chapter 2 probability 1.pdf

7/26/2019 Chapter 2 Probability 1.pdf

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Chapter 2: Probability

- INDEPENDENT AND DEPENDENT EVENTS- TOTAL PROBABILITY RULE

- BAYES TH OR M


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2.1 Introduction of Probability

What is Probability?

A measure of the likeliness that an event will occur.

Quantified a number between 0 and 1

Value near 0 - the event is unlikely to happen

Value near 1 - the event is likely to happen


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Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a chance, so we can expect 50Heads.

But when we actually try it we might get 48 heads, or 55 heads ... oranything really, but in most cases it will be a number near 50.


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Some Basic Terms and Concepts

Experiment A process that when performed,

results in one and only one of many observations.

Outcome The observation/ possible result of an

experiment. (1 outcome from 1 trial)

Sample space set of all possible outcomes of a

statistical experiment and presented by the symbol,

S.

Sample points The elements of a sample space.

Event A set of outcomes of an experiment (a

subset of the sample space)


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Example:

An experiment is carried out: Toss a coin twice.

Possible Outcomes: 1H & 1T, 2H & 0T, 0H & 2T

Sample space: S = {HT, HH, TT}

Sample points

Possible events: no tail obtained, no head obtained,

at least one tail etc.


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Sample space1. Venn Diagram: a picture that consists of all possible outcomes for anexperiment

2. Tree Diagram: each outcome is represented by a branch of tree


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SymbolsA event A occurring

P(A) probability of event A occurring

A, A, ~A event A not occuringP(A) @ P(A) @ P(~A) probaility of event A not occuring


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2.2 Definition of Event

Complement Event: an event with subset of all elements ofS that are not in A and is denoted by the symbol A' .

+ 1All outcomes that are NOT the event.

Examples:

Intersection Event: intersection of two events A and B

denoted by the symbol

A B, is the event containing all

elements that are common to A and B.

( )


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2.2 Definition of Event (cont.)

Union of Event-The union of the two events A and B,

denoted by the symbol A B , is the event containing allelements that belong to A or B or both. + ( )

Mutually Exclusive Events: Two events A and B are mutually

exclusive or disjoint if A B or , that is, if A and Bhave no element in common. () +

Mutually Non-Exclusive Events: Two events A and B aresaid to be mutually non exclusive events if both the events

A and B have at least one common outcome between them.


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ExampleIf S = {p, q, r, s, t, u, v, w, x, y} and A = {q, s, t, v, x}, B = {p, r, v, w, y},C = {r, u, w, y} and D = {p, t, x}, list the elements of the sets

corresponding to the following events:

(a) AC(b) AB

(c) C

(d) (AB) D(e) (B C)


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2.3 Concept of Probability

Probability is the likelihood of the occurrence of anevent that is measured by using numerical value.

Equally likely outcomes is two or more outcomes that

have the same probability of occurrence.

Theorem:

If an experiment can result in any one of different equallylikely outcomes and if exactly of these outcomescorrespond to event, then the probability of event is

no. of outcomes included in the event

sample size


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Definition

The probability of an event A is the sum of the weights of allsample points in A. Therefore,

0 P(A) 1, and P(S) = 1

For an impossible event, M: P(M) = 0

For a sure event, C: P(C) = 1


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Example 2.1:

1) A bag contains 10 red marbles. A marble is drawn randomlyfrom the bag. Find the probability that the marble drawn is

a) Red

b) Black

2) Ali has a set of eight cards numbered 1 to 8. A card is drawn

randomly from the set of cards. Find the probability that the

number drawn is

a) 8

b) Not 8


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3) Roll a dice three times and see whether you obtain no. 6.

Find the probability of each of the following events:

a) You obtain at least one no. 6.

Ans: 91/216

b) You obtain at least two no. 6.Ans: 2/27

c) You obtain at most two no. 6.

Ans: 215/216

d) You will not obtain no. 6 exactly once.

Ans: 47/72


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2.4 Independent and Dependent Events

Two events are said to be independent if the

occurrence of one does not affect the probability of the

occurrence of the others.

Two events are dependent if the outcome of the first

affects the outcome of second so that the probability is

changed.


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Multiplication Rule for Independent Events

The probability of the intersection of two independentevents A and B is

P(A B) = P(A)P(B)


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Example:

Independent events: Probability of Tom and Tony score

A in Mathematics are 0.7 and 0.9, respectively. What isthe probability that both of them fail to score A in

Mathematics?

Ans: 0.03

Dependent events: A bag contains 3 white balls and 2

black balls. Tom draws a ball without placing it back to

the bag and followed by drawing the second ball. What

is the probability that the second ball drawn is in white

colour?Ans: 3/5


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2 5 Conditional probability

Probability that an event A will occur given that another

event B has already occurred:

() , > 0.

Independent Events

The probability of the intersection of two independent

events and is


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Example 2.2:

1) The probability that an automobile being filled with gasolinealso needs an oil change is 0.25; the probability that it needs

a new oil filter is 0.40; the probability that both the oil and

the filter need changing is 0.14.

a) Are the events that needs an oil change and oil filterchange independent?

b) If the oil has to be changed, what is the probability that a

new oil filter is needed?

c) If the new oil filter is needed, what is the probability that

the oil has to be changed?


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2) The probability that the head of household ishome when a telemarketing representative calls

is 0.4. Given that the head of house is home, the

probability that goods will be bought from the

company is 0.3. Find the probability that thehead of the house is home and goods are bought

from the company.


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3) The probability of closing the ith relay in the circuit shown

below are

If all relays function independently, what is the probabilitythat a current flows between A and B?

Circuit 1 2 3 4 5

P(closure) 0.7 0.6 0.65 0.65 0.97


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2.6 Total Probability Rule

A collection of sets , , 3, 4, , such that 3 is said to be exhaustive.

Assume , , 3, 4, , are mutually exclusive andexhaustive sets, then for any eventA of S,

=

=

(|) .

+ + + (|)


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Example 2.3:

In a certain assembly plant, three machines, , and 3,make 30%, 45%, and 25%, respectively, of the products. It

is known from the past experience that 2%, 3%, and 2% of

the products made by each machine, respectively, aredefective. A finished product is randomly selected. What is

the probability that it is defective?


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2.7 Bayes Theorem

If the events , , , constitute a partitionof the sample space S such that () 0 for 1,2,,, then for any event in such thatP(A) 0 ,

| ( )=

(|)= (|)for 1,2, , .


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How to get the Formula?

From previous, | and

Then,

If , , , are mutually exclusive and exhaustive events and is anyevent, then

+ + +


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Example 2.4:

1) A manufacturing firm employs 3 analytical plans for the

design and development of a particular product. For

cost reasons, all three are used in varying times. In fact,

plans 1, 2 or 3 are used for 30%, 20% and 50% of theproducts, respectively. The defect rate given plan 1, 2

and 3 are 0.01, 0.03 and 0.02, respectively. If a random

product was observed and found to be defective, which

plan was most likely used and thus responsible?


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2) A construction company employs two sales engineers.

Engineer 1 does the work of estimating cost for 70% ofjobs bid by the company. Engineer 2 does the work for

30% of jobs bid by the company. It is known that the

error rate for engineer 1 is such that 0.02 is the

probability of an error when he does the work,

whereas the probability of an error in the work of

engineer 2 is 0.04. Suppose a bid arrives and a serious

error occurs in estimating cost. Which engineer would

you guess did the work? Explain and show all work.


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ExerciseIt is known from past experience that the probability ofselecting an adult over 60 years of age who aresmokers is 0.35. Of those adults over 60 years of agewho are smokers, 55% of them have heart attack. Ofthose adults over 60 years of age who arenonsmokers, 12% of them have heart attack. What isthe probability of selecting one of these adults withheart attack is found to be a nonsmoker? What is theprobability of selecting one of these adults withoutheart attack is found to be a smoker?

chapter 2 probability 1.pdf

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