Transcript Ch 5

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Chapter
Five
McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
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Chapter Five
A Survey of Probability Concepts
GOALS
When you have completed this chapter, you will be able to:
ONE
Define probability.
TWO
Describe the classical, empirical, and subjective approaches
to probability.
THREE
Understand the terms: experiment, event, outcome,
permutations, and combinations.
Goals
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Chapter Five
continued
A Survey of Probability Concepts
GOALS
When you have completed this chapter, you will be able to:
FOUR
Define the terms: conditional probability and joint
probability.
FIVE
Calculate probabilities applying the rules of addition and the
rules of multiplication.
SIX
Use a tree diagram to organize and compute probabilities.
Goals
Chapter Five
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continued
A Survey of Probability Concepts
SEVEN
Calculate a probability using Bayes’ theorem.
Goals
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Movie
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There are three definitions of probability: classical,
empirical, and subjective.
The
Classical
definition
applies when
there are n
equally likely
outcomes.
The Empirical
definition applies
when the number
of times the event
happens is
divided by the
number of
observations.
Subjective
probability is
based on
whatever
information is
available.
Definitions continued
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Movie
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An Outcome is
the particular
result of an
experiment.
An Event is
the collection
of one or more
outcomes of an
experiment.
Experiment: A fair die is cast.
Possible outcomes: The
numbers 1, 2, 3, 4, 5, 6
One possible event: The
occurrence of an even
number. That is, we collect
the outcomes 2, 4, and 6.
Definitions continued
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Events are Mutually
Exclusive if the
occurrence of any one
event means that none
of the others can occur
at the same time.
Mutually exclusive:
Rolling a 2 precludes
rolling a 1, 3, 4, 5, 6
on the same roll.
Events are Independent
if the occurrence of one event
does not affect the occurrence
of another.
Independence: Rolling a 2
on the first throw does not
influence the probability of
a 3 on the next throw. It is
still a one in 6 chance.
Mutually Exclusive Events
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Events are Collectively Exhaustive
if at least one of the events must occur
when an experiment is conducted.
Collectively Exhaustive Events
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Throughout her
teaching career
Professor Jones has
awarded 186 A’s out
of 1,200 students.
What is the
probability that a
student in her
section this
semester will
receive an A?
This is an example of the
empirical definition of
probability.
To
find the probability a
selected student earned an A:
186
P( A) 
 0.155
1200
Example 2
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Examples of subjective probability are:
estimating the probability the
Washington Redskins will win
t h e S u p e r B o w l t h i s y e a r.
es t i m at i n g t h e p r o b a b i l i t y
mortgage rates for home loans
will top 8 percent.
Subjective Probability
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If two events
A and B are mutually
exclusive, the
Special Rule of
Addition states that the
Probability of A or B
occurring equals the sum of
their respective
probabilities.
P(A or B) = P(A) + P(B)
Basic Rules of Probability
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New England Commuter Airways recently supplied
the following information on their commuter flights
from Boston to New York:
Arrival
Frequency
Early
100
On Time
800
Late
75
Canceled
25
Total
1000
Example 3
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If A is the event that a
flight arrives early,
then P(A) = 100/1000
= .10.
If B is the event that a
flight arrives late, then
P(B) = 75/1000 = .075.
The probability that a flight is either early or late
is:
P(A or B) = P(A) + P(B) = .10 + .075 =.175.
Example 3 continued
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The Complement Rule is used to determine the
probability of an event occurring by subtracting the
probability of the event not occurring from 1.
If P(A) is the probability of event A and P(~A) is
the complement of A,
P(A) + P(~A) = 1 or P(A) = 1 - P(~A).
The Complement Rule
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A Venn
Diagram illustrating the complement rule
would appear as:
A
~A
The Complement Rule continued
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Recall example 3. Use the complement
rule to find the probability of an early
(A) or a late (B) flight
If C is the event that a
flight arrives on time, then
P(C) = 800/1000 = .8.
If D is the event that a
flight is canceled, then
P(D) = 25/1000 = .025.
Example 4
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P(A or B)
C
.8
= 1 - P(C or D)
= 1 - [.8 +.025]
=.175
D
.025
~(C or D) = (A or B)
.175
Example 4 continued
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If A and B are two
events that are not
mutually exclusive,
then P(A or B) is
given by the
following formula:
P(A or B) = P(A) + P(B) - P(A and B)
The General Rule of Addition
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The Venn Diagram illustrates this rule:
B
A and B
A
The General Rule of
Addition
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In a sample of 500 students, 320 said they had a
stereo, 175 said they had a TV, and 100 said they had
both. 5 said they had neither.
Stereo
320
Both
100
TV
175
EXAMPLE 5
If a student is selected at
random, what is the
probability that the student
has only a stereo or TV?
What is the probability
that the student has both a
stereo and TV?
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P(S or TV) = P(S) + P(TV) - P(S and TV)
= 320/500 + 175/500 – 100/500
= .79.
P(S and TV) = 100/500
= .20
Example 5 continued
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A Joint Probability measures the likelihood
that two or more events will happen concurrently.
An example would
be the event that a
student has both a
stereo and TV in his
or her dorm room.
Joint Probability
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The Special Rule of Multiplication
requires that two events A and B are
independent.
Two
events A and B are independent if the
occurrence of one has no effect on the probability of
the occurrence of the other.
This
rule is written:
P(A and B) = P(A)P(B)
Special Rule of Multiplication
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Stock price $
Chris owns two stocks,
IBM
5-year stock prices
GE
IBM and General
45
Electric (GE). The
40
35
probability that IBM
30
25
stock will increase in
20
15
value next year is .5 and
10
5
the probability that GE
0
1
2
3
4
5
stock will increase in
Year
value next year is .7.
Assume the two stocks
are independent. What
is the probability that
P(IBM and GE) = (.5)(.7) = .35
both stocks will increase
in value next year?
Example 6
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What
is the
probability
that at least
one of these stocks
increases in value in
the next year?
This means that
either one can
increase or
both.
P(at least one)
= P(IBM but not GE)
+ P(GE but not IBM)
+ P(IBM and GE)
(.5)(1-.7)
+ (.7)(1-.5)
+ (.7)(.5)
= .85
Example 6 continued
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A Conditional Probability is the
probability of a particular event occurring,
given that another event has occurred.
The probability of
event A occurring
given that the event
B has occurred is
written P(A|B).
Conditional Probability
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The General
Rule of
Multiplication is
used to find the joint
probability that two
events will occur.
It states that for two
events A and B, the
joint probability that
both events will happen
is found by multiplying
the probability that
event A will happen by
the conditional
probability of B given
that A has occurred.
General Multiplication Rule
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The joint probability,
P(A and B), is given by the
following formula:
P(A and B) = P(A)P(B/A)
or
P(A and B) = P(B)P(A/B)
General Multiplication Rule
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The Dean of the School of Business at Owens University
collected the following information about undergraduate
students in her college:
Major
Accounting
Male
170
Female
110
Total
280
Finance
120
100
220
Marketing
160
70
230
Management
150
120
270
Total
600
400
1000
Example 7
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If a student is selected at random, what is the
probability that the student is a female (F)
accounting major (A)?
P(A and F) = 110/1000.
Given that the student is a
female, what is the
probability that she is an
accounting major?
P(A|F) = P(A and F)/P(F)
= [110/1000]/[400/1000] = .275
Example 7 continued
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A Tree Diagram is
useful for
portraying
conditional and
joint probabilities.
It is particularly
useful for
analyzing business
decisions involving
several stages.
Example
8: In a bag
containing 7 red
chips and 5 blue
chips you select 2
chips one after the
other without
replacement.
Construct a tree
diagram showing this
information.
Tree Diagrams
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6/11
7/12
5/12
R2
R1
5/11
B2
7/11
R2
B1
4/11
B2
Example 8 continued
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Bayes’ Theorem is a method
for revising a probability
given additional information.
It is computed using the
following formula:
P( A1 ) P( B / A1 )
P( A1 | B) 
P( A1 ) P( B / A1 )  P( A2 ) P( B / A2 )
Bayes’ Theorem
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Duff Cola Company
recently received several
complaints that their bottles
are under-filled. A
complaint was received
today but the production
manager is unable to
identify which of the two
Springfield plants (A or B)
filled this bottle. What is
the probability that the
under-filled bottle came
from plant A?
Example 9
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The following table summarizes the Duff
production experience.
% of total
production
% of
underfilled
bottle
A
5.5
3.0
B
4.5
4.0
Example 9 continued
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P( A) P(U / A)
P( A / U ) 
P( A) P(U / A)  P( B) P(U / B)
.55 (. 03)

 .4783
.55 (. 03)  .45 (.04 )
The likelihood the bottle was filled in
Plant A is reduced from .55 to .4783.
Example 9 continued
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The Multiplication
Formula
indicates
that if there are m ways
of doing one thing and n
ways of doing another
thing, there are m x n
ways of doing both.
Example 10: Dr.
Delong has 10
shirts and 8 ties.
How many shirt
and tie outfits
does he have?
(10)(8) = 80
Some Principles of Counting
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A Permutation is any arrangement of r
objects selected from n possible objects.
Note: The order of arrangement is important in
permutations.
n
n!
Pr 
( n  r )!
Some Principles of Counting
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A Combination
is the number of
ways to choose r
objects from a
group of n objects
without regard to
order.
n!
nCr 
r! (n  r )!
Some Principles of Counting
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There are 12 players
on the Carolina
Forest High School
basketball team.
Coach Thompson
must pick five
players among the
twelve on the team to
comprise the starting
lineup. How many
different groups are
possible? (Order
does not matter.)
12!
12C 5 
 792
5! (12  5)!
Example 11
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Suppose that in
addition to
selecting the
group, he must
also rank each of
the players in
that starting
lineup according
to their ability
(order matters).
12!
 95,040
12 P 5 
(12  5)!
Example 11 continued