Revision Lecture (4) (1)

7/24/2019 Revision Lecture (4) (1)

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Prof. Amina Saleh

1

Probability TheoryMaSc467

Lectures notes prepared forLevel 6 students at

College of Computer Science and InformationDepartments of:

Computer Sciences & Network and Communication Systems

Professor Dr. Amina A. Saleh Dr. Hibato -Allah El-Bahnasawy


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Revision

2

Revision Includes:

1. Sample space (discrete andcontinuous),

2. Events,

3. Probability function, Axioms ofprobability,

4. Conditional probabilities, andIndependent events.



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1. Sample Space

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Definition:

Random experiment: An experiment is

said to be random if it can beperformed a large number of timesunder the same conditions resulting inone and only one unpredictable

outcome out of several possibleoutcomes.



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Each performance of a randomexperiment is called a trial

The result of a performance (i.e. a trial)is called an outcome.

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Sample Space Continue



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5

The set of all possible outcomes of a random

experiment is called a sample space. Every

element in the sample space is called asample point (i.e. the sample point is a

possible outcome of the experiments).

A sample space is denoted by S.

Reference: Sahoo , Applied probability; pages 11-12




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6

Examples

1. Describe the sample space of rolling a die

2. Describe the sample space of rolling a pairof dice

3. Describe the sample space of observing andcounting traffic accidents on somecrossroads over a period of one week.

4. Describe the sample space of measuring thetemperature on a certain day during thesummer.




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Finite sample space: contains

finite number of sample points(examples 1and 2).

Infinite sample space: containsinfinitely many sample points

(examples 3 and 4).


Finite and Infinite Sample Space



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8

Sample Space ContinueDiscrete and Continuous Sample Space

Discrete sample space: contains

finite or countably infinite samplepoints (examples 1, 2 and 3).

Continuous sample space:

contains uncountable number ofsample points (example 4).



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2. Event

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Quick Definition: A subset of thesample space is called an event.

Events are denoted by capitalletters;A, B, C,

OrA1,A2,A3,



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EventContinue

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Facts about events:

S is a subset of itself, called a

certain event i.e. The empty set is a subset of any

set, called impossible event, i.e.

Any event contains singleton

outcome is called simple eventProfessor Dr. Amina A. Saleh Dr. Hibato -Allah El-Bahnasawy


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EventContinue

Various types of events

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Equally likely events: if one ofthem cannot be expected in

preference to the otherComposite events: the union oftwo or more events.

The compound events: theintersection between two or moreevents

rofessor Dr. Amina A. Saleh Dr. Hibato -Allah El-Bahnasawy


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Various types

of eventsContinue Mutually exclusive (disjoint) events:

two events are called mutuallyinclusive if the occurrence of one ofthem prevents the occurrence ofthe other.

Exhaustive events: the union ofthem is equal to the sample space.

12



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Various types

of eventsContinue Mutually exclusive and Exhaustive

events: nevents are said to beMutually exclusive and Exhaustiveif they are pairwise mutuallyexclusive and their union is equal

to the sample space

13



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Various types

of eventsContinueExamples

1. A pair of dice is rolled. Two possible events

are rolling a number greater than 8 androlling an even number. Are these twoevents mutually exclusive events?

2. A pair of dice is rolled. Two possible eventsare rolling a number less than 5 and rollinga number which is a multiple of 5. Are thesetwo events mutually exclusive?

14



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3. A pair of dice is rolled. Two possible events are rolling a

number which is a multiple of 3 and rolling a number

which is a multiple of 5. Are these two events mutually

exclusive?4. A pair of dice is rolled and the resulting number is odd.

Which of the following events is the complement of this

event?

A. A number greater than 8 is rolled.B. An even number is rolled.

C. A number less than 5 is rolled.

D. AA multiple of 5 is rolled.

Various types

of eventsContinue



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Event Space

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A collection F of subsets of Ssatisfying the

following three rules:

(a) S F ;

(b) if A F then Ac F

; and

(c) ifAj F for j =1, 2,3,.then F

is called an event space or a -field.Professor Dr. Amina A. Saleh Dr. Hibato -Allah El-Bahnasawy


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Event Space

Examples

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Example 1

Let

Then an event space is

F



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Examples

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Example 6Let and the two events of

interest:

Then an event space is

F ==

}Note:



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3.Probability Function

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Definition:

Let Sbe the sample space of a random experiment. A probability

function P : F ( [0, 1] is a set function which assigns real numbers to

the various events of Ssatisfying

(P1): P(A) 0 for all eventAF,

(P2): P(S) = 1,

(P3): IfA1,A2,A3, ...,Ak, ..... are mutually disjoint events of S; then

The above three conditions are called axioms of a probabilitymeasure



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Important Theorems*

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Theorem 1If is a empty set (that is an impossible event,

then .Note: If the probability of an event is zero, thatdoes not mean the event is empty (or impossible),similarly ifAis an event with probability 1, then itdoes not meanAis the sample space S.

* Sahoo pages 13- 17



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Important Theorems Continue

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Theorem 2Let {A1,A2, ...,An} be a finite collection of nmutuallydisjoint events; that is . Then



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Theorem 3IfAis an event of a discrete sample space S, then

the probability ofAis equal to the sum of the

probabilities of its elementary events.Example:

If a fair coin (i.e. each outcome is equally likely) is

tossed twice, what is the probability of getting atleast one head? (Example 1.17 page 15)




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Corollary :If S is a finite sample space with nsampleelements andA is an event in Swith melements,then the probability ofAis given by P(A) = m/n.

Example:A die is loaded in such a way that the probabilityof the face with j dots turning up is proportional to

j for j = 1, 2, ..., 6. What is the probability, in one rollof the die, that an odd number of dots will turnup? (Example 1.18. page 16)




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Some Properties of the

Probability Function*

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IfAbe any event of the sample space S, thenP(Ac) = 1 P(A)

IfA B S, then

P(A) P(B). IfAis any event in S, then

0 P(A) 1. IfAand Bare any two events, then

P(A B) = P(A) + P(B) P(A B).

* Sahoo pages 17-21



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Examples

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Example 1.19. page 20If P(A) = 0.25 and P(B) = 0.8, then show that

0.05 P(A B) 0.25.

Example 1.20. page 21Let A and B be events in a sample space S such thatP(A) = 1/2= P(B) and P(Ac Bc) = 1/3. Find P(A Bc).



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Exercises

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1. If a fair coin is tossed twice, what is theprobability of getting heads on both coin?

2. Discuss and criticize the following

for the probabilities of three mutually exclusiveeventsA, B, C

3. Let A1,A2, andA3be three events in a sample

space Ssuch that:P(A1)=3 P(A2) and P(A2) = 5 P(A3)Find P(A1); P(A2); and P(A3)



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Exercises Continue

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4. Consider the experiment of spinning thespinner shown below twice and note the area

on which the arrow stops. Find the probability

that the outcome of the two spins is:a.) red both times

b.) white first, blue second

c.) white first, red secondd.) not blue both times



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4. Conditional probabilities

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Definition

Let Sbe a sample space associated with a random

experiment. The conditional probability of an

eventA, given that event Bhas occurred, is defined

by

provided P(B) > 0.

This conditional probability functionP

(A

|B

)satisfies all three axioms of a probability function.

Reference: Sahoo page 28



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Conditional probabilitiesContinue

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Example 2.1. Page 28 (Sahoo)A drawer contains 4 black, 6 brown, and 8 olivesocks. Two socks are selected at random from thedrawer. (a) What is the probability that both socksare of the same color (event A)? (b) What is theprobability that both socks are olive (event B) if itis known that they are of the same color?

Remark:N(S) = 153, N(A)= 49, N(B)= 28Then P(A) = 49/153 , P(B) = 28/ 153 = P(A B) since B Ahence, P(B|A)=28/49.rofessor Dr. Amina A. Saleh Dr. Hibato -Allah El-Bahnasawy


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Independent Events

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Definition*:Two eventsAand Bof a sample space Sare calledindependent if and only if

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

Theorem*: Let A,B S. If A and B are independent andP(B) > 0, then

P(A| B) = P(A) .

Example 2.5 page 32:N(S)=10, N(A)= 4, N(B)= 5, N(A B)=2. Are A and B independent?Answer: Yes (Why?)

*Reference: Sahoo page 33



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Theorem 1:If A and B are independent events. Then Acand Bare independent. Similarly A and Bcareindependent. (show that Ac and Bcare

independent)Theorem 2:Two possible mutually exclusive events are alwaysdependent (that is not independent).

Theorem 3:Two possible independent events are not mutuallyexclusive.

Independent Events Continue

Reference: Sahoo page 33 - 35



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Examples:1. Probability that a boy will pass an exam is 3/5

and that for a girl it is 2/5. What is theprobability that at least one of them will passthe exam?Answer: 19/25

2. If four whole numbers taken at random are

multiplied together, find the probability thatthe last digit in the product is 1, 3, 7 or 9 .Answer: 16/625



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3. The probability that a 50 years old man will bealive at 60 is 0.83 and the probability that a 45

years old woman will be alive at 55 is 0.87.

what is the probability that a man who is 50and wife who is 45 will be alive 10 years hence?

Answer: 0.7221




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4. Three groups of children contain respectively 3girls and 1 boy; 2 girls and 2 boys; 1 girl and 3

boys. One child selected at random from each

group, what is the probability that the threeselected consist of 1 girl and 2 boys?

Answer: 13/32




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Independent and dependentevents: Final Words

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Two events are said to be independentif the result of the

second event is not affected by the result of the first event.

If A and B are independent events, the probability of both

events occurring is the product of the probabilities of theindividual events

If the result of one event IS affected by the result of another event, the

events are said to be dependent.

If A and B are dependent events, the probability of both events occurringis the product of the probability of the first event and the probability of the

second event once the first event has occurred.



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Preliminary definitions and theorem:

Definition: Let S be a set and let P= {A1,A2 ,Am}

be a collection of subsets of S. The collection Piscalled a partition of Sif(a)(b)

Bayes Theorem



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Bayes Theorem Continue

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Theorem:

If the events constitute a partition ofthe sample space S and = 1, 2, ...,m,

then for any eventAin S . then



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Bayes Theorem Continue

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Bayes Theorem:

If the events constitute a partition ofthe sample space S and = 1, 2, ...,m,

then for any eventAin S such that

Proof and Examples Sahoo page 37+



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References

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1. Sahoo, P. Probability and MathematicalStatistics; 2008


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Revision Lecture (4) (1)