Probability Rules

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Probability Rules

• Chapter 15

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Review of Approaches to Probability
1.) What is the probability that the Dow Jones
Industrial Average will exceed 12,000? Which
approach to probability would you use to answer
this question? 2.) The National Center for Health
Statistics reports that of 883 deaths, 24
resulted from an automobile accident, 182 from
cancer, and 333 from heart disease. What is the
probability that a particular death is due to an
automobile accident? Which approach to
probability did you use to answer this
question? 3.)One card will be randomly selected
from a standard 52-card deck. What is the
probability the card will be an Ace? Which
approach to probability did you use to answer
this question?
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Review of Basic Probability Rules
Complement Rule P(A) 1 P(Ac) Addition Rule
for Mutually Exclusive Events P(A or B) P(A)
P(B) Multiplication Rule for Independent
Events P(A and B) P(A)P(B) At Least One
Rule P(At least one) 1 P(none)
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Venn Diagram Mutually Exclusive Events
One card is drawn from a standard deck of cards.
What is the probability that it is an ace or a
nine?
A
B
Ace
nine
Events A and B are mutually exclusive. A card
can be either an Ace or a nine, but can not be
both
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Venn DiagramEvents that are Not Mutually
Exclusive
One card is drawn from a standard deck of cards.
What is the probability that it is red or an ace?
Red
Ace
Both red and an ace
These events are not mutually exclusive as it is
possible for a card to be both red and an ace
(ace of hearts, ace of diamonds)
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Probabilities of Events that Are Not Mutually
Exclusive
Find P(Red or Ace)
0.5
0.077
Red
Ace
0.038
P(Red) P(Ace) P(Both Red and Ace) 0.5
0.077 -0.038 0.539
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General Addition Rule
Used when events are not mutually
exclusive. P(A or B) P(A) P(B) P(A and
B) Example The Illinois Tourist Commission
selected a sample of 200 tourists who visited
Chicago during the past year. The survey
revealed that 120 tourists went to the Sears
Tower, 100 went to Wrigley Field and 60 visited
both sites. What is the probability of selecting
a person at random who visited both the Sears
Tower and Wrigley Field?
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General Addition Rule (continued)
P(Sears Tower) 120/200 0.6 P(Wrigley Field)
100/200 0.5 P(Both) 60/200 0.3
0.3
0.6
0.5
S
W
P(S or W) P(S) P(W) P(S and W) 0.6
0.5 - 0.3 0.8
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General Addition Rule (continued)
What is the probability that a randomly selected
person visited either the Sears Tower or Wrigley
Field but NOT both? P(S or W but NOT both) P(S
or W) P(S and W) 0.8 0.3 0.5 Second
approach P(S and Wc) 0.6 0.3 0.3
P(W and SC) 0.5 0.3 0.2 P(S or W but NOT
both) P(S and Wc) P(W and SC) 0.3
0.2 0.5
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General Addition Rule (continued)
What is the probability that a randomly selected
tourist went to neither location? P(neither
location) 1 P(either location) 1 P(S
or W) 1 0.8 0.2
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Conditional Probabilities
Here is a contingency table that gives the counts
of ECO 138 students by their gender and political
views. (Data are from Fall 2005 Class Survey)


P(Female) 77/137 0.562 P(Female and Liberal)
30/137 0.219 What is the probability that a
selected student has moderate political views
given that we have selected a female?
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Conditional Probabilities (continued)
What is the probability that a selected student
has moderate political views given that we have
selected a female?
P(Moderate Female) 24/77 0.311 Conditional
probability, P (BA) the probability of event B
given event A.
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Conditional Probabilities (continued)
Formal Definition P(BA) P(A and B)
P(A) Example P(Moderate and Female)
P(Female) (24/137) / (77/137) 0.175 /
0.562 0.311
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Multiplication Rule
Multiplication Rule for Independent events P(A
and B) P(A) P(B) Independent the
occurrence of one event has no effect on the
probability of the occurrence of another
event. Example A survey by the American
Automobile Association (AAA) revealed that 60
percent of its members made airline reservations
last year. Two members are selected at random.
What is the probability both made airline
reservations last year? P(R1 and R2)
P(R1)P(R2) (0.6)(0.6) .36
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General Multiplication Rule
Use when events are Dependent. P(A and B)
P(A) P(BA) For two events A and B, the joint
probability that both events will happen is found
by multiplying the probability event A will
happen by the conditional probability of event B
occurring.
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General Multiplication Rule (continued)
Example A county welfare agency employs 10
welfare workers who interview prospective food
stamp recipients. Periodically the supervisor
selects, at random, the forms completed by two
workers to audit for illegal deductions. Unknown
to the supervisor, three of the workers have
regularly been giving illegal deductions to
applicants. What is the probability that both of
the two workers chosen have been giving illegal
deductions?
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General Multiplication Rule (continued)

• Solution Define the following two events
• A First worker selected gives illegal
  deductions
• B Second worker selected gives illegal
  deductions
• We want to find the probability that both A and B
  occur.
• To find the P(A) consider the following Venn
  Diagram.
• I worker with illegal deductions N worker not
  giving illegal D

Each observation in the sample space is equally
likely. P (A) P(I1) P(I2) P(I3)
1/10 1/10 1/10 3/10 or 0.30
N1 N2 N3 N4 N5 N6 N7
I1 I2 I3
A
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General Multiplication Rule (continued)

• To find the conditional probability, P(BA), we
  need to make changes to the sample space.
  Remember our assumption is that the first worker
  selected is giving illegal deductions.

P (BA) P(I1) P(I2) 1/9 1/9
2/9 Substituting P(A) and P(BA) into the formula
for the general multiplication rule, we find P(A
and B) P(A)P(BA) (3/10) (2/9) 6/90
1/15 or 0.067
N1 N2 N3 N4 N5 N6 N7
I1 I2
BA
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Tree Diagram
A tree diagram is a display of conditional events
or probabilities that is helpful in thinking
through conditioning.
N (6/9)
N and N (7/10)(6/9) 42/90
N
(7/10)
N and I (7/10)(3/9) 21/90
I (3/9)
(3/10)
N (7/9)
I and N (3/10)(7/9) 21/90
I
I (2/9)
I and I (3/10)(2/9) 6/90
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Independent Events?
Again, events are independent when the outcome of
one event does not influence the probability of
the other. Events A and B are independent
whenever P(BA) P(B) In the case of
independent events the general multiplication
rule reduces to the simple multiplication
rule. P(A and B) P(A) P(BA) P(A) P(B)
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Exploring Independence
Is the probability of being liberal independent
of gender for ECO 138 students?
In other words, does P(Liberal Female)
P(Liberal)? P(LiberalFemale) 30/77
0.39 P(Liberal) 47/137 0.343 Because these
probabilities are not equal, we can be pretty
sure that liberal political views are not
independent of the students gender
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Lets Try Some Examples
1.) Two cards are drawn without replacement.
What is the probability they are both aces? 2.)
What is the probability of getting 5 hearts in a
row? 3.) I draw one card and look at it. I
tell you that it is red. What is the probability
it is a heart? And what is the probability it is
red, given that it is a heart?
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Examples
4.) Are red card and spade independent?
Mutually exclusive? 5.) Are face card and
king independent? Mutually exclusive?
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Example - Travel
Suppose the probability that a U.S. resident has
traveled to Canada is 0.18, to Mexico is 0.09,
and to both countries is 0.04. Whats the
probability that an American chosen at random
has A.) traveled to Canada but not Mexico? B.)
traveled to either Canada or Mexico? C.) not
traveled to either country? D.) Are travel to
Mexico and Canada mutually
exclusive events? E.) Are travel to Mexico and
Canada independent events? Explain.
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Example - Sick Cars
Twenty percent of cars that are inspected have
faulty pollution control systems. The cost of
repairing a pollution control system exceeds 100
about 40 of the time. When a driver takes her
car in for inspection, whats the probability
that she will end up paying more than 100 to
repair the pollution control system?
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Example - Health
The probabilities that an adult American man has
high blood pressure and/or high cholesterol are
shown in the table
A.) What is the probability that a man has both
conditions? B.) Whats the probability that he
has high blood pressure? C.) Whats the
probability that a man with high blood pressure
has high cholesterol? D.)Are high blood pressure
and high cholesterol independent?
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Example - Absenteeism
A companys records indicate that on any given
day about 1 of their day shift employees and 2
of the night shift employees will miss work.
Sixty percent of the employees work the day
shift. A.) Is absenteeism independent of shift
worked? Explain. B.) What percent of
employees are absent on any given day?
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Example - Blood Type
The American Red Cross says that about 45 of the
U.S. population has Type O blood, 40 Type A, 11
Type B, and the rest Type AB. Among four
potential donors, what is the probability
that A.) All are Type O? B.) No one is Type
AB? C.) They are not all Type A? D.) At least one
person is Type B?
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Graduation
A private college report contains these
statistics 70 of incoming freshmen attended
public schools. 75 of public school students who
enroll as freshmen eventually graduate 90 of
other freshmen eventually graduate. A.) Is there
any evidence that a freshmans chances to
graduate may depend upon what kind of high school
the student attended? Explain. B.) What percent
of freshman eventually graduate?
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Assignment

• Read Chapter 16 (Random Variables) by Wednesday,
  March 8
• Try the following exercises
• 1,3,5,7,9,13,15,19,25,31, 35 and 43
• Quiz 3 Wednesday, March 8
• Covers chapters 14 and 15
• Bring pencil, calculator, UID card