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HAWKES LEARNING SYSTEMS
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Copyright © 2010 by Hawkes Learning
Systems/Quant Systems, Inc.
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Chapter 6
Probability, Randomness, and
Uncertainty
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
Sections 6.1-6.4 Classical Probability
Objectives:
• To learn the basic vocabulary used in counting techniques.
• To solve problems using classical probability.
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Probability, Randomness, and Uncertainty
math courseware specialists
Section 6.1 Important Definitions
Probability:
• In every day usage the word probably describes an event or
circumstance the speaker believes will occur. At the same time the
speaker reserves the possibility that it may not occur.
• Probability is used to quantify uncertainty.
• Games of chance, such as tossing a coin, provide a way to
demonstrate some of the fundamental laws (rules) of probability.
• Statistically speaking, the playing of the game is called an
experiment.
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Section 6.1 Important Definitions
Random Experiment:
• A random experiment is defined as any activity or phenomenon
that meets the following conditions:
i.
ii.
iii.
There is a distinct outcome for each trial of the
experiment.
The outcome of the experiment is uncertain.
The set of all distinct outcomes of the experiment can
be specified and is called the sample space.
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Section 6.1 Important Definitions
Events:
• A simple event is any member of the sample space.
If you were to toss a single coin there are only 2 simple events
{Head, Tail} in the experiment.
• An event is a set of simple events.
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Section 6.1 Important Definitions
Sample Space:
• The sample space of the experiment contains every possible
outcome that could occur in the experiment.
• For a coin tossing experiment the sample space would be
S = {Head, Tail}.
• The sample space may be called the outcome set.
• The result of a random experiment must be a simple event.
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
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Section 6.1 Important Definitions
Example:
Experiment 1: Toss a coin three times and observe the number of
heads.
Have we met the three conditions of a random experiment?
i.
ii.
iii.
There will only be one outcome since it is not possible to have
exactly one and exactly two heads on the same trial.
The outcome will be unknown before tossing the three coins.
The sample space can be specified and is composed of eight
simple events,
S={TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}
This experiment meets the conditions of a random experiment.
An event could be obtaining more than one head, which involves
the following set of simple events, {THH, HTH, HHT, HHH}.
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
Section 6.1 Important Definitions
Example:
Experiment 2: Roll a die and observe the number of dots on the
uppermost surface.
Have we met the three conditions of a random experiment?
Solution:
i.
ii.
iii.
There will only be one outcome.
The value of the outcome is not known.
The sample space can be specified and is composed of simple
events,
S={1, 2, 3, 4, 5, 6}.
This experiment meets the conditions of a random experiment.
An event could be rolling an even number, which would be given by
the set of simple events, {2, 4, 6}.
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
Section 6.1 Important Definitions
Example:
Experiment 3: Draw a card from a well shuffled deck consisting of 13
hearts, 13 clubs, 13 spades, and 13 diamonds and observe the suit of
the card.
Have we met the three conditions of a random experiment?
Solution:
i.
ii.
iii.
There will only be one outcome.
The suit is unknown since the card will be drawn at random.
The sample space consists of the set of outcomes,
S = {heart, club, spade, diamond}.
This experiment meets the conditions of a random experiment.
An event could be drawing either a spade or a club, which would be
given by the set of simple events, {spade, club}.
HAWKES LEARNING SYSTEMS
Probability, Randomness,
and Uncertainty
Ch. 6 Probability,
Randomness, and Uncertainty
math courseware specialists
Section 6.2 Interpreting
Probability:
Relative
Frequency
6.2 Interpreting
Probability:
Relative
Frequency
Relative Frequency:
• Someone who wanted the probability of getting a head on the
toss of a coin could toss a coin a large number of times and
observe the number of times that a head appeared.
• The probability could be computed as the number of times a
head was observed divided by the number of times the coin
was flipped. This is the relative frequency interpretation of
probability.
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Section 6.2 Interpreting Probability: Relative Frequency
Relative Frequency:
• If an experiment is performed n times, under identical conditions,
and the event A happens k times, the relative frequency of A is
k
Relative Frequency of A  .
n
• If the relative frequency stabilizes as n increases, then the relative
frequency is said to be the probability of A.
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Section 6.2 Interpreting Probability: Relative Frequency
Relative Frequency of Heads
Relative Frequency:
This is a graph of the first 42 flips of the coin. The relative frequency after 42
flips is 36%. This means in 42 flips, the coin landed on heads 36% of the time.
More coin tosses are needed to stabilize the percentage.
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Section 6.2 Interpreting Probability: Relative Frequency
Relative Frequency of Heads
Relative Frequency:
Here the flips start at 200 flips. By flip 296, there are 141 heads and 155
tails which equate to a .4764 probability (or relative frequency) of heads.
Although this is slightly less than expected, such a percentage is
reasonable considering the randomness of the coin toss.
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Section 6.2 Interpreting Probability: Relative Frequency
Relative Frequency of Heads
Relative Frequency:
0
It is clear in this chart that from 1350 to 1450 flips the percentage
converges on some point close to .5 and remains very stable.
At flip number 1450, there are 718 heads and 732 tails which equate to
a .4952 probability (or relative frequency) of heads.
HAWKES LEARNING SYSTEMS
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Section 6.2 Interpreting Probability: Relative Frequency
Statistical Regularity:
Will the observed relative frequency ever reach .5?
• No mathematical or physical law requires the observed relative
frequency to ever reach some predetermined level. But if the
probability of observing a head is really .5, the observed relative
frequency should closely approach the value after a large number of
flips.
k 718
 .4952
Relative frequency of the event “getting a head”: A  
n 1450
• The relative frequency of a head seems to converge to the expected
relative frequency. We call this statistical regularity.
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Section 6.3 Interpreting Probability: Subjective Approach
Subjective Approach:
• An example of subjective probability is a weatherman predicting
the weather.
• The local weather man on channel 2 predicts there is an 80%
chance of rain, but the weatherman on channel 4 predicts there is a
66% chance of rain, while lastly on channel 5 the weatherman
predicts there is a 33% chance of rain.
Who more accurate?
• It is actually possible that all three weatherman are accurate.
How is this possible?
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Section 6.3 Interpreting Probability: Subjective Approach
Subjective Approach:
• If 80% of the days channel 2’s weatherman predicted an 80%
chance of rain it actually rained then channel two has very
accurate weatherman.
• If 66% of the time it rained when the weatherman at channel 4
predicted a 66% chance of rain, he is as accurate as channel 2.
• Lastly if channel 5’s weatherman predicted a 33% chance of
rain and it indeed rained 33% of the time he made that
prediction, he is as accurate as channels 2, and 4.
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Section 6.3 Interpreting Probability: Subjective Approach
Subjective Approach:
• The subjective viewpoint regards the probability of an event as a
measure of the degree of belief that an event has occurred.
• Someone’s degree of belief in some event will depend on his or
her life experiences.
• The subjective approach must allow for differences in the
degree of belief among reasonable people.
• Subjective probability is criticized since it is not a universally
accepted criterion. Two reasonable people might examine the
same data and reach different conclusions about their degree of
belief about some proposition.
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Section 6.4 Interpreting Probability: Classical Approach
Classical Approach:
• Probability can be measured as a simple proportion: the
number of simple events that compose the event divided by
the number of simple events in the sample space. Namely, the
probability if an event A , denoted P (A), is given by
P(A) 
number of simple events in A
total number of simple events in the sample space
HAWKES LEARNING SYSTEMS
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math courseware specialists
Section 6.4 Interpreting Probability: Classical Approach
Example:
In experiment 1 we tossed a coin three times and the number of
heads was observed. The sample space consisted of 8 simple
events {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}. Let A be
the event of getting at least one head.
What is the probability of A?
Solution:
Since the event A consists of 7 simple events,
{TTH,THT, THH, HTT, HTH, HHT, HHH}, and there are 8 equally
likely simple events in the sample space,
7
P(A)  .
8
HAWKES LEARNING SYSTEMS
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math courseware specialists
Sections 6.5-6.11 Basic Probability Rules
Objectives:
• To calculate the conditional probability of a situation.
• To correctly apply the appropriate probability rule to a given
situation.
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Section 6.5 What is Probability?
What is Probability?:
• There are several conflicting ideas which seek to define the
interpretation of probability. In fact, there is a substantial conflict
of ideas between the notions of statistical regularity and degree
of belief.
• The theory of probability does not require the interpretation of
probabilities, just as in geometry the interpretation of points,
lines, and planes are irrelevant.
• Fortunately probability must obey certain laws no matter how
probability is defined.
HAWKES LEARNING SYSTEMS
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Probability,
Randomness,
and Uncertainty
Ch. 6 Probability,
Randomness,
and Uncertainty
Section
Some
of Probability
6.6 6.6
Some
Laws Laws
of Probability
Probability Laws:
Probability Law Number 1
A probability of zero means the event cannot happen.
Probability Law Number 2
A probability of one means the event must happen.
Probability Law Number 3
All probabilities must be in between zero and one inclusively.
Probability Law Number 4
The sum of the probabilities of all simple events equals one.
Probability, Randomness, and Uncertainty
HAWKES LEARNING SYSTEMS
math courseware specialists
Section 6.7 What’s the Connection Between Probability and
Statistics?
Probability and Statistics:
• Most of the time when statisticians are working with samples,
they try to deduce from the samples the population parameters.
• The process of making judgments about population parameters is
called statistical inference.
• Because samples are random, there is no guarantee that the
sample accurately represents the population.
• If a sample is not representative, then using the sample mean as
a guess (inference) of the population mean would not be very wise.
• Probability is used to assess the quality of our inference.
• All statistical conclusions must be endowed with a degree of
uncertainty.
• Because probability is used to assess the reliability of sample
inferences, it is the foundation of all inferential statistics.
HAWKES LEARNING SYSTEMS
Ch. 6 Probability,
Randomness,
and Uncertainty
Probability,
Randomness,
and Uncertainty
math courseware specialists
6.8 Probability
and Business
Section
6.8 Probability
and Business
Probability and Business:
• The probability concept also has many applications in business.
When a manager wonders if dropping a bid price by 5% will
increase the probability of winning the bid, she is thinking about
chance.
• Probability is also used as a criterion in designing and evaluating
product reliability, evaluating insurance, inventory management,
project management and in the study of queuing theory (a
probability analysis of waiting lines).
• A special kind of statistician called an actuary has emerged to
assist in the development of insurance models which quantify
uncertainty and aid in business decisions.
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Section 6.9 Other Probability Rules
Compound Event:
• A compound event is an event that is defined by
combining two or more events.
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Section 6.9 Other Probability Rules
Compound Events:
• Suppose the marketing director for Sports Illustrated believed
that anyone who possessed an income greater than $50,000
and/or subscribed to more than one other sports magazine could
potentially be a good prospect for a direct mail marketing
campaign.
• Let the events
A = {annual income is greater than $50,000} and
B = {subscribes to more than one other sports magazine}.
• There are several different types of compound events. To illustrate
the concepts consider the two events A and B.
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Section 6.9 Other Probability Rules
Union:
• The union of the events A and B is the set of all outcomes that
are included in A or B or both, and is denoted A  B .

• Notice that the union includes all the points in both A and/or B.
HAWKES LEARNING SYSTEMS
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Section 6.9 Other Probability Rules
Intersection:
• The intersection of the events A and B is the set of all
outcomes that are included in both A and B.

.
• The symbol for intersection is
• Notice that the intersection includes only points in both A and B.
This is the set the marketing director is interested in.
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Section 6.9 Other Probability Rules
Complement:
• The complement of an event A is the set of all outcomes in the
sample space that are not in A.
A
The compliment of
A is the shaded
region.
• The complement of a set will be written as Ac.
• Notice the complement of A includes all points that are not in A.
• For the event A = {annual income is greater than $50,000}, the
complement of A would be
Ac = {annual income is less than or equal to $50,000}.
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Section 6.9 Other Probability Rules
Mutually Exclusive:
• Two points are mutually exclusive if they have no points in
common.
A
B
• Mutual exclusivity is also called disjointedness. The figure above
represents two disjoint events.
• The set relationships of complement, union, and intersection give
rise to a number of probability laws.
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Section 6.9 Other Probability Rules
Example:
Probability Law Number 5
The probability of Ac is given by P(Ac )  1  P(A)
Consider the event A = {annual income is greater than $50,000}.
Suppose the probability of A was .08.
What is the probability of observing someone whose
income is less than or equal to $50,000?
Solution:
P(annual income is less than or equal to $50,000)  P(A )
c
 1 P A
 1  .08
 .92.
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Section 6.9 Other Probability Rules
Probability Law Number 6:
Union of Mutually Exclusive Events
If the events A and B are mutually exclusive, then
P(A  B)  P(A)  P(B).
A
B
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Section 6.9 Other Probability Rules
Probability Law Number 7:
The Addition Rule
For any two events A and B,
P(A  B)  P(A)  P(B)  P(A  B)
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Section 6.10 Conditional Probability
Conditional Probability:
• The probability that one event will occur, given that some event
has occurred or is certain to occur, is a conditional probability.
Example:
• Consider the question of whether cigarette smoking harms those
who are indirectly exposed to smoke.
• Suppose that 3 percent of women who do not smoke die of
cancer. However, if a non-smoking woman is married to a smoker,
the probability of dying of cancer is .08. This is a conditional
probability, because the sample space is being limited by some
condition – in this case limited to wives of smoking husbands.
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Section 6.10 Conditional Probability
Probability Law Number 8:
• The conditional probability of A, given that B has occurred is
P(A  B)
P(A | B) 
.
P(B)
• The events A and B can be reversed in the preceding rule to
compute P (B | A) .
• The notation P (A | B) is read as “the probability of A given B.”
The vertical bar within a probability statement will always mean
given.
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Section 6.11 Independence
Independence:
• Two events, A and B, are independent if and only if
P(A | B)  P(A) or P(B | A)  P(B)
• An extremely important concept in statistical analysis is
independence.
• Independence describes a special kind of relationship between
two events.
• Two events are said to be independent if knowledge of one
event does not provide information about the other event’s
occurrence.
• If two events are not independent, they are dependent.
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Section 6.11 Independence
Probability Law Number 9:
Multiplication Rule for Independent Events
If two events, A and B, are independent, then
P(A  B)  P(A)  P(B).
If n events, A1,A2,…,An, are independent, then
P(A1  A2  ...  An )  P(A1 )  P(A2 )  ...  P(A n ).
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Section 6.12a Basic Counting Rules
Objectives:
• To learn the five basic counting rules.
• To differentiate between permutation and combination.
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Section 6.12 Counting
Fundamental Rule for Counting:
• E1 is an event with n1 possible outcomes and E2 is an event
with n2 possible outcomes. The number of ways the events
can occur in sequence is n1  n2. This principle can be applied
for any number of events occurring in sequence.
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Section 6.12 Counting
Example:
A local office supply store offers ballpoint pens from three
different manufacturers. Each manufacturer’s pens come in
either red, blue, black, or green and either fine or medium tip is
available for each color. How many different pens does the
store carry?
Solution:
Applying the fundamental rule for counting:
3

4

2

 number of 

  color of ink   types of tips 
 manufacturers 
24
 different pens 
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Section 6.12 Counting
n Factorial:
• The product 5  4  3  2 1 is a special type of product called a
factorial.
• Suppose n is a positive whole number. Then,
n!  n   n 1   n  2  ...  3  2 1.
• Note: 0! = 1 by definition.
• n! is read as “n factorial”.
• Using this notation, 5!  5  4  3  2 1  120.
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Section 6.12 Counting
Permutation:
• A permutation is a specific order or arrangement of objects in
a set. There are n! permutations of n unique objects.
• The number of permutations of n unique objects taken k at a
time is:
n!
P 
.
 n  k !
n
k
• You may see also see this denoted as n Pk or P  n, r  .
• If given n objects, with n1 alike, n2 alike, …, nk alike, then the
number of distinguishable permutations of all n objects is
n!
.
 n1 !n2 !n3 !...nk !
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Section 6.12 Counting
Combination:
• A combination is a collection or grouping of objects where
the order is not important.
• The number of combinations of n unique objects taken k at a
time is:
n!
C 
.
 n  k  !k !
n
k
• It is important to remember that permutations are used when
order is important and combinations are used when order is not
important.