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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?
2. What is the probability of A B?
3. What is the probability of A B?
4. 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?