Presentation Chapter 7 MTH1022 Rev#01

7/28/2019 Presentation Chapter 7 MTH1022 Rev#01

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Module Outcomes:MO1

Identify basic mathematical concepts, skills and

mathematical techniques for algebra, calculus and data

handling.

MO2Apply the mathematical calculations, formulas,

statistical methods and calculus techniques for problem

solving in industry.

MO3 Analyse calculus and statistical problems in industry.


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LEARNING OUTCOMESAt the end of this chapter, student should be able to :

understand the concepts of experiments, outcomes, samplespaces and events

define probability

understand basic laws of probability

calculate the probabilities using the rules of probability.

develop contingency table and use a tree diagram to organizeprobabilities


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Definition of ProbabilityProbability is a measure of howhigh is the possibility for anevent to occur.


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Basic concepts in Probability

Definition Example

An experiment is a situationinvolving chance or probability that

leads to results called outcomes.Toss a fair dice.

An outcome is the result of a singletrial of an experiment.The possible outcomes are no. 1,2,3,4, 5 and

6.

Asample space is the set of allpossible outcomes of the experiment

Notation of Sample Space : SNumber of Sample Space : n(S)

S = {1,2,3,4,5,6}

n(S) = 6

An event is one or more outcomes ofan experiment.

One event of this experiment is no. 1.


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Examples of experiments and their possible

associated outcomes and sample spaces:

Experiment Outcomes Sample space

Flip two coins HH, HT, TH, TT S = {HH,HT,TH,TT}

Role a dice 1, 2, 3, 4, 5, 6 S = {1, 2, 3, 4, 5, 6}

Play a game Win (W), Lose (L), Draw (D) S = {Win (W), Lose (L), Draw (D)}


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HOW TO FIND OUT THE SAMPLE SPACE/ NUMBER

OF SAMPLE SPACE

Techniques

LIST

TABLE

TREEDIAGRAM

VENNDIAGRAM


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VENN DIAGRAM FOR


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VENN DIAGRAM FOR


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VENN DIAGRAM FOR =

A B


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Definition of Classical ProbabilitySuppose S is a sample space and each outcome in S isequally likely to occur. If A is an event (a subset of S)then the probability of A is:

=

=


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Example 1A card is drawn from a pack of 52 cards randomly. Find

the probability that:

It is a four card

It is a black card


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Solution Let F : the event of getting a four card

B : the event of getting a black card

n(F) = 4, n(B) = 26, n(S) = 52 =

()

()=

=

=()

()=

=


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Basic probability rules:

The range of values of a probability is 0 () 1.

If = 1, eventA is sure to occur

IfP(A) = 0, eventA will not occur

For n mutually exclusive events () = 1.

If A is the complement of A, then P(A) = 1 P(A)

The complement of an eventA, is a sample space S,consist of all outcomes ofSwhich are not the outcomesofA.


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CONTINUE

AA


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Example 2 There are red and blue balls in a bag. If the probability

of choosing red ball is

, what is the probability of

getting blue balls?


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Solution Let : the event of getting blue ball

: the event of not getting blue ball

= 1 ()

= 1

=


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COMPOSITE EVENTS For any two events A and B, the probability of either

event A or B happening is denoted ( ), where

= + ( )

( ) represents the probability of both events Aand B happening together.

If A and B are mutually exclusive events, then = . This implies that if A and B are mutually

exclusive events, then = 0.


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Example 3 Probability X and Y are such that:

=

, =

and =

. Find ( ).


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Solution = +

=

+

=


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Example 4 If A and B are two events and P(A) = 0.6, P(B) = 0.3

and = 0.8 find

a. ( )

b.

c. ( )


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Solutiona. = +

0.8 = 0.6 + 0.3 = 0.1

From the Venn diagram:

0.1 0.20.5

o.2


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Solutionb. = 0.5

c. = 0.2 + 0.2 = 0.4


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Example 5 The given table shows the number of male and female

students in a class who wear glasses. If the student ischosen at random from that class, calculate the

probability that a female student or a student whowears glasses is chosen.

Wear glasses Do not wearglasses

Male 5 10

Female 9 11


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SOLUTION Let M: event for male students

F : event for female studentsWG : event for students who wear glassesDWG : event for students who do not wear glasses

Question : probability for female student or a student who wears glasses.

From the table:n(M) = 15, n(F) = 20, n(WG) = 14, n(DWG) = 21, n(S) = 35Thus,

= +

=

+

=20

35+

14

35

9

35

=25

35=

5

7


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Mutually exclusive events Two events are said to be mutually exclusive if they

cannot occur at the same time.

1 3 5

2 4 6

A B


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CONTINUE IfA and B are mutually excluxive events, then

= + = 0


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INDEPENDENT AND CONDITIONAL

EVENTS INDEPENDENT EVENT : the outcomes of an

experiments of event A do not influence the outcomesof event B

= () ()

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Presentation Chapter 7 MTH1022 Rev#01