Dear 4E/5N Students

1
General Instructions

• Dear 4E/5N Students
• You are expected to do the following for the
  Maths module
• Read and study the slides on Probability
• Explore the websites given
• Complete the one task given on the worksheet and
  submit them to your math teacher on 5 February
  2008.
• Happy Enjoy-Learning!
• Best Regards
• Your Math Teachers

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Probability
3
Introduction

• In this e-lesson, you will learn to
• solve simple probability problems in Part One
• use possibility diagrams and tree diagrams to
  solve probability problems involving combined
  events in Part Two

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Introduction

• Probability Theory was first used to solve
  gambling problems.
• Lotteries have always been a magnet to those who
  dream of instant riches. Toto, 4-D and Singapore
  Sweep are some of the favourite games of chance
  among Singaporeans.
• However, do you know that you have a one-in-8.1
  million chance of winning the first prize for
  Toto, and the odds for striking any of the
• first three prizes in the Singapore Sweep
• and 4-D are one in 3 million and one in
• 10,000 respectively? Do you know how
• to calculate the odds?

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Introduction

• Probability Theory has since been widely used in
  areas like business, finance, science and
  industry, and has become a powerful branch of
  mathematics.
• We often make statements involving probability or
  chance in our daily life. Some examples of these
  statements are
• It will probably rain today.
• It is unlikely that we will win the
  championship. There is a high chance that you
  will find him
• in the canteen.
• It is impossible to pass the test!

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Part One
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Experiments
Probability is defined as the likelihood of an
occurrence of a special event. In probability,
an experiment is an operation or a process with
a result or an outcome whose occurrence depends
on chance. Some examples of an experiment are
Example 1 Tossing a coin Example 2 Tossing a dice
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Sample Space
An experiment can result in several possible
outcomes. The set of all possible outcomes is
called the sample space or probability space,
S. Example 1 Tossing a coin Possible Outcomes
Head or Tail ? S Head, Tail Example
2 Tossing a dice Possible Outcomes 1 or 2 or 3
or 4 or 5 or 6 ? S 1, 2, 3, 4, 5, 6
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Events
An event, E is a particular result of an
experiment. Hence, E contains some or all of the
possible outcomes in S. Example 1 Tossing a
coin Let E be the event of getting a tail. ?
E Tail Example 2 Tossing a dice Let E
be the event of getting a number less than 5
on the dice. ? E 1, 2, 3, 4
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Simple Probability
The probability of an event E occurring is given
by where n(E) is the number of outcomes in
E and n(S) is the total number of possible
outcomes in S.
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Simple Probability
Example 1 Tossing a coin S Head, Tail and
n(S) 2 E Tail and n(E) 1 Example
2 Tossing a dice S 1, 2, 3, 4, 5, 6 and
n(S) 6 E 1, 2, 3, 4 and n(E) 4
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Simple Probability

• The probability of any event occurring lies
  between
• 0 and 1 inclusive, i.e. 0 P(E) 1.
• Do you know why?
• Some important notes
• If P(E) 0, then the event cannot possibly
  occur.
• If P(E) 1, then the event will certainly
  occur.
• Probability of an event E not occurring
• 1 probability of an event occurring
• i.e. P(E) 1 P(E)

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Sample Question
Question In an experiment, a card is drawn
from a pack of 52 playing cards. (a) What
is the total number of possible outcomes
of this experiment? (b) What is the
probability of drawing (i) a black card,
(ii) a green card,
(iii) a red ace, (iv)
a heart, (v) a
card which is not a heart?
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Solution to Sample Question
(a) Total number of possible outcomes, n(S)
52. (b)(i) P(drawing a black card) (ii)
P(drawing a green card) (iii)
P(drawing a red ace)
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Solution to Sample Question
(b)(iv) P(drawing a heart) (v) P(drawing
a card which is not a heart) 1
P(drawing a heart)
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Part Two
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Possibility Diagrams and Tree Diagrams
Possibility diagrams and tree diagrams are used
to list all possible outcomes of a sample space
in a systematic and effective manner. These
diagrams are useful for finding the probabilities
of combined events.
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An example of possibility diagrams
Two coins are tossed together. The possibility
diagram below shows all the possible outcomes
S Each
represents an outcome.
HT,
HH,
TT
TH,
1st coin
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An example of tree diagrams
Two coins are tossed together. The tree diagram
below shows all the possible outcomes
Each outcome is obtained by tracing along a
branch from left to right. The probability of
each outcome is obtained by multiplying the
probabilities along the respective branch. The
total probability of all possible outcomes is
¼¼¼¼ 1.
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Sample Question
Question A box contains three cards numbered 1,
3, 5. A second box contains
four cards numbered 2, 3, 4, 5.
A card is chosen at random from
each box. (a) Show all
the possible outcomes of the
experiment using a possibility diagram or
a tree diagram.
(b) Calculate the probability that
(i) the numbers on the cards are
the same, (ii) the numbers
on the cards are odd, (iii)
the sum of the two numbers on the
cards is more than 7.
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Solution to Sample Question
(a) Using a possibility diagram
S (1,2), (1,3), (1,4), (1,5), (3,2), (3,3),
(3,4), (3,5), (5,2), (5,3), (5,4), (5,5) n(S)
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(b)(i) P(both numbers are the same)

(b)(ii) P(both numbers are odd)
(b)(iii) P(sum gt 7)
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Solution to Sample Question
(a) Using a tree diagram
(b)(i) P(both numbers are the same)
P(3,3) or (5,5)
1st box
2nd box
(b)(ii) P(both numbers are odd)
P(1,3) or (1,5) or (3,3) or
(3,5) or (5,3) or (5,5)
(b)(iii) P(sum gt 7) P(3,5)
or (5,3) or (5,4) or (5,5)
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Websites http//mathforum.org/dr.math/faq/faq.p
rob.intro.htmlhttp//regentsprep.org/Regents/mat
h/math-a.cfma6http//www.bbc.co.uk/schools/ks3b
itesize/maths/handling_data/index.shtml
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Assignment
Task Complete the Multiple Choice Questions. Do
your work on foolscap paper and show your working
clearly.
NOTE Submit your assignment to your Math Teacher
on 05 February 2008.
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References1. Lee, P. Y., Fan, L. H., Teh, K.
S. and Looi, C. K. (2002) New Syllabus
Mathematics 4 Singapore Shing Lee Publishers Pte
Ltd.2. Tay, C. H. (2003) New Mathematics Counts
for Secondary 5 Normal (Academic) Singapore
Federal Publications.
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End of e-Lesson