Chapter 1_ Introduction To Probability

Chapter One Introduction to Probability

DISCRETE RANDOM VARIABLES Learning Objectives: At the end of the lecture, you will be able to : -

describe types of events and random variables

calculate their probability distribution and their cumulative distribution

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Basic Definitions: Random Experiment and Outcomes Random Experiment

Outcomes/ Equally likely Possibilities

Toss a fair coin once

Head, Tail

Roll a fair die once

1,2,3,4,5,6

Take a test

Pass, Fail

Select a worker

Male, female

Outcomes can be represented by Venn diagram or tree diagram. Event: collection of one or more of the outcomes of an experiment. 3

Sample Spaces and Events Example1.

Roll a die

Sample space: S = {1, 2, 3, 4, 5, 6}

Simple

events (or outcomes):

E1: observe a 1= {1}

E3 = {3}

E4 = {4}

E2 = {2}

E5 = {5}

E6 = {6}

Compound

events: A : observe an odd number = {1, 3, 5} B : observe a number greater than or equal to 4 = {4, 5, 6}

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Example 1 

Toss a coin three times and note the number of heads

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The lifetime of a machine (in days)

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The working state of a machine

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The number of calls arriving at a telephone exchange during a specific time interval

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Example 2: 



Each message in a digital communication system is classified as to whether it is received within the time specified by the system design. If 3 messages are classified, what is an appropriate sample space for this experiment? To generate the sample space, we can use a tree diagram

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Types of events: Complementary events: The complement of event A, is the event that includes all the outcomes for an experiment that are not in A. Intersection of events ( A ∩ B) : The collection of all outcomes that are common to both event A and event B. Union of events (A U B) : The collection of all outcomes that belong to either event A or to event B or to both event A and event B. Mutually Exclusive events: Events that cannot occur together.

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Definition of Probability A probability P is a rule ( or function) which assigns a number between 0 and 1 to each event and satisfies: 

0 ≤ P(E) ≤ 1 for any event E



P(Ø ) = 0 , P(S) = 1,

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N u m b e r o f p s i b l t e s i n v t A P(A )T onA tSalquaykpobiles

Calculating Probability

Probability: a numerical measure of the chance/likelihood that a specific event will occur. Classical way of finding probability

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

The probability of the complement of any event A is given as

P ( A ')  1  P ( A) 

Example: If P(rain tomorrow) = 0.6 then P(no rain tomorrow) = 0.4



Other notations for complement: Ac or Ā 10

Examples : Generate the sample space using tree diagram a)A jar

contains 5 red sweets and 3 blue sweets. Two sweets are drawn at random i) with replacement and ii) without replacement. b)Kamil

has the option of taking one of three routes to work A, B or C. The probability of taking route A is 30%, and B is 15%. The probability of being late for work if he goes by route A is 10% and similarly by route B is 5% and route C is 2%. c)A normal

six sided fair die is thrown until a five is scored and then no more throws are made. The process continues up to a maximum of three throws.

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General Addition Law 8 blue marbles, 5 blue cubes, 10 green marbles and 7 green cubes Total sample space : 30 objects P( Cubes or green)

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General Addition Law Let A and B be two events defined in a sample space

S.

P(A  B)  P(A)  P(B)  P(A  B) P ( A)  5 / 25, P ( B )  6 / 25, P ( A  B )  2 / 25  P ( A  B )  5 / 25  6 / 25  2 / 25  9 / 25 13

General Addition Law Let A and B be two mutually exclusive, events defined in a sample space S.

P(A  B)  0

P(A  B)  P(A)  P(B) This can be expanded to consider more than two mutually exclusive events. 14

Example 1: Samples of building materials from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows: Conforms

Supplier

Yes

No

R

30

10

S

22

8

T

25

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Let A denote the event that a sample is from supplier R, and B denote the event that a sample conforms to the specifications. If sample is selected at random, determine the following probabilities:  (a) P(A) (b) P(B) (c) P(B’)  (d) P(AUB) (e) P(AB) (f) P(AUB’) 15

Solution

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Example 2. In a residential suburb, 60% of all households subscribe to the metro newspaper published in a nearby city, 80% subscribe to the local paper, and 50% of all households subscribe to both papers. Draw a Venn diagram for this problem. If a household is selected at random, what is the probability that it subscribes to:  

a) at least one of the two newspapers b) exactly one of the two newspapers

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Solution

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Example 3 A system consists of two components. The probability that the second component functions satisfactorily is 0.9, the probability that at least one of the two components does so is 0.96, and the probability that both components do so is 0.75. What is the probability that a) the first component functions satisfactorily? b) neither the first nor the second component function satisfactorily? c) the second one functions in a satisfactory manner given that the first component does also? 19

Conditional Probability A jar contains 5 red sweets and 3 blue sweets. Two sweets are drawn at random without replacement. Draw a tree diagram for the problem.

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Conditional Probability   

Let A and B be two events defined in a sample space S. The conditional probability of A, given that B has already occurred, is denoted as P ( A | B) or P ( A / B ). Important note: a common mistake is to assume that the “/” indicates division. It does not indicate this. It denotes “given”. The probability of A “given “ B.

P( A  B) P( A | B)  With the condition that P(B) > 0 P( B)



Likewise,

P( A  B) P( B | A)  P ( A)

, P(A) > 0 21

Example 5:

Sarah goes to work either by one of two routes, A or B. The probability of going by route A is 30%. If she goes by route A, the probability of being late for work is 5% and if she goes by route B, the probability of being late is 10%. a)Find the probability that she is late for work b)Given that Sarah is late for work, find the probability that she went via route A.

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Example 3 ( revisit) A system consists of two components. The probability that the second component functions satisfactorily is 0.9, the probability that at least one of the two components does so is 0.96, and the probability that both components do so is 0.75. What is the probability that a) the first component functions satisfactorily? b) neither the first nor the second component function satisfactorily? c) the second one functions in a satisfactory manner given that the first component does also? 23

Example 4: Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are given as: Shock resistance

Scratch resistance

high

low

high

70

10

low

16

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Let A denote the event that a disk has high shock resistance and B denote the event that a disk has high scratch resistance. If sample is selected at random, determine the following probabilities: (a) P(A)

(b) P(B)

(c) P(A|B) (d) P(B|A)

(d ) Are events A and B independent? 24

Independent Events A jar contains 5 red sweets and 3 blue sweets. Two sweets are drawn at random with replacement. Draw a tree diagram for the problem

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Independent Events 

Two events A and B are independent if

P ( A | B)  P ( A)



Example: - Roll a fair die, consider P(A) = 1/2 - Event A = { 2,4,6} P(A|B) = 2/3 - Event B = { 4,5, 6} - Are events A and B independent? 26

Multiplicative Law of Probability and Independence For two events A and B

P ( A  B )  P( A | B ). P( B ) Definition: Events A and B are independent if and only if

P ( A  B )  P ( A). P( B ) If events A1, .., Ak are independent then,

P( A1  A2  ...  Ak )  P ( A1 ) P( A2 )  P( Ak ) 27

Example 5 Ali and Kamil are sometimes late for class. Let A = the event that Ali is late for class and K = the event that Kamil is late for class Given that P(A) = 0.25 , P( A and K) = 0.15 and P ( A’ and K’) = 0.7 On a randomly selected day, find the probability that a)At least one of Ali or Kamil are late for class b)Kamil is late for class 28

c) Given that Ali is late for class, find the probability that Kamil is late The professor suspects that Ali being late for class and Kamil being late for class are linked in some way. d)Determine whether or not A and K are statistically independent e)Based on your results in part (d), comment on the professor’s suspicion 29

The Law of Total Probability

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The Law of Total Probability 

Suppose B1, B2 ,…, Bn are mutually exclusive and exhaustive in S, then for any event A n

n

i 1

i 1

P( A)   P( A  Bi )   P ( A | Bi ) P( Bi )

S

A A∩ B1 B1

A∩ B2

B2

B1 A∩ B3

B3

A∩ B4

B4 31

Bayes’ Theorem

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Bayes’ Theorem Suppose B1, B2,…, Bn are mutually exclusive and exhaustive (whose union is S). Let A be an event such that P(A) > 0. Then for any event Bj , j =1, 2, …, n,

P ( A | Bk ) P ( Bk ) P ( Bk | A)  P ( A) P ( Bk | A) 

P ( A | Bk ) P ( Bk ) n

 P( A | B i 1

i

) P ( Bi ) 33

Example 1. A store stocks light bulbs from three suppliers. Suppliers A, B, and C supply 10%, 20%, and 70% of the bulbs respectively. It has been determined that company A’s bulbs are 1% defective while company B’s are 3% defective and company C’s are 4% defective. If a bulb is selected at random and found to be defective, what is the probability that it came from supplier B? Solution

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Example 2. A particular city has three airports. Airport A handles 50% of all airline traffic, while airports B and C handle 30% and 20%, respectively. The rates of losing a baggage in airport A, B and C are 0.3, 0.15 and 0.4 respectively.

If a passenger arrives in the city and losses a baggage, what is the probability that the passenger arrives at airport B?

What is the probability that a customer loses a baggage and at airport C

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Example 3: In a certain assembly plant, three machines, B1, B2, B3, make 30%, 45% and 25%, respectively, of the products. It is known from past experience that 2%,3% and 2% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective? Solution:

If a product was chosen randomly and found to be defective, what is the probability that it was made by machine B3?

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