4.2 - Triola textbook 5.2 - Sullivan textbook

1. Elementary Statistics
Chapter 4:
Probability
4.2 Addition Rule and
Multiplication Rule
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2. Chapter 4: Probability
4.1 Basic Concepts of Probability
4.2 Addition Rule and Multiplication Rule
4.3 Complements and Conditional Probability, and Bayes’ Theorem
4.4 Counting
4.5 Probabilities Through Simulations (available online)
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Objectives:
• Determine sample spaces and find the probability of an event, using classical probability or empirical probability.
• Find the probability of compound events, using the addition rules.
• Find the probability of compound events, using the multiplication rules.
• Find the conditional probability of an event.
• Find the total number of outcomes in a sequence of events, using the fundamental counting rule.
• Find the number of ways that r objects can be selected from n objects, using the permutation rule.
• Find the number of ways that r objects can be selected from n objects without regard to order, using the combination rule.
• Find the probability of an event, using the counting rules.

3. Probability: It is the science of uncertainty, allowing us to control the likelihood that a statistical inference is correct. (Likelihood of occurrence of an event)
A simple event is an outcome or an event that cannot be further broken down into simpler components.
A sample space is the set of all possible outcomes or simple events of a probability experiment.
How to interpret probability values, which are expressed as values between 0 and 1. A small probability, such as 0.001, corresponds to an event that rarely occurs.
Odds and their relation to probabilities. Odds are commonly used in situations such as lotteries and gambling.
Notation: P(A) = the probability of event A , 0 ≤ P(A) ≤ 1
The sum of the probabilities of all the outcomes in the sample space is 1: 𝑃𝑖 = 1
Impossible Set: If an event E cannot occur (i.e., the event contains no members in the sample space), its probability is 0. P(A) = 0
Sure (Certain) Set: If an event E is certain, then the probability of E is 1. P(E) = 1
Complementary Events: 𝐴: consists of all outcomes that are not included in the outcomes of event A. 𝑃 𝐴 = 1 − 𝑃(𝐴)
The actual odds against event A: O 𝐴 =
𝑃 𝐴
𝑃(𝐴)
, (expressed as form of a:b or “a to b”); a and b are integers.
The actual odds in favor of event A: O 𝐴 =
𝑃 𝐴
𝑃 𝐴
(the reciprocal of the actual odds against the even) If the odds against A are a:b, then the odds in favor of A are b:a.
Payoff odds against event A =
net profit
amount bet
net profit = (Payoff odds against event A)(amount bet)
Value: Give the exact fraction or decimal or round off final decimal results to 3 significant digits.
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Recall: 4.1 Basic Concepts of Probability

4. Recall: 3 CommonApproaches to Finding the Probability of an Event
•Classical probability uses sample spaces to determine the numerical probability that an event will
happen and assumes that all outcomes in the sample space are equally likely to occur. (confirm that the
outcomes are equally likely)
•Empirical probability (Relative Frequency Approximation of Probability) Conduct (or observe) a
procedure and count the number of times that event A occurs. P(A) is then approximated as follows:
•Subjective probability: P(A), the probability of event A, is estimated by using knowledge of the
relevant circumstances. (A subjective probability can be estimated in the absence of historical data.) uses
a probability value based on an educated guess or estimate, employing opinions and inexact information.
Experiments that have neither equally likely outcomes nor the potential of being repeated are assigned by
subjective probability. Examples: weather forecasting, predicting outcomes of sporting events
•Simulations: Sometimes none of the preceding three approaches can be used. A simulation of a
procedure is a process that behaves in the same ways as the procedure itself so that similar results are
produced. Probabilities can sometimes be found by using a simulation. 4
 
 
 
# of desired outcomes
, ( )
Total # of possible outcomes
n E s
P E orP A
n S n
  
 
( ) frequency of desired class
( ) ,
(Procedure Repeated) Sum of all frequencies
n A f
P A OrP E
n n
  
Example: P (Even numbers in
roll of a die) = 3/6 = 1/2
Example: P (A randomly selected student
from a Statistics class is a female) =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑒𝑚𝑎𝑙𝑒 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
𝑇𝑜𝑡𝑎𝑙

5. Addition rule: A tool to find P(A or B), which is the probability that either
event A occurs or event B occurs (or they both occur) as the single outcome of a
procedure. The word “OR” in the Addition rule is associated with the addition
of probabilities.
Multiplication rule: A tool to find P(A and B), which is the probability that
event A occurs and event B occurs. The word “and” in the multiplication rule
is associated with the multiplication of probabilities.
Compound Event: A compound event is any event combining two or more
simple events.
4.2 Addition Rule and Multiplication Rule Key Concept
( ) ( ) 1P A P A 
( ) 1 ( )P A P A 
( ) 1 ( )P A P A 
Complementary Events: Rules
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6. Addition Rule (Union): P(A or B) = P(in a single trial, event A occurs or event B occurs or they both occur)
To find P(A or B), add the number of ways event A can occur and the number of ways event B can occur, but add
in such a way that every outcome is counted only once. P(A or B) is equal to that sum, divided by the total
number of outcomes in the sample space.
P(A or B) = P(A) + P(B) − P(A and B)
where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a
procedure.
Disjoint (or mutually exclusive): Events A and B are disjoint (or mutually exclusive) if they cannot occur at
the same time. (That is, disjoint events do not overlap.) P (A & B) = 0
𝐴 𝑎𝑛𝑑 𝐴 must be disjoint.
4.2 Addition Rule and Multiplication Rule Key Concept
Venn
Diagram
for
Events
That Are
Not
Disjoint
Venn
Diagram
for
Disjoint
Events
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7. Example 1
Determine which events are mutually
exclusive (disjoint) and which are not,
when a single die is rolled.
a. Getting an odd number and getting an
even number
b. Getting a 3 and getting an odd number
c. Getting an odd number and getting a
number less than 4
d. Getting a number greater than 4 and
getting a number less than 4
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Mutually Exclusive (Disjoint)
a. Getting an odd number: 1, 3, or 5
Getting an even number: 2, 4, or 6
Mutually Exclusive
b. Not Mutually Exclusive
c. Not Mutually Exclusive
d. Getting a number greater than 4: 5 or 6
Getting a number less than 4: 1, 2, or 3
Mutually Exclusive

8. Assume the probability of randomly selecting someone who has
sleepwalked is 0.3, so P(sleepwalked) = 0.3. If a person is randomly
selected, find the probability of getting someone who has not sleepwalked.
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P(has not sleepwalked) = 1 − P(sleepwalked)
= 1 − 0.3 = 0.7
Example 2 Complementary Events: Rules ( ) 1 ( )P A P A 

9. Example 3
In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3
physicians are females. If a staff person is selected, find the probability
that the subject is a nurse or a male.
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Addition Rules ( ) ( ) ( ) ( )P A B P A P B P A B    
Staff Females Males Total
Nurses
Physicians
8
5
       Nurse or Male Nurse Male Male NurseP P P P  
7 1
3 2
Total 10 3 13
8 3 1 10
13 13 13 13
   
P(A or B) = P(A) + P(B) − P(A and B)

10. P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)
P(B | A) represents the probability of event B occurring after it is assumed that event A has
already occurred.
To find the probability that event A occurs in one trial and event B occurs in another trial,
multiply the probability of event A by the probability of event B, but be sure that the
probability of event B is found by assuming that event A has already occurred.
Independence and the Multiplication Rule
Independent: Two events A and B are independent if the occurrence of one does not affect
the probability of the occurrence of the other. (Several events are independent if the
occurrence of any does not affect the probabilities of the occurrence of the others.) If A and B
are not independent, they are said to be dependent.
Sampling: In the world of statistics, sampling methods are critically important, and the
following relationships hold:
Sampling with replacement: Selections are independent events.
Sampling without replacement: Selections are dependent events.
P(A ∩ B) = P(A)  P(B)
P(A and B) = P(A)  P(B | A)
Multiplication Rule
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11. Example 4
50 test results from the subjects who use
drugs are shown below:
a. If 2 of these 50 subjects are randomly
selected with replacement, find the
probability the first selected person had a
positive test result and the second
selected person had a negative test result.
b. Repeat part (a) by assuming that the
two subjects are selected without
replacement.
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Multiplication Rules
Tested Frequency
+ 45
− 5
Total 50
P(1st selection is positive
and 2nd is negative)
Formal Multiplication Rule
P(A and B) = P(A)  P(B | A)
a. P(A ∩ B) = P(A)  P(B) =
45
50
∙
5
50
=
9
100
P(1st selection is positive and 2nd is negative)
b. P(A ∩ B) = P(A)  P(B | A)=
45
50
∙
5
49
= 0.0918

12. Treating Dependent Events and Independent
5% Guideline for Cumbersome Calculations:
When sampling without replacement and the sample size is no more than 5% of the
size of the population, treat the selections as being independent (even though they
are actually dependent).
Example 5
Three adults are randomly selected without replacement from the 247,436,830
adults in the United States. Also assume that 10% of adults in the United States
use drugs. Find the probability that the three selected adults all use drugs.
Without replacement: The three events are dependent.
The sample size of 3 < 5% (247,436,830) ⇾ Assume independent
P(all 3 adults use drugs) = P(first uses drugs) · P(second uses drugs) · P(third uses drugs)
= (0.10)(0.10)(0.10) = 0.001 12
Given: P (D) = 0.10

13. Redundancy: Important Application of the Multiplication Rule
The principle of redundancy is used to increase the reliability of many
systems.
Our eyes have passive redundancy in the sense that if one of them fails, we
continue to see.
An important finding of modern biology is that genes in an organism can
often work in place of each other.
Engineers often design redundant components so that the whole system
will not fail because of the failure of a single component.
Modern aircraft are now highly reliable, and one design feature
contributing to that reliability is the use of redundancy, whereby critical
components are duplicated so that if one fails, the other will work. For
example, the Airbus 310 twin-engine airliner has three independent
hydraulic systems, so if any one system fails, full flight control is
maintained with another functioning system. 13

14. For a typical flight, the probability of an aircraft’s engine hydraulic system for
failure is 0.002.
a. If the aircraft had only one hydraulic system, what is the probability
that the aircraft’s flight control would work for a flight?
b. An aircraft has 3 independent hydraulic systems, what is the
probability that on a typical flight, control can be maintained with a working
hydraulic system?
Example 5
a. P(1 hydraulic system does not fail) = 1 − P(failure) = 1 − 0.002 = 0.998
b. With 3 independent hydraulic systems, flight control will be maintained if
the three systems do not all fail.
P(All 3 failing) = 0.002 · 0.002 · 0.002 = 0.000000008.
It follows that the probability of maintaining flight control is as follows:
P(Not All 3 failing) = 1 − 0.000000008 = 0.999999992
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