2Probability_Set Theory Notes

8/9/2019 2Probability_Set Theory Notes

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PROBABILITY AND SET THEORY


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PROBABILITY

The term probability refers to the study of

randomness and uncertainty. In situations in

which a number of outcomes may occur, the

theory of probability provides methods for

quantifying the chance or likelihood

associated with various outcomes


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Experiment

The process of making an observation or takingmeasurement

Ex: Tosing a die and observing the number on the up

face of the die.

Tossing a coin once, twice, or four times.

Observing the model of vehicle you see on your nextglance towards the parking lot.

How long it will take you to eat your next lunch.


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Event

An outcome of an experiment. This outcome

may be:

Simple Eventex: Heads in a coin toss)

Complex Event

ex: Heads comes out at least once in 3consecutive tosses


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Sample Space

The sample space of an experiment is the

collection of all its simple events.

Experiments and their Sample Spaces1. Tossing of a Coin

2. Tossing Three Coins

3. Throwing Two Dice


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PROBABILITY AXIOMS

AXIOM 1: 0 P(A) 1

AXIOM 2: P(S) = 1

AXIOM 3: = 1 AXIOM 4: P(O) = 0

)(!SiiEP


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SOME RELATIONS FROM SET THEORY

An event can be thought of as a set. As a set, we

may use relationships and results from

elementary set theory to study events. The

following operations will be used to construct

new events from given sets.

Union (denoted by AB and read A or B)

Intersection (denoted by AB and read Aand B)

Complement (read A-prime or not A)


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SOME RELATIONS FROM SET THEORY

The following operations will be used to construct new

events from given sets.

Definitions:

Union: The union of two events A and B (denoted by

AB and read A or B) is the event consisting of all

outcomes that are either in A or in B or in both events.

Intersection: The intersection of two events A and B

(denoted by AB and read A and B) is the event

consisting of all outcomes that are in both A and B.

Complement: The complement of an event A, denoted

by A (read A-prime or not A), is the set of all

outcomes in the universal set S that are not contained in

A.


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Venn Diagram

Sets can be represented as a Venn Diagram: a

rectangle that includes circles depicting the

subsets, named after the English logician John

Venn (1834-1923).


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Example #1S = {A, B, C, D, E, F, G, H, I, J}

A = {A, C, E, G, I}

B = {B, D, F, H, J}

C = {A, B, E, F}

D = {B, H, I, J}

Find:1. A D2. A B3. A C4. D5. (C D)A6. (A D)7. A C D


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PROBABILITY LAWS RELATED TO SET THEORY

1. P(AB)= P(A) + P(B) P(AB)2. P(AB)= P(A) + P(B) if A and B are mutually

exclusive: P(AB) = 0 (no common

occurrence)3. P(A) = 1 - P(A) or P(A) = 1 - P(A)

4. P(AB) = P(A) x P(B) if A and B are

independent to each other. Two events Aand B are independent if the probability of

one event is unaffected by the occurrence of

the other.


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Contingency Table

Events A A Probability

B P(AB) P(A

B) P(B)

B P(AB) P(AB) P(B)

Probability P(A) P(A) 1.00


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Example #2

Events A and B have the followingprobability structure:

P(AB) = 0.4 P(AB) = 0.2 P(AB) = 0.3

1. What is the probability of A B?




5. Are A and B independent events?


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Example #3

The probability that computer A will break down on aparticular day is P(A) = 1/50; similarly, for computer B,

P(B) = 1/100. Assuming independence, on a particular

day,

1. What is the probability that both will break down?2. What is the probability that at most one will break

down?

3. What is the probability that neither will break down?

4. What is the probability that one or the other will

break down?

5. What is the probability that exactly one will break

down?


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Assignment

The probability that a life insurance salesmanfollowing up a magazine lead will make a sale is

0.60. A salesman has two leads on a certain day.

Assuming independence, what is the probability

that

1. He will sell both?

2. He will sell exactly one policy?

3. He will sell at least one policy?

4. He will sell at least one policy if he has three

leads?


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Conditional Probabilities

For experiments with two or more events of interest, attention isoften directed not only at the probabilities of individual events

but also at the probability of an event occurring conditional on

the knowledge that another event has occurred. It measures the

probability that event B occurs when it is known that that event

A occurs.

P(B/A) =

Without Replacement: P(AB) = P(A) * P(B/A) or

P(AB) = P(B) * P(A/B)

With Replacement: P(AB) = P(A) x P(B)

)(

)P(A

AP

B


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Example #4

A box contains 4 red marbles and 3 yellowmarbles, draw 2 marbles from the box with

replacement. What is the probability that

1. Both marbles drawn are red?

2. A yellow marble is drawn given that

the 1st marble drawn is red?


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Example #5

A box contains 4 red marbles and 3 yellowmarbles, draw 2 marbles from the box

without replacement. What is the

probability that1. Both marbles drawn are red?

2. A yellow marble is drawn given that

the 1st marble drawn is red?


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Example #6

A candidate runs for two political offices, A and B. Heassigns 0.30 as the probability of being elected to both,

0.60 as the probability of being elected to A if he is

elected to B, and 0.8 as the probability of being elected

to B if he is elected to A.1. What is the probability of being elected to A?

2. What is the probability of being elected to B?

3. What is the probability of being elected to neither office?

4. What is the probability of being elected to at least one ofthe offices?

5. What is the probability of being elected to exactly one

office?


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AssignmentThe Guessight Employment Agency administers a Verbal

Comprehension test and a Verbal Reasoning test to each of itsapplicants. On the Verbal Comprehension test, a score above 14

is considered passing, and on the Verbal Reasoning test, a score

of19 is considered passing. From the agencys records, it has

been determined that 10% of the applicants fail the Verbal

Comprehension test, 12% fail the Verbal Reasoning test and 20%fail at least one of the tests.

1. What is the probability that a randomly selected applicant passes both

tests?

2. What is the probability that an applicant will fail in only one of the test?

3. If an applicant randomly selected passed the Verbal Reasoning test, what

is the probability that he also passed the Verbal Comprehension test?

4. Three applicants selected at random failed the Verbal Comprehension

test. What is the probability that exactly one of the 3 applicants passed

the Verbal Reasoning test?


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Assignment

A firm uses two components A and B for its

electronic subsystem. The probability that A

functions is 90%, while the probability that B

functions is 95%. Assume that A and B are not

independent and that the probability that both willfunction is 88%. What is the probability that:

1. At least one component will function?

2. Component A functions if B does not

function?

3. Component B functions if it is known that

exactly one component is functioning?


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Example #7

Suppose that colored balls are distributed in 3

indistinguishable boxes as follows:

A box is selected at random from which a ball isselected at random. The ball selected is green.

What is the probability that Box A is selected?

Box A Box B Box CYellow 3 2 1Green 5 4 3

2 3 2


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Example #8

The Wash White Company has 3 machines A, B and C,which produce the spare part. Machine A produces

60% of the total volume and produces 80% acceptable

parts and 20% rejects. Machine B and C each produce

20% of the total volume. Machine B produces 60%acceptable parts and 40% rejects. Machine C produces

50% acceptable parts and 50% rejects. Three elements

were sampled from a production lot and all were found

to be acceptable. What is the probability that thesamples were produced by Machine A?


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Assignment

The probability that a person has a disease X is

P(X) = 0.001. The probability that medical

examination will indicate the disease if a person

has it is P(I/X) = 0.8, and the probability that

examination will indicate the disease if a person

does not have is P(I/X) = 0.02. What is the

probability that a person has the disease ifmedical examination so indicates?


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Assignment

As a result of past hiring procedures, a company finds that60% of its employees are good workers and 40% are poor

workers. The personnel manager believes that the

proportion of good workers can be increased by designing a

test to be administered to job applicants, and hiring those

who pass. A consulting firm supplies the test and offers to

administer it for a fee to applicants. Because of the cost, it is

decided to determine first how well the test discriminates

between good and poor workers before trying it on current

employees. It is found that 80% of the good workers and

40% of the poor workers pass the test? Should the

personnel manager adopt this new system of hiring? Why?

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2Probability_Set Theory Notes

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