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Chapter 7
Randomness and
Probability
Copyright © 2012 Pearson Education.
7.1 Random Phenomena and Probability
With random phenomena, we can’t predict the individual
outcomes, but we can hope to understand characteristics
of their long-run behavior.
For any random phenomenon, each attempt, or trial, generates
an outcome.
We use the more general term event to refer to outcomes or
combinations of outcomes.
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7.1 Random Phenomena and Probability
Sample space is a special event that is the collection of all
possible outcomes.
We denote the sample space S or sometimes Ω.
The probability of an event is its long-run relative frequency.
Independence means that the outcome of one trial doesn’t
influence or change the outcome of another.
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7.1 Random Phenomena and Probability
The Law of Large Numbers (LLN) states that if the events are
independent, then as the number of trials increases, the longrun relative frequency of an event gets closer and closer to a
single value.
Empirical probability is based on repeatedly observing the
event’s outcome.
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7.2 The Nonexistent Law of Averages
Many people confuse the Law of Large numbers with the socalled Law of Averages that would say that things have to even
out in the short run.
The Law of Averages doesn’t exist.
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7.3 Different Types of Probability
Model-Based (Theoretical) Probability
The (theoretical) probability of event A can be computed with
the following equation:
# outcomes in A
P( A) 
total # of outcomes
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7.3 Different Types of Probability
Personal Probability
A subjective, or personal probability expresses your uncertainty
about the outcome.
Although personal probabilities may be based on experience,
they are not based either on long-run relative frequencies or on
equally likely events.
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7.4 Probability Rules
Rule 1
If the probability of an event occurring is 0, the event can’t
occur.
If the probability is 1, the event always occurs.
For any event A,
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0  P( A) . 1
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7.4 Probability Rules
Rule 2: The Probability Assignment Rule
The probability of the set of all possible outcomes must be 1.
P(S)  1
where S represents the set of all possible outcomes and is
called the sample space.
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7.4 Probability Rules
Rule 3: The Complement Rule
The probability of an event occurring is 1 minus the probability
that it doesn’t occur.
P ( A)  1  P ( A C )
where the set of outcomes that are not in event A is called the
“complement” of A, and is denoted AC.
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7.4 Probability Rules
For Example: Lee’s Lights sell lighting fixtures. Lee records
the behavior of 1000 customers entering the store during one
week. Of those, 300 make purchases. What is the probability
that a customer doesn’t make a purchase?
If P(Purchase) = 0.30
then P(no purchase) = 1 – 0.30 = 0.70
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7.4 Probability Rules
Rule 4: The Multiplication Rule
For two independent events A and B, the probability that
both A and B occur is the product of the probabilities of the
two events.
P( A and B)  P( A)  P(B)
provided that A and B are independent.
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7.4 Probability Rules
For Example: Whether or not a caller qualifies for a platinum
credit card is a random outcome. Suppose the probability of
qualifying is 0.35. What is the chance that the next two callers
qualify?
Since the two different callers are independent, then
P( A and B)  P( A)  P(B)
P(customer 1 qualifies and customer 2 qualifies)
= P(customer 1 qualifies) x P(customer 2 qualifies)
= 0.35 x 0.35
= 0.1225
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7.4 Probability Rules
Rule 5: The Addition Rule
Two events are disjoint (or mutually exclusive) if
they have no outcomes in common.
The Addition Rule allows us to add the probabilities of disjoint
events to get the probability that either event occurs.
P( A or B)  P( A)  P(B)
where A and B are disjoint.
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7.4 Probability Rules
For Example: Some customers prefer to see the
merchandise but then make their purchase online. Lee
determines that there’s an 8% chance of a customer making a
purchase in this way. We know that about 30% of customers
make purchases when they enter the store. What is the
probability that a customer who enters the store makes no
purchase at all?
P(purchase in the store or online)
= P (purchase in store) + P(purchase online)
= 0.30 + 0.09 = 0.39
P(no purchase) = 1 – 0.39 = 0.61
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7.4 Probability Rules
Rule 6: The General Addition Rule
The General Addition Rule calculates the probability that either
of two events occurs. It does not require that the events be
disjoint.
P( A or B)  P( A)  P(B)  P( A and B)
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7.4 Probability Rules
For Example: Lee notices that when two customers enter
the store together, their behavior isn’t independent. In fact,
there’s a 20% they’ll both make a purchase. When two
customers enter the store together, what is the probability that
at least one of them will make a purchase?
P(Both purchase) = P(A purchases or B purchases)
= P(A purchases) + P(B purchases)
– P(A and B both purchase)
= 0.30 + 0.30 – 0.20 = 0.40
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7.4 Probability Rules
Example: Car Inspections
You and a friend get your cars inspected. The event of your
car’s passing inspection is independent of your friend’s car. If
75% of cars pass inspection what is the probability that
Your car passes inspection?
Your car doesn’t pass inspection?
Both cars pass inspection?
At least one of two cars passes?
Neither car passes inspection?
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7.4 Probability Rules
Example (continued): Car Inspections
You and a friend get your cars inspected. The event of your
car’s passing inspection is independent of your friend’s car. If
75% of cars pass inspection what is the probability that
Your car passes inspection?
P(Pass) = 0.75
Your car doesn’t pass inspection?
P(PassC) = 1-0.75 = 0.25
Both cars pass inspection?
P(Pass)P(Pass) = (0.75)(0.75) = 0.5625
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7.4 Probability Rules
Example (continued): Car Inspections
You and a friend get your cars inspected. The event of your
car’s passing inspection is independent of your friend’s car. If
75% of cars pass inspection what is the probability that
At least one of two cars passes?
1 – (0.25)2 = 0.9375 OR
0.75 + 0.75 – 0.5625 = 0.9375
Neither car passes inspection?
1 – 0.9375 = 0.0625
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7.5 Joint Probability and Contingency Tables
Events may be placed in a contingency table such as the
one in the example below.
As part of a Pick Your Prize Promotion, a store invited
customers to choose which of three prizes they’d like to win.
The responses could be placed in the following contingency
table:
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7.5 Joint Probability and Contingency Tables
Marginal probability depends only on totals found in
the margins of the table.
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7.5 Joint Probability and Contingency Tables
In the table below, the probability that a respondent
chosen at random is a woman is a marginal probability.
P(woman) = 251/478 = 0.525.
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7.5 Joint Probability and Contingency Tables
Joint probabilities give the probability of two events occurring
together.
P(woman and camera) = 91/478 = 0.190.
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7.5 Joint Probability and Contingency Tables
Each row or column shows a conditional distribution given
one event.
In the table above, the probability that a selected customer
wants a bike given that we have selected a woman is:
P(bike|woman) = 30/251 = 0.120.
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7.6 Conditional Probability
In general, when we want the probability of an event from a
conditional distribution, we write P(B|A) and pronounce it “the
probability of B given A.”
A probability that takes into account a given condition is called a
conditional probability.
P( A and B)
P (B | A ) 
P ( A)
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7.6 Conditional Probability
Rule 7: The General Multiplication Rule
The General Multiplication Rule calculates the probability
that both of two events occurs. It does not require that the
events be independent.
P( A and B)  P( A)  P(B | A)
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7.6 Conditional Probability
What is the probability that a randomly selected customer
wants a bike if the customer selected is a woman?
P( A and B)
P (B | A ) 
 P(bike|woman)
P ( A)
P(bike and woman) 30 478


 0.12
P(woman)
251 478
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7.6 Conditional Probability
Are Prize preference and Sex independent? If so,
P(bike|woman) will be the same as P(bike). Are they equal?
P(bike|woman)= 30/251 = 0.12
P(bike) = 90/478 = 0.265
0.12 ≠ 0.265
Since the two probabilites are not equal, Prize preference and
Sex and not independent.
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7.6 Conditional Probability
Events A and B are independent whenever P(B|A) = P(B).
Independent vs. Disjoint
For all practical purposes, disjoint events cannot be
independent.
Don’t make the mistake of treating disjoint events as if they
were independent and applying the Multiplication Rule for
independent events.
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7.7 Constructing Contingency Tables
If you’re given probabilities without a contingency table, you
can often construct a simple table to correspond to the
probabilities and use this table to find other probabilities.
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7.7 Constructing Contingency Tables
A survey classified homes into two price categories (Low and
High). It also noted whether the houses had at least 2 bathrooms
or not (True or False). 56% of the houses had at least 2
bathrooms, 62% of the houses were Low priced, and 22% of the
houses were both. Translating the percentages to probabilities,
we have:
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7.7 Constructing Contingency Tables
The 0.56 and 0.62 are marginal probabilities, so they go in the
margins.
The 22% of houses that were both Low priced and had at least 2
bathrooms is a joint probability, so it belongs in the interior of the
table.
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7.7 Constructing Contingency Tables
Because the cells of the table show disjoint events, the
probabilities always add to the marginal totals going across rows
or down columns.
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7.7 Constructing Contingency Tables
Example: Online Banking
A national survey indicated that 30% of adults conduct their
banking online. It also found that 40% are under the age of 50,
and that 25% are under the age of 50 and conduct their banking
online.
What percentage of adults do not conduct their banking online?
What type of probability is the 25% mentioned above?
Construct a contingency table showing joint and marginal
probabilities.
What is the probability that an individual who is under the age of
50 conducts banking online?
Are Banking online and Age independent?
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7.7 Constructing Contingency Tables
Example (continued): Online Banking
A national survey indicated that 30% of adults conduct their
banking online. It also found that 40% are under the age of 50,
and that 25% are under the age of 50 and conduct their banking
online.
What percentage of adults do not conduct their banking online?
100% – 30% = 70%
What type of probability is the 25% mentioned above?
Marginal
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7.7 Constructing Contingency Tables
Example (continued): Online Banking
A national survey indicated that 30% of adults conduct their
banking online. It also found that 40% are under the age of 50,
and that 25% are under the age of 50 and conduct their banking
online.
Construct a contingency table showing joint and marginal
probabilities.
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7.7 Constructing Contingency Tables
Example (continued): Online Banking
A national survey indicated that 30% of adults conduct their
banking online. It also found that 40% are under the age of 50,
and that 25% are under the age of 50 and conduct their banking
online.
What is the probability that an individual who is under the age of
50 conducts banking online?
0.25/0.40 = 0.625
Are Banking online and Age independent?
No. P(banking online|under 50) = 0.625, which is not equal to
P(banking online) = 0.30.
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• Beware of probabilities that don’t add up to 1.
• Don’t add probabilities of events if they’re not disjoint.
• Don’t multiply probabilities of events if they’re not independent.
• Don’t confuse disjoint and independent.
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What Have We Learned?
Apply the facts about probability to determine whether an
assignment of probabilities is legitimate.
•
Probability is long-run relative frequency.
•
Individual probabilities must be between 0 and 1.
•
The sum of probabilities assigned to all outcomes must be
1.
Understand the Law of Large Numbers and that the common
understanding of the “Law of Averages” is false.
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What Have We Learned?
Know the rules of probability and how to apply them.
• The Complement Rule says that P( A)  1  P( AC ) .
• The Multiplication Rule for independent events says
P( A and B)  P( A)  P(B) that provided events A and B are
independent.
• The General Multiplication Rule says that
P( A and B)  P( A)  P(B | A)
• The Addition Rule for disjoint events says that
P( A or B)  P( A)  P(B) provided events A and B are disjoint.
• The General Addition Rule says that
.)
P( A or B)  P( A)  P(B)  P( A and B
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What Have We Learned?
Know how to construct and read a contingency table.
Know how to define and use independence.
• Events A and B are independent if P(B | A)  P(B)
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