Chapter OneIntroduction to Probability

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DISCRETE RANDOM VARIABLESDISCRETE RANDOM VARIABLES

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

Learning Objectives:

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Basic Definitions:

Random Experiment and Outcomes

Outcomes can be represented by Venn diagram or tree diagram.

Event: collection of one or more of the outcomes of an experiment.

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

Sample Spaces and Events

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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

The lifetime of a machine (in days)

The working state of a machine 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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Calculating Probability

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

Classical way of finding probability

n(S)

n(A)

iespossibilitlikelyequallyofNumberTotal

AeventiniespossibilitofNumber)(

AP

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

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

Other notations for complement: Ac or Ā

( ') 1 ( )P A P A

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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.

General Addition Law

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8 blue marbles, 5 blue cubes, 10 green marbles and 7 green cubes

Total sample space : 30 objects

P( Cubes or green)

General Addition Law

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Let A and B be two events defined in a sample space S.

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

259252256255

252256255

////)(

/)(,/)(,/)(

BAP

BAPBPAP

General Addition Law

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P(A B) 0

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

Let A and B be two mutually exclusive, events defined in asample space S.

This can be expanded to consider more than two mutually exclusive events.

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:

 

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’)

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Conforms

Yes No

SupplierR 30 10

S 22 8

T 25 5

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Solution

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?

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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.

Likewise, , P(A) > 0

( )( | )

( )

P A BP A B

P B

With the condition that P(B) > 0

( )( | )

( )

P A BP B A

P A

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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?

Example 4: Disks of polycarbonate plastic from a supplier are analyzed for scratch

and shock resistance. The results from 100 disks are given as:

 

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

Shock resistance

high low

Scratch resistance

high 70 10

low 16 4

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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

Example: - Roll a fair die, consider - Event A = { 2,4,6} - Event B = { 4,5, 6} - Are events A and B independent?

)()|( APBAP

P(A) = 1/2

P(A|B) = 2/3

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Multiplicative Law of Probability and Independence

( ) ( ). ( )P A B P A P B

( ) ( | ). ( )P A B P A B P B For two events A and B

Definition: Events A and B are independent if and only if

If events A1, .., Ak are independent then,

1 2 1 2( ... ) ( ) ( ) ( )k kP A A A P A P A P A

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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

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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

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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

1 1

( ) ( ) ( | ) ( )n n

i i ii i

P A P A B P A B P B

B1

SA

A∩ B1

A∩ B3A∩ B4A∩ B2

B1 B2B3

B4

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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,

)(

)()|()|(

AP

BPBAPABP kk

k

n

iii

kkk

BPBAP

BPBAPABP

1

)()|(

)()|()|(

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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?