Assigning Probabilities and Probability Relationships Chapter 4 BA 201 Assigning Probabilities Assigning Probabilities Basic Requirements for Assigning Probabilities 1 The probability assigned to each experimental ID: 368807
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Slide1
Introduction to ProbabilityAssigning Probabilities and Probability Relationships
Chapter 4
BA 201Slide2
Assigning ProbabilitiesSlide3
Assigning Probabilities
Basic Requirements for Assigning Probabilities
1. The probability assigned to each experimental
outcome must be between 0 and 1, inclusively.
0
<
P(Ei) < 1 for all i
where:
E
i
is the
i
th
experimental
outcome and
P
(
E
i
) is its probabilitySlide4
Assigning Probabilities
Basic Requirements for Assigning Probabilities
2. The sum of the probabilities for all experimental
outcomes must equal 1.
P
(
E
1) + P(E2) + . . . + P(
E
n
) = 1
where:
n
is the number of experimental outcomesSlide5
Assigning Probabilities
Classical Method
Relative Frequency Method
Subjective Method
Assigning probabilities based on the assumption
of
equally likely outcomes
Assigning probabilities based on experimentation or historical data
Assigning probabilities based on
judgmentSlide6
Classical Method
If
an experiment has
n
possible outcomes, the classical method would assign a probability of 1/n
to each outcome.
Experiment: Rolling a dieSample Space: S = {1, 2, 3, 4, 5, 6}Probabilities:
Each
sample point has a
1/6
chance of
occurring
1/6+1/6+1/6+1/6+1/6+1/6 = 1
Example: Rolling a DieSlide7
Relative Frequency Method
Number of
Polishers Rented
Number
of Days
0
1
234 4 618
10
2
Lucas
Tool Rental would like to assign probabilities
to the number of car polishers it rents each day.
Office records show the following frequencies of daily
rentals for the last 40 days.
Lucas
Tool RentalSlide8
Each probability assignment is given by dividing
the frequency (number of days) by the total frequency
(total number of days).
Relative Frequency Method
4/40
Probability
Number of
Polishers Rented
Number
of Days
0
1
2
3
4
4
6
18
10
2
40
0.10
0.15
0.45
0.25
0.05
1.00
Example: Lucas Tool RentalSlide9
Subjective Method
When economic conditions and a company’s
circumstances change rapidly it might be
inappropriate to assign probabilities based solely on
historical data.
We can use any data available as well as our
experience and intuition, but ultimately a probability
value should express our degree of belief that the experimental outcome will occur.
The best probability estimates often are obtained by
combining the estimates from the classical or relative
frequency approach with the subjective estimate.Slide10
Subjective Method
An analyst made the following probability estimates.
Exper
. Outcome
Net Gain
or
Loss
Probability(10, 8)
(10,
-
2)
(5, 8)
(5,
-
2)
(0, 8)
(0,
-
2)
(
-
20, 8)
(
-
20,
-
2)
$18,000 Gain
$8,000 Gain
$13,000 Gain
$3,000 Gain
$8,000 Gain
$2,000 Loss
$12,000 Loss
$22,000 Loss
0.20
0.08
0.16
0.26
0.10
0.12
0.02
0.06
Example: Bradley InvestmentsSlide11
An
event
is a collection of sample points.
The
probability of any event
is equal to the sum of the probabilities of the sample points in the event.
If we can identify all the sample points of an experiment and assign a probability to each, we can compute the probability of an event.Events and Their ProbabilitiesSlide12
Events and Their Probabilities
Event
M
= Markley Oil Profitable
M
= {(10, 8), (10,
-
2), (5, 8), (5, -2)}P(M) = P(10, 8) + P
(10,
-
2) +
P
(5, 8) +
P
(5,
-
2)
=
0.20
+
0.08
+
0.16
+
0.26
=
0
.
7
0
Bradley
InvestmentsSlide13
Events and Their Probabilities
Event
C
= Collins Mining Profitable
C
= {(10, 8), (5, 8), (0, 8), (
-
20, 8)}P(C) = P
(10, 8) +
P
(5, 8) +
P
(0, 8) +
P
(
-
20, 8)
=
0.20
+
0.16
+
0.10
+
0.02
=
0.48
Example: Bradley InvestmentsSlide14
Probability RelationshipsSlide15
Some Basic Relationships of Probability
There
are some
basic probability relationships
that
can be used to compute the probability of an eventwithout knowledge of all the sample point probabilities.
Complement of an Event
Intersection of Two EventsMutually Exclusive EventsUnion of Two EventsSlide16
The complement of
A
is denoted by
A
c
.
The
complement of event A is defined to be the event consisting of all sample points that are not in A.
Complement of an Event
Event
A
A
c
Sample
Space
SSlide17
The union of events
A
and
B
is denoted by
A
B The union of events A and B is the event containing
all sample points that are in
A
or
B
or both.
Union of Two Events
Sample
Space
S
Event
A
Event
BSlide18
Union of Two Events
Event
M
= Markley Oil Profitable
Event
C
= Collins Mining Profitable
M C = Markley Oil Profitable
or
Collins Mining Profitable (or both)
M
C
= {(10, 8), (10,
-
2), (5, 8), (5,
-
2), (0, 8), (
-
20, 8)}
P
(
M
C)
=
P
(10, 8) +
P
(10,
-
2) +
P
(5, 8) +
P
(5,
-
2)
+
P
(0, 8) +
P
(
-
20, 8)
=
0.20
+
0.08
+
0.16
+
0.26
+
0.10
+ 0.02= 0.82
Example: Bradley InvestmentsSlide19
The intersection of events
A
and
B
is denoted by
A
The intersection of events A and B is the set of all
sample points that are in both
A
and
B
.
Sample
Space
S
Event
A
Event
B
Intersection of Two Events
Intersection of
A
and
BSlide20
Intersection of Two Events
Event
M
= Markley Oil Profitable
Event
C
= Collins Mining Profitable
M C = Markley Oil Profitable
and
Collins Mining Profitable
M
C
= {(10, 8), (5, 8)}
P
(
M
C)
=
P
(10, 8) +
P
(5, 8)
=
0.20
+
0.16
= 0.36
Bradley
InvestmentsSlide21
Mutually Exclusive Events
Two events are said to be
mutually exclusive
if the
events have no sample points in common.
Two events are mutually exclusive if, when one event
occurs, the other cannot occur.
Sample
Space
S
Event
A
Event
BSlide22
Mutually Exclusive Events
If events
A
and
B
are mutually exclusive,
P(A
B = 0. The addition law for mutually exclusive events is:
P
(
A
B
) =
P
(
A
) +
P
(
B
)Slide23
Practice Probability RelationshipsSlide24
Practice
Based on the Venn diagram, describe the following:
A
B
CSlide25
Practice
Out of forty students, 14 are taking English and 29 are taking Chemistry. Five students are in both classes.
Draw a Venn Diagram depicting this relationship.Slide26
Practice
Students are either undergraduate students or graduate students.
Draw a Venn Diagram depicting this relationship.Slide27
Scenario
Outcome
Probability
O
1
0.10
O2
0.30O30.05O40.15O
5
0.20
O
6
0.05
O
7
0.10
O
8
0.05
Event
Outcomes
E
1
O
1,
O
3
E
2
O
1,
O
4,
O
5,
O
6
E
3
O
2
E
4
O
7,
O
8
E
5
O
4,
O
5,
O7E6O1, O
7
A statistical experiment has the following outcomes, along with their probabilities and the following events, with the corresponding outcomes.
Outcomes
EventsSlide28
Probabilities of Events
What is the probability of E
1
[P(E
1
)]?Slide29
Probabilities of Events
What is the probability of E
2
[P(E
2
)]?Slide30
Probabilities of Events
What is the probability of E
3
[P(E
3
)]?Slide31
Probabilities of Events
What outcomes are in the complement of E
2
[E
2
c
]?Slide32
Probabilities of Events
What outcomes are in the union of E
3
and E
4
? (E
3
⋃ E4)? What is the probability of E3 ⋃ E4 [P(E
3
⋃
E
4
)]?Slide33
Probabilities of Events
What outcomes are in the intersection of E
2
and E
5
(E
2
⋂ E5)? What is the probability of E2 ⋂ E5 [P(E
2
⋂
E
5
)]?Slide34
Probabilities of Events
Based on the available information, are E
2
and E
4
mutually exclusive? What is the probability of E
2
⋂ E4 [P(E2 ⋂ E4)]? What is the probability of E
2
⋃
E
4
[P(E
2
⋃
E
4
)]?Slide35