# Probability Rules.ppt

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```					Probability Rules
Chapter 15

1
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
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?

2
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)
3
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                         4
Venn Diagram:
Events 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)                                                       5
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
6
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?

7
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                        8
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

9
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

10
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)

Political views
Liberal        Moderate Conservative Total
Gender Male        17                29          14   60
Female      30                24          23   77
Total       47                53          37 137

P(Female) = 77/137 = 0.562

P(Female and Liberal) = 30/137 = 0.219
What is the probability that a selected student has moderate       11
political views given that we have selected a female?
Conditional Probabilities (continued)
What is the probability that a selected student has moderate
political views given that we have selected a female?

Political views
Liberal   Moderate Conservative Total
Gender Male        17           29          14   60
Female      30           24          23   77
Total       47           53          37 137

P(Moderate | Female) = 24/77 = 0.311

Conditional probability, P (B|A) – the probability of event B
given event A.
12
Conditional Probabilities (continued)
Formal Definition:
P(B|A) = P(A and B)
P(A)
Example: P(Moderate and Female)
P(Female)
=(24/137) / (77/137)
= 0.175 / 0.562
= 0.311
13
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

14
General Multiplication Rule

Use when events are Dependent.

P(A and B) = P(A) * P(B|A)

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.
15
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?
16
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

N1 N2 N3                          Each observation in the sample
I1        I2
space is equally likely.
N4 N5 N6                I3
A                  P (A) = P(I1) + P(I2) + P(I3)
N7
= 1/10 + 1/10 + 1/10
= 3/10 or 0.30              17
General Multiplication Rule (continued)
To find the conditional probability, P(B|A), we need to make changes
to the sample space. Remember our assumption is that the first
worker selected is giving illegal deductions.

P (B|A) = P(I1) + P(I2)
N1 N2 N3                             = 1/9 + 1/9 = 2/9
N4 N5 N6        I1
N7                         Substituting P(A) and P(B|A) into
I2
B|A             the formula for the general
multiplication rule, we find
P(A and B) = P(A)P(B|A)
= (3/10) * (2/9)
= 6/90 = 1/15 or 0.067

18
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)
I = (3/9)
N and I = (7/10)(3/9) = 21/90

(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
19
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(B|A) = 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(B|A)
= P(A) * P(B)

20
Exploring Independence
Is the probability of being liberal independent
of gender for ECO 138 students?
Political views
Liberal   Moderate Conservative Total
Gender Male        17           29          14   60
Female      30           24          23   77
Total       47           53          37 137

In other words, does P(Liberal | Female) = P(Liberal)?
P(Liberal|Female) = 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 student’s   21
gender
Let’s 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?                   22
Examples
4.) Are “red card” and “spade” independent?
Mutually exclusive?

5.) Are “face card” and “king” independent?
Mutually exclusive?

23
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. What’s 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   24
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, what’s the
probability that she will end up paying more than
\$100 to repair the pollution control system?

25
Example - Health
The probabilities that an adult American man has high blood
pressure and/or high cholesterol are shown in the table:
Blood Pressure
Cholesterol          High      OK

High   0.11     0.21

OK     0.16     0.52

A.) What is the probability that a man has both conditions?
B.) What’s the probability that he has high blood pressure?
C.) What’s the probability that a man with high blood
pressure has high cholesterol?
D.)Are high blood pressure and high cholesterol
independent?                                                  26
Example - Absenteeism
A company’s 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?                        27
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?

28
A private college report contains these statistics:
70% of incoming freshmen attended public schools.
75% of public school students who enroll as freshmen
90% of other freshmen eventually graduate.

A.) Is there any evidence that a freshman’s chances to
graduate may depend upon what kind of high school the
student attended? Explain.

B.) What percent of freshman eventually graduate?

29
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

30

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