2.
If you toss no heads you lose $10. What is the financial expectation for playing the game?
FE = 0.25  $6 + 0.5  $2 + 0.25  $10 = ???
Example 4
A dice game is played where if you roll a two or three, you win $5. What would the penalty need to
be for not rolling a two or three to make the financial expectation of the game zero.
Outcome
Win
Lose
Financial Outcome
$5
$x
Probability of an outcome
1/3
2/3
F.E. = 1/3  $5 + 2/3  $x = 0
etc……
Class Work
Exercise 7K (p243)
Q1, 3, 5-6, 7, 9, 11, 13, 15, 17, 20.
PB3 – Multi Stage Events
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