Dear 4E/5N Students You are expected to do the following for the Maths module:

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

General InstructionsGeneral Instructions

ProbabilityProbability

IntroductionIntroduction

In this e-lesson, you will learn to In this e-lesson, you will learn to

solve simple probability problems solve simple probability problems inin Part OnePart One

use possibility diagrams and tree use possibility diagrams and tree diagrams to solve probability diagrams to solve probability problems involving combined events problems involving combined events inin Part TwoPart Two

IntroductionIntroductionProbability Theory was first used to solve gambling Probability Theory was first used to solve gambling problems. problems.

Lotteries have always been a magnet to those who Lotteries have always been a magnet to those who dream of instant riches. Toto, 4-D and Singapore dream of instant riches. Toto, 4-D and Singapore Sweep are some of the favourite games of chance Sweep are some of the favourite games of chance among Singaporeans.among Singaporeans.

However, do you know that you have aHowever, do you know that you have a one-in-8.1 one-in-8.1 millionmillion chance of winning the first prize for Toto, and chance of winning the first prize for Toto, and the odds for striking any of the the odds for striking any of the first three prizes in the Singapore Sweep first three prizes in the Singapore Sweep and 4-D areand 4-D are one in 3 millionone in 3 million and and one in one in 10,00010,000 respectively? Do you know how respectively? Do you know how to calculate the odds?to calculate the odds?

IntroductionProbability Theory has since been widely Probability Theory has since been widely used in areas like business, finance, science used in areas like business, finance, science and industry, and has become a powerful and industry, and has become a powerful branch of mathematics.branch of mathematics.

We often make statements involving We often make statements involving probability orprobability or chancechance in our daily life. Some in our daily life. Some examples of these statements are: examples of these statements are: ‘‘It will It will probably probably rain today.’ rain today.’ ‘‘It is It is unlikelyunlikely that we will win the that we will win the championship.’ ‘There is a high championship.’ ‘There is a high chancechance that that you will find him you will find him in the canteen.’ in the canteen.’ ‘‘It isIt is impossible impossible to pass the test!’ to pass the test!’

Part OnePart One

ExperimentsExperimentsProbabilityProbability is defined as theis defined as the likelihoodlikelihood of an of an occurrence of a special event. occurrence of a special event.

In probability, anIn probability, an experiment experiment is an operation is an operation or or a process with a result or an outcome whose a process with a result or an outcome whose occurrence depends onoccurrence depends on chancechance. . Some Some examples of an experiment are:examples of an experiment are:

Example 1 Tossing a coin

Example 2 Tossing a dice

Sample SpaceSample SpaceAn experiment can result in several possible An experiment can result in several possible outcomes. The set of all possible outcomes is outcomes. The set of all possible outcomes is called thecalled the sample spacesample space or or probability spaceprobability space, , SS..

Example 1Example 1 Tossing a coinTossing a coinPossible Outcomes: Head or TailPossible Outcomes: Head or Tail S = {Head, Tail}S = {Head, Tail}

Example 2Example 2 Tossing a diceTossing a dicePossible Outcomes: 1 or 2 or 3 Possible Outcomes: 1 or 2 or 3 or 4 or 5 or 6or 4 or 5 or 6 S = {1, 2, 3, 4, 5, 6}S = {1, 2, 3, 4, 5, 6}

EventsEventsAn An eventevent, , EE is a particular result of an is a particular result of an experiment. Hence,experiment. Hence, EE contains some or all of contains some or all of the possible outcomes inthe possible outcomes in SS..

Example 1Example 1 Tossing a coinTossing a coinLet E be the event of getting a tail.Let E be the event of getting a tail. E = {Tail}E = {Tail}

Example 2Example 2 Tossing a diceTossing a diceLet E be the event of getting a

number less than 5 on the dice.

E = {1, 2, 3, 4}

Simple ProbabilitySimple Probability

The probability of an event E occurring is given The probability of an event E occurring is given by by

where n(E) is the number of outcomes in E and where n(E) is the number of outcomes in E and n(S) is the total number of possible outcomes n(S) is the total number of possible outcomes in S. in S.

n(S)n(E)

P(E)

Simple ProbabilitySimple ProbabilityExample 1Example 1 Tossing a coinTossing a coin

S = {Head, Tail} and n(S) = 2S = {Head, Tail} and n(S) = 2E = {Tail} and n(E) = 1E = {Tail} and n(E) = 1

Example 2Example 2 Tossing a diceTossing a diceS = {1, 2, 3, 4, 5, 6} and n(S) = 6S = {1, 2, 3, 4, 5, 6} and n(S) = 6E = {1, 2, 3, 4} and n(E) = 4E = {1, 2, 3, 4} and n(E) = 4

.21

P(E)

.32

64

P(E)

Simple ProbabilitySimple ProbabilityThe probability of any event occurring lies The probability of any event occurring lies between between 0 and 1 inclusive, i.e.0 and 1 inclusive, i.e. 0 ≤ P(E) ≤ 10 ≤ P(E) ≤ 1..Do you know why? Do you know why?

Some important notes:Some important notes: If P(E) = 0, then the event cannot possibly If P(E) = 0, then the event cannot possibly occur.occur. If P(E) = 1, then the event will certainly occur.If P(E) = 1, then the event will certainly occur.

Probability of an event E Probability of an event E not not occurringoccurring = 1 – probability of an event occurring = 1 – probability of an event occurring i.e.i.e. P(E’) = 1 – P(E)P(E’) = 1 – P(E)

Question: In an experiment, a card is drawn Question: In an experiment, a card is drawn from a pack of 52 playing from a pack of 52 playing

cards. cards.

(a) What is the total number of (a) What is the total number of possible outcomes of this possible outcomes of this experiment? experiment?

(b) What is the probability of (b) What is the probability of drawingdrawing

(i) a black card, (i) a black card, (ii) a green card, (ii) a green card, (iii) a red ace,(iii) a red ace,

(iv) a heart, (iv) a heart, (v) a card which is not a heart?(v) a card which is not a heart?

Sample QuestionSample Question

.21

5226

n(S)card) n(black

Solution to Sample QuestionSolution to Sample Question

(a) Total number of possible outcomes, n(S) (a) Total number of possible outcomes, n(S) = 52.= 52.

(b)(i) P(drawing a black card) =(b)(i) P(drawing a black card) =

(ii) P(drawing a green card) = (ii) P(drawing a green card) =

(iii) P(drawing a red ace) = (iii) P(drawing a red ace) =

0.520

n(S)card) n(green

.261

522

n(S)ace) n(red

.43

41

5213

11

.41

5213

n(S)n(heart)


(b)(iv) P(drawing a heart) =(b)(iv) P(drawing a heart) =

(v) P(drawing a card which is not a heart) (v) P(drawing a card which is not a heart) = 1 – P(drawing a heart) = 1 – P(drawing a heart)

Part TwoPart Two

Possibility Diagrams and Tree DiagramsPossibility Diagrams and Tree Diagrams

Possibility diagrams and tree Possibility diagrams and tree diagrams are used to list all diagrams are used to list all

possible outcomes of a sample possible outcomes of a sample space in a systematic and effective space in a systematic and effective

manner.manner.

These diagrams are useful for These diagrams are useful for finding the probabilities of finding the probabilities of

combined events. combined events.

An example of possibility diagramsAn example of possibility diagrams

Two coins are tossed together. The Two coins are tossed together. The possibility diagram below shows all the possibility diagram below shows all the possible outcomes:possible outcomes:

1st coin

2nd

coin

H T

H

T

S = { }

Each represents an outcome.

HT,HH, TTTH,

Eg. P(getting 2 heads)

= P(HH)

= .

41

An example of tree diagramsAn example of tree diagramsTwo coins are tossed together. The tree diagram Two coins are tossed together. The tree diagram below shows all the possible outcomes:below shows all the possible outcomes:

T

H

H

T

H

1st coin

½

T

2nd coin

½

½

½

½

½

Outcome

HH

HT

TH

TT

Probability

P(HH) = ½ ½ = ¼

P(HT) = ½ ½ = ¼

P(TH) = ½ ½ = ¼

P(TT) = ½ ½ = ¼

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 ¼+¼+¼+¼ = 1.

Sample QuestionSample QuestionQuestion: 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.


(a) Using a possibility diagram:(a) Using a possibility diagram:

2nd

box

1st box

1

2

3

5

3

4

5

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) = 12

(b)(i) P(both numbers are the same)

=

.31

124

.61

122

(b)(ii) P(both numbers are odd)

=

(b)(iii) P(sum > 7) =

.21

126


(a) Using a tree diagram:(a) Using a tree diagram:

1st box 2nd box

(b)(i) P(both numbers are the same)

= P[(3,3) or (5,5)]

= .

61

122

241

31

41

31

41

31

(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)]

=

.21

126

641

31

(b)(iii) P(sum > 7)

= P[(3,5) or (5,3) or (5,4) or (5,5)]

= .

31

124

441

31

1

3

5

23

54

23

54

23

54

31

41

Websites: Websites:

http://mathforum.org/dr.math/faq/faq.probhttp://mathforum.org/dr.math/faq/faq.prob.intro.html.intro.html

http://regentsprep.org/Regents/math/math-http://regentsprep.org/Regents/math/math-a.cfm#a6a.cfm#a6

http://www.bbc.co.uk/schools/ks3bitesize/http://www.bbc.co.uk/schools/ks3bitesize/maths/handling_data/index.shtmlmaths/handling_data/index.shtml

Task: Task: Complete the Multiple Choice Complete the Multiple Choice Questions. Do your work on foolscap Questions. Do your work on foolscap paper and show your working clearly.paper and show your working clearly.

AssignmentAssignment

NOTE: Submit your assignment to your Math Teacher

on 05 February 2008.

References:References:

1. Lee, P. Y., Fan, L. H., Teh, K. S. 1. Lee, P. Y., Fan, L. H., Teh, K. S. and Looi, C. K. (2002) and Looi, C. K. (2002) New Syllabus New Syllabus Mathematics 4Mathematics 4 Singapore: Shing Lee Singapore: Shing Lee Publishers Pte Ltd.Publishers Pte Ltd.

2. Tay, C. H. (2003) 2. Tay, C. H. (2003) New Mathematics New Mathematics Counts for Secondary 5 Normal Counts for Secondary 5 Normal (Academic)(Academic) Singapore: Federal Singapore: Federal Publications. Publications.

End End ofof e-Lesson e-Lesson

Documents

Dear 4E/5N Students You are expected to do the following for the Maths module: