Chapter 6
Probability
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Dealing with Random Phenomena
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A random phenomenon is a situation in which we know
what outcomes could happen, but we don’t know which
particular outcome did or will happen.
Each occasion upon which we observe a random
phenomenon is called a trial. Example: flipping a coin.
At each trial, we note the value of the random
phenomenon, and call it an outcome. Example: tails
When we combine outcomes, the resulting combination is
an event. Example: Flipping a coin twice and getting both
tails.
The collection of all possible outcomes is called the
sample space. Example: List all the possible outcomes of
flipping two coins.
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Independent Events - the outcome of one trial
doesn’t influence or change the outcome of
another.
 Example: Are the following independent
events?
a. coin flips
b. Rolling a die or a pair of dice.
c. Having an IPod and have an ITunes gift card.
d. Your grade in AP statistics and your grade in
trigonometry.
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The Law of Large Numbers
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The Law of Large Numbers (LLN) says that the
long-run relative frequency of repeated
independent events gets closer and closer to a
single value.
We call the single value the probability of the
event.
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Modeling Probability
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The probability of an event is the number of
outcomes in the event divided by the total number
of possible outcomes.
P(A) =
# of outcomes in A
# of possible outcomes
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Formal Probability
1. Two requirements for a probability:
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A probability is a number between 0 and 1.
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For any event A, 0 ≤ P(A) ≤ 1.
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Formal Probability (cont.)
2. Probability Assignment Rule:
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The probability of the set of all possible
outcomes of a trial must be 1.
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P(S) = 1 (S represents the set of all possible
outcomes.)
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Formal Probability (cont.)
3. Complement Rule:
 The set of outcomes that are not in the event
A is called the complement of A, denoted AC.
 The probability of an event occurring is 1
minus the probability that it doesn’t occur:
P(A) = 1 – P(AC)
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Example: When we arrive at the intersection of
Altama and Community Rd. the probability the
light is green is about 35% of the time. If
P(green) = .35, what is the probability the light
isn’t green when you get to Altama and
Community?
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Formal Probability (cont.)
4. Addition Rule:
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Events that have no outcomes in common
(and, thus, cannot occur together) are called
disjoint (or mutually exclusive).
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Formal Probability (cont.)
4. Addition Rule (cont.):
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For two disjoint events A and B, the
probability that one or the other occurs is
the sum of the probabilities of the two
events.
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P(A  B) = P(A) + P(B), provided that A
and B are disjoint.
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Example: Knowing the P(green) = .35 and P(yellow) =
.04, when we get to the light at the corner of Altama and
Community:
a. What is the probability the light is green or yellow?
Written P(green  yellow).
b. When you get to the light are the events of the light
being green and yellow disjoint or independent?
c. What is the probability the light is red?
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Formal Probability (cont.)
5. Multiplication Rule:
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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.
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P(A  B) = P(A)  P(B), provided that A and
B are independent.
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Example: Opinion polling organizations contact their
respondents by telephone. In 1990s this method could
reach about 60% of US households. By 2003, the contact
rate had risen to 76%. We can reasonably assume each
household’s response to be independent of the others.
What is the probability that …
a. The interviewer successfully contacts the next
household on the list?
b. The interviewer successfully contacts both of the next
two households on the list?
c. The interviewer’s first successful contact is the third
household on the list?
d. The interviewer makes at least one successful contact
among the next five households on the list?
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Formal Probability (cont.)
5. Multiplication Rule (cont.):
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Two independent events A and B are not
disjoint, provided the two events have
probabilities greater than zero:
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Formal Probability (cont.)
5. Multiplication Rule:
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Many Statistics methods require an
Independence Assumption, but assuming
independence doesn’t make it true.
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Always Think about whether that assumption
is reasonable before using the Multiplication
Rule.
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Formal Probability - Notation
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In this text we use the notation P(A  B) and
P(A  B).
In other situations, you might see the following:
 P(A or B) instead of P(A  B)
 P(A and B) instead of P(A  B)
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The General Addition Rule
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When two events A and B are disjoint, we can
use the addition rule for disjoint events from
Chapter 14:
P(A  B) = P(A) + P(B)
However, when our events are not disjoint, this
earlier addition rule will double count the
probability of both A and B occurring. Thus, we
need the General Addition Rule.
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The General Addition Rule (cont.)
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General Addition Rule:
 For any two events A and B,
P(A  B) = P(A) + P(B) – P(A  B)
The following Venn diagram shows a situation in
which we would use the general addition rule:
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Example: A survey of college students found that
56% live in a campus residence hall, 62%
participate in a campus meal program and 42%
do both. What is the probability that a randomly
selected student either lives or eats on campus?
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Example: Draw a Venn Diagram for the previous
example. What is the probability that a randomly
selected student:
a. Lives off campus and doesn’t have a meal
program?
b. Lives in a residence hall but doesn’t have a
meal program?
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Example: Draw a Venn Diagram for the previous
example. What is the probability that a randomly
selected student:
a. Lives off campus and doesn’t have a meal
program?
b. Lives in a residence hall but doesn’t have a
meal program?
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Disjoint
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No two events have any outcomes in common,
then you can add the individual event probabilities
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Independent
 Knowing one event’s probability does not
change the probability that another event
occurs, then you can multiply the probabilities
 Flipping a coin
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Independent ≠ Disjoint
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Give an example of disjoint events:
Give an example of independent events:
Disjoint events cannot be independent.
A common error is to treat disjoint events as if
they were independent, and apply the
Multiplication Rule for independent events—don’t
make that mistake.
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Example: It is reported that typically as few as 10% of random
phone calls result in a completed interview. Reasons are
varied, but some of the most common include no answer,
refusal to cooperate and failure to complete the call. Which of
the following events are independent, disjoint or neither
independent nor disjoint?
a. A = Your telephone number is randomly selected.
B = You’re not at home at dinner time when they call.
b. A = As a selected subject, you complete the interview.
B = As a selected subject, you refuse to cooperate.
c. A = You are not at home when they call at 11 am.
B = You are employed full time.
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Example: It is reported that typically as few as 10% of random
phone calls result in a completed interview. Reasons are
varied, but some of the most common include no answer,
refusal to cooperate and failure to complete the call. Which of
the following events are independent, disjoint or neither
independent nor disjoint?
a. A = Your telephone number is randomly selected.
B = You’re not at home at dinner time when they call.
b. A = As a selected subject, you complete the interview.
B = As a selected subject, you refuse to cooperate.
c. A = You are not at home when they call at 11 am.
B = You are employed full time.
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Conditional Probability
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The probability of an event under the
condition that some preceding event
has occurred
 P(A l B)=P(A and B)
P(B)
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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 has already occurred.”
A probability that takes into account a given condition is
called a conditional probability.
P(B|A) P(A  B)
P(A)
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The General Multiplication Rule
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When two events A and B are independent, we
can use the multiplication rule for independent
events from Chapter 14:
P(A  B) = P(A) x P(B)
However, when our events are not independent,
this earlier multiplication rule does not work.
Thus, we need the General Multiplication Rule.
P(A  B) = P(A)  P(B|A)
or
P(A  B) = P(B)  P(A|B)
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Conditional Probability
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You toss two coins. What is the probability that you toss two
heads given that you have tossed at least 1 head?
 P(A) = The two coins come up heads
 P(B) = There is a least one head
 Different outcomes of two coins?
 (H,H)-(T,H)-(H,T)-(T,T)
 P(B)=3/4
 P(A and B) = ¼
 P(A l B) = P(A and B)/P(B)
 P(A l B) = (¼)/(3/4)
 P(A l B) = 1/3
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Conditional Probability
A neighborhood lets families have two pets.
They can have two dogs, two cats, or one
of each. What is the probability that the
family will have exactly 2 cats if the second
pet is a cat?
 P(A) = Two cats = 1/3
 P(B) = At least one cat=2/3
 P (A l B) = (1/3)/(2/3)=1/2
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Conditional Probability
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Two number cubes are tossed. Find the probability
that the numbers showing on the cubes match
given that their sum is greater than five.
P(A) = Cubes Match
 P(A) = 1/6
P(B) = Sum is greater than 5
 P(B)=26/36
P(A and B) = 4/36
P(A l B) = (4/36)/(26/36)
P(A l B) = 2/13
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Conditional Probability
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One card is drawn from a standard deck of
cards. What is the probability that it is a
queen given that it is a face card?
 P(A) = Queen = 4/52
 P(B) =Face card = 12/52
 P(A and B) = 4/52
 P(A l B) = (4/52)/(12/52)
 P(A l B) = 1/3
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Practice
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Try problems6.54-6.55 on page 369
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Drawing Without Replacement
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Sampling without replacement means that once
one individual is drawn it doesn’t go back into the
pool.
 We often sample without replacement, which
doesn’t matter too much when we are dealing
with a large population.
 However, when drawing from a small
population, we need to take note and adjust
probabilities accordingly.
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Example: You draw two cards from a deck and
place them in your hand. What is the probability
both cards are black face cards?
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Example: You draw two cards from a deck and
place them in your hand. What is the probability
both cards are black face cards?
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Tree Diagrams
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A tree diagram helps us think through conditional
probabilities by showing sequences of events as
paths that look like branches of a tree.
Making a tree diagram for situations with
conditional probabilities is consistent with our
“make a picture” mantra.
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Tree Diagrams (cont.)
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Figure 15.5 is an
example of a tree
diagram and shows how
we multiply the
probabilities of the
branches together.
All the final outcomes
are disjoint and must
add up to one.
We can add the final
probabilities to find
probabilities of
compound events.
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Reversing the Conditioning
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Reversing the conditioning of two events is rarely intuitive.
Suppose we want to know P(A|B), and we know only
P(A), P(B), and P(B|A).
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We also know P(A  B), since
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P(A  B) = P(A) x P(B|A)
From this information, we can find P(A|B):
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P(A|B) P(A  B)
P(B)
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Bayes’s Rule
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When we reverse the probability from the
conditional probability that you’re originally given,
you are actually using Bayes’s Rule.
P A | B P B 
P B | A  
C
C
P A | B P B   P A | B P B

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