Probability

2-2
Basics of Probability …
◼Probability is the study of randomness.
◼A Probability is a Numerical Measure
of the Likelihood of Occurrence of an event;
more it is , more is the likelihood of happening of
an event.

2-3
Probability is:
⚫ A quantitative measure of uncertainty
⚫ A measure of the strength of belief in the
occurrence of an uncertain event
⚫ A measure of the degree of chance or
likelihood of occurrence of an uncertain
event
⚫ Measured by a number between 0 and 1 (or
between 0% and 100%)

2-4
What makes to define PROBABILITY?
PROBABILITY
SAMPLE
SPACE
SAMPLE
OUTCOME
RANDOM
EXPERIMENT
EVENT

2-5
RANDOM EXPERIMENT
◼ … is a process that lead to more than one possible outcome and
each outcome is uncertain.
◼ … is a process leading to at least two possible outcomes with
uncertainty as to which will occur.
◼ … is any procedure
1. that can be repeated, theoretically or otherwise, an infinite number of
times;
2. that has a well-defined set of possible outcomes; and
3. the outcome that will prevail in a given experience is uncertain.

2-6
• Process that leads to one of several possible
outcomes , e.g.:
✓ Coin toss
• Heads, Tails
✓ Throw die
• 1, 2, 3, 4, 5, 6
✓ Pick a card
◼ AH, KH, QH, ...
• Each trial of an experiment has a single observed
outcome.
• The precise outcome of a random experiment is
unknown before a trial.
Random Experiment

2-7
SAMPLE OUTCOME
◼Each of the potential eventualities of a
random experiment is defined as sample
outcome.
◼Possible outcomes of a random experiment is
called Sample Outcome.

2-8
SAMPLE SPACE
◼The totality of all eventualities of a random
experiment is called Sample Space.
◼Complete set of all possible outcomes of a
random experiment is called Sample Space.

2-9
EVENT
◼… is any subset of a sample space.
◼… is any designated collection of sample
outcomes, including individual outcomes, the
entire sample space, and the null set.
◼… is a subset of basic outcomes of the sample
space.

2-10
⚫ Sample Space
✓ Set of all possible outcomes (universal set) for a given
experiment
⚫ E.g.: Roll a regular six-sided die
◼ S = {1,2,3,4,5,6}
Events : Definition

2-11
⚫ Sample Space
experiment
◼ S = {1,2,3,4,5,6}
⚫ Event
✓ Collection of outcomes having a common characteristic
⚫ E.g.: Even number
◼ A = {2,4,6}
◼ Event A occurs if an outcome in the set A occurs
Events : Definition

2-12
⚫ Sample Space
experiment
◼ S = {1,2,3,4,5,6}
⚫ Event
✓ Collection of outcomes having a common characteristic
⚫ E.g.: Even number
◼ A = {2,4,6}
◼ Event A occurs if an outcome in the set A occurs
⚫ Probability of an event
✓ Sum of the probabilities of the outcomes of which it consists
⚫ P(A) = P(2) + P(4) + P(6)
Events : Definition

2-13
Events may be …
• Simple, Joint or Compound.
•Discrete or continuous.

2-14
Types of Probability
⚫ Objective or Classical Probability
✓based on equally-likely events
✓based on long-run relative frequency of events
✓not based on personal beliefs
✓ is the same for all observers (objective)
✓examples: toss a coin, throw a die, pick a card

2-15
• For example:
✓ Throw a die
• Six possible outcomes {1,2,3,4,5,6}
• If each is equally-likely, the probability of each is 1/6 = 0.1667 =
16.67%
◼
• Probability of each equally-likely outcome is 1 divided by the number of
possible outcomes
P e
n S
( )
( )
=
1
Equally-likely Probabilities
(Hypothetical or Ideal Experiments)

2-16
• For example:
✓ Throw a die
• Six possible outcomes {1,2,3,4,5,6}
• If each is equally-likely, the probability of each is 1/6 = 0.1667 =
16.67%
◼
• Probability of each equally-likely outcome is 1 divided by the number of
possible outcomes
✓ Event A (even number)
• P(A) = P(2) + P(4) + P(6) = 1/6 + 1/6 + 1/6 = 1/2
• for e in A
P A P e
n A
n S
( ) ( )
( )
( )
=
= = =

3
6
1
2
P e
n S
( )
( )
=
1
Equally-likely Probabilities
(Hypothetical or Ideal Experiments)

2-17
Types of Probability (Continued)
⚫ Subjective Probability
✓based on personal beliefs, experiences, prejudices,
intuition - personal judgment
✓different for all observers (subjective)
✓examples: Elections, new product introduction,
snowfall

2-18
⚫ Set - a collection of elements or objects of
interest
✓Empty set (denoted by )
⚫ a set containing no elements
✓Universal set (denoted by S)
⚫ a set containing all possible elements
✓Complement (Not). The complement of A is
⚫ a set containing all elements of S not in A
( )
A
Basic Definitions

2-19
Complement of a Set
A
A
S
Venn Diagram illustrating the Complement of an event

2-20
✓ Intersection (And)
– a set containing all elements in both A and B
✓ Union (Or)
– a set containing all elements in A or B or both
( )
A B

( )
A B

Basic Definitions (Continued)

2-21
A B

Sets: A Intersecting with B
A
B
S

2-22
Sets: A Union B
A B

A
B
S

2-23
Pick a Card: Sample Space
Event ‘Ace’
Union of
Events ‘Heart’
and ‘Ace’
Event ‘Heart’
The intersection of the
events ‘Heart’ and ‘Ace’
comprises the single point
circled twice: the ace of hearts
P Heart Ace
n Heart Ace
n S
( )
( )
( )


=
=
=
16
52
4
13
P Heart
n Heart
n S
( )
( )
( )
= = =
13
52
1
4
P Ace
n Ace
n S
( )
( )
( )
= = =
4
52
1
13
P Heart Ace
n Heart Ace
n S
( )
( )
( )


= =
1
52
Hearts Diamonds Clubs Spades
A A A A
K K K K
Q Q Q Q
J J J J
10 10 10 10
9 9 9 9
8 8 8 8
7 7 7 7
6 6 6 6
5 5 5 5
4 4 4 4
3 3 3 3
2 2 2 2

2-24
• Mutually exclusive or disjoint sets
–sets having no elements in common, having no
intersection, whose intersection is the empty set
• Partition
–a collection of mutually exclusive sets which
together include all possible elements, whose
union is the universal set
Basic Definitions (Continued)

2-25
Mutually Exclusive or Disjoint Sets
A
B
S
Sets have nothing in common

2-26
Sets: Partition
A1
A2
A3
A4
A5
S

2-27
Basic Rules for Probability

2-28
⚫ Range of Values for P(A):
⚫ Complements - Probability of not A
⚫ Intersection - Probability of both A and B
✓ Mutually exclusive events (A and C) :
1
)
(
0 
 A
P
P A P A
( ) ( )
= −
1
P A B n A B
n S
( ) ( )
( )
 = 
P A C
( )
 = 0

2-29
• Union - Probability of A or B or both (rule of unions)
✓Mutually exclusive events: If A and B are mutually exclusive, then
P A B n A B
n S
P A P B P A B
( ) ( )
( )
( ) ( ) ( )
 =  = + − 
)
(
)
(
)
(
0
)
( B
P
A
P
B
A
P
so
B
A
P +
=

=

(Continued)

2-30
Sets: P(A Union B)
)
( B
A
P 
A
B
S

2-31
Example 1
1. ShopperTrak is a hidden electric eye designed to count
the number of shoppers entering a store. When two
shoppers enter a store together, one walking in front of the
other, the following probabilities apply: There is a 0.98
probability that the first shopper will be detected, a 0.94
probability that the second shopper will be detected, and a
0.93 probability that both of them will be detected by the
device. What is the probability that the device will detect at
least one of two shoppers entering together?

2-32
Example 2
2. Following are age and gender data for 20 midlevel managers
at a service company: 34 F, 49 M, 27 M, 63 F, 33 F, 29 F, 45 M,
46 M, 30 F, 39 M, 42 M, 30 F, 48 M, 35 F, 32 F, 37 F, 48 F, 50 M,
48 F, 61 F. A manager must be chosen at random to serve on a
companywide committee that deals with personnel problems.
What is the probability that the chosen manager will be either a
woman or over 50 years old or both?

2-33
• Conditional Probability - Probability of A given B
✓Independent events:
0
)
(
,
)
(
)
(
)
( 

= B
P
where
B
P
B
A
P
B
A
P
P AB P A
P B A P B
( ) ( )
( ) ( )
=
=
Conditional Probability

2-34
Rules of conditional probability:
If events A and D are statistically independent:
so
so
P AB P A B
P B
( ) ( )
( )
=  P A B P AB P B
P B A P A
( ) ( ) ( )
( ) ( )
 =
=
P AD P A
P D A P D
( ) ( )
( ) ( )
=
=
P A D P A P D
( ) ( ) ( )
 =
Conditional Probability (continued)

2-35
AT& T IBM Total
Telecommunication 40 10 50
Computers 20 30 50
Total 60 40 100
Counts
AT& T IBM Total
Telecommunication .40 .10 .50
Computers .20 .30 .50
Total .60 .40 1.00
Probabilities
Contingency Table - Example

2-36
AT& T IBM Total
Telecommunication 40 10 50
Computers 20 30 50
Total 60 40 100
Counts
AT& T IBM Total
Telecommunication .40 .10 .50
Computers .20 .30 .50
Total .60 .40 1.00
Probabilities
Probability that a project
is undertaken by IBM
given it is a
telecommunications
project:
Contingency Table - Example

2-37
example
3. A financial analyst believes that if interest rates
decrease in a given period, then the probability
that the stock market will go up is 0.80. The
analyst further believes that interest rates have a
0.40 chance of decreasing during the period in
question. Given the above information, what is
the probability that the market will go
up and interest rates will go down during the
period in question?

2-38
example
4. An investment analyst collects data on stocks and notes whether or not
dividends were paid and whether or not the stocks increased in price over a given period.
Data are presented in the following table.
Price No Price
Increase Increase Total
Dividends paid 34 78 112
No dividends paid 85 49 134
Total 119 127 246
a. If a stock is selected at random out of the analyst’s list of 246 stocks, what is
the probability that it increased in price?

2-39
example
Price No Price
Total 119 127 246
b. If a stock is selected at random, what is the probability that it paid dividends?

2-40
example
Price No Price
Total 119 127 246
c. If a stock is randomly selected, what is the probability that it both increased
in price and paid dividends?

2-41
example
Price No Price
Total 119 127 246
d. What is the probability that a randomly selected stock neither paid dividends nor
increased in price?

2-42
example
Price No Price
Total 119 127 246
increased in price?
e. Given that a stock increased in price, what is the probability that it also paid
dividends?

2-43
example
Price No Price
Total 119 127 246
increased in price?
dividends?
f. If a stock is known not to have paid dividends, what is the probability that it
increased in price?

2-44
example
Price No Price
Total 119 127 246
increased in price?
dividends?
f. If a stock is known not to have paid dividends, what is the probability that it
increased in price?
g. What is the probability that a randomly selected stock was worth holding
during the period in question; that is, what is the probability that it increased
in price or paid dividends or did both?

2-45
Example
The probability that a consumer will be exposed to an
advertisement for a certain product by seeing a
commercial on television is 0.04. The probability that the
consumer will be exposed to the product by seeing an
advertisement on a billboard is 0.06. The two events,
being exposed to the commercial and being exposed to
the billboard ad, are assumed to be independent. (a)
What is the probability that the consumer will be exposed
to both advertisements? (b) What is the probability that
he or she will be exposed to at least one of the ads?

2-46
0976
.
0
0024
.
0
06
.
0
04
.
0
)
(
)
(
)
(
)
(
)
0024
.
0
06
.
0
*
04
.
0
)
(
)
(
)
(
)
=
−
+
=
−
+
=
=
=
=
B
T
P
B
P
T
P
B
T
P
b
B
P
T
P
B
T
P
a



Events Television (T) and Billboard (B) are
assumed to be independent.
Independence of Events –
Example

2-47
The probability of the union of several independent events
is 1 minus the product of probabilities of their complements:
P A A A An P A P A P A P An
( ) ( ) ( ) ( ) ( )
1 2 3
1
1 2 3
    = −
 
The probability of the intersection of several independent events
is the product of their separate individual probabilities:
P A A A An P A P A P A P An
( ) ( ) ( ) ( ) ( )
1 2 3 1 2 3
    =
 
Product Rules for Independent Events

2-48
Example
5. A package of documents needs to be sent to a given
destination, and delivery within one day is important. To
maximize the chances of on-time delivery, three copies of
the documents are sent via three different delivery
services. Service A is known to have a 90% on-time
delivery record, service B has an 88% on-time delivery
record, and service C has a 91% on-time delivery record.
What is the probability that at least one copy of the
documents will arrive at its destination on time?

2-49
P A P A B P A B
( ) ( ) ( )
=  + 
In terms of conditional probabilities:
More generally (where Bi make up a partition):
P A P A B P A B
P AB P B P AB P B
( ) ( ) ( )
( ) ( ) ( ) ( )
=  + 
= +
P A P A B
i
P AB
i
P B
i
( ) ( )
( ) ( )
= 

= 
The Law of Total Probability and
Bayes’ Theorem
The law of total probability:

2-50
Example
An analyst believes the stock market has a 0.75
probability of going up in the next year if the economy
should do well, and a 0.30 probability of going up if the
economy should not do well during the year. The
analyst further believes there is a 0.80 probability that
the economy will do well in the coming year. What is
the probability that the stock market will go up next year
(using the analyst’s assessments)?

2-51
Example
7. A drug manufacturer believes there is a 0.95 chance that the Food
and Drug Administration (FDA) will approve a new drug the company
plans to distribute if the results of current testing show that the drug
causes no side effects. The manufacturer further believes there is a
0.50 probability that the FDA will approve the drug if the test shows
that the drug does cause side effects. A physician working for the
drug manufacturer believes there is a 0.20 probability that tests will
show that the drug causes side effects. What is the probability that
the drug will be approved by the FDA?

2-52
Bayes’ Theorem…
◼Bayes' Theorem relates the conditional and
marginal probabilities of two random events.
It is often used to compute posterior
probabilities given observations.
It provides a mechanism to REVISE our priori
probabilities in the light of NEW INFORMATION!!!

2-53
• Bayes’ theorem enables you, knowing just a little
more than the probability of A given B, to find the
probability of B given A.
• Based on the definition of conditional probability
and the law of total probability.
P B A
P A B
P A
P A B
P A B P A B
P AB P B
P AB P B P AB P B
( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
=
=
+
=
+


 
Applying the law of total
probability to the denominator
Applying the definition of
conditional probability throughout
Bayes’ Theorem

2-54
Bayes’ Theorem
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A
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A
P
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B
A
P
B
P
B
A
P
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A
P
=

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A
P
B
P
B
A
P
A
B
P
A
P
A
B
P
B
P
B
A
P
+
=
=
=
So:
The above formula is referred to as Bayes’ theorem. It is
extremely useful in decision analysis when using
information.

2-55
Bayes’ Theorem - General

=

= n
i
i
i
i
i
i
i
E
A
P
E
P
E
A
P
E
P
A
P
A
E
P
A
E
P
1
)
|
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&
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(

2-56
Seeing it in Tabular form…
EVENTS
PRIORI
PROBABILITIES
CONDITIONAL
PROBABILITIES
JOINT
PROBABILITIES
POSTERIOR
PROBABILITIES

2-57
Example
Consider a test for an illness. The test has a known reliability:
1. When administered to an ill person, the test will indicate so
with probability 0.92.
2. When administered to a person who is not ill, the test will
erroneously give a positive result with probability 0.04.
Suppose the illness is rare and is known to affect only 0.1% of
the entire population. If a person is randomly selected from the
entire population and is given the test and the result is positive,
what is the posterior probability (posterior to the test result) that
the person is ill?

2-58
• Given a partition of events B1,B2 ,...,Bn:
P B A
P A B
P A
P A B
P A B
P A B P B
P A B P B
i
i i
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
1
1
1
1 1
=

=



=

Applying the law of total
probability to the denominator
Applying the definition of
conditional probability throughout
Bayes’ Theorem Extended

2-59
⚫ An economist believes that during periods of high economic growth, the U.S.
dollar appreciates with probability 0.70; in periods of moderate economic
growth, the dollar appreciates with probability 0.40; and during periods of
low economic growth, the dollar appreciates with probability 0.20.
⚫ During any period of time, the probability of high economic growth is 0.30,
the probability of moderate economic growth is 0.50, and the probability of
low economic growth is 0.20.
⚫ Suppose the dollar has been appreciating during the present period. What is
the probability we are experiencing a period of high economic growth?
Bayes’ Theorem Extended -
Example

2-60
The Joint Probability Table
⚫ A joint probability table is similar to a contingency
table , except that it has probabilities in place of
frequencies.
⚫ The joint probability for Example is shown on the
next slide.
⚫ The row totals and column totals are called
marginal probabilities.

2-61
The Joint Probability Table:
Example
⚫ The joint probability table for Example is
summarized below.
High Medium Low Total
$
Appreciates
$Depreciates
Total 0.30 0.5 0.20 1.00
Marginal probabilities are the row totals and the column totals.

2-62
The Joint Probability Table:
Example
⚫ The joint probability table for Example is
summarized below.
High Medium Low Total
$
Appreciates
0.21 0.2 0.04 0.45
$Depreciates
0.09 0.3 0.16 0.55
Total 0.30 0.5 0.20 1.00
Marginal probabilities are the row totals and the column totals.

2-63
Example
Q8: When the economic situation is “high,” a certain economic indicator
rises with probability 0.6. When the economic situation is “medium,” the
economic indicator rises with probability 0.3. When the economic
situation is “low,” the indicator rises with probability 0.1. The economy is
high 15% of the time, it is medium 70% of the time, and it is low 15% of
the time. Given that the indicator has just gone up, what is the
probability that the economic situation is high?

2-64
Example
Q9. Saflok is an electronic door lock system made in Troy, Michigan,
and used in modern hotels and other establishments. To open a door,
you must insert the electronic card into the lock slip. Then a green light
indicates that you can turn the handle and enter; a yellow light indicates
that the door is locked from inside, and you cannot enter. Suppose that
90% of the time when the card is inserted, the door should open
because it is not locked from inside. When the door should open, a
green light will appear with probability 0.98. When the door should not
open, a green light may still appear (an electronic error) 5% of the time.
Suppose that you just inserted the card and the light is green. What is
the probability that the door will actually open?

2-65
Example 10
Q10. An aircraft emergency locator transmitter (ELT) is a device
designed to transmit a signal in the case of a crash. The
Altigauge Manufacturing Company makes 80% of the ELTs, the
Bryant Company makes 15% of them, and the Chartair
Company makes the other 5%. The ELTs made by Altigauge
have a 4% rate of defects, the Bryant ELTs have a 6% rate of
defects, and the Chartair ELTs have a 9% rate of defects (which
helps to explain why Chartair has the lowest market share).
a. If an ELT is randomly selected from the general population of
all ELTs, find the probability that it was made by the Altigauge
Manufacturing Company.
b. If a randomly selected ELT is then tested and is found to be
defective, find the probability that it was made by the Altigauge
Manufacturing Company.

Probability

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