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Chapter 14
From Randomness to
Probability
Copyright © 2009 Pearson Education, Inc.
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.
In general, each occasion upon which we observe a random
phenomenon is called a trial.
At each trial, we note the value of the random phenomenon, and
call it an outcome.
When we combine outcomes, the resulting combination is an
event.
The collection of all possible outcomes is called the
sample space.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 2
The Law of Large Numbers
First a definition . . .
 When thinking about what happens with
combinations of outcomes, things are simplified if
the individual trials are independent.
 Roughly speaking, this means that the outcome
of one trial doesn’t influence or change the
outcome of another.
 For example, coin flips are independent.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 3
The Law of Large Numbers (cont.)
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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.
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We call the single value the probability of the
event.
Because this definition is based on repeatedly
observing the event’s outcome, this definition of
probability is often called empirical probability.
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Copyright © 2009 Pearson Education, Inc.
Slide 1- 4
Foundation of Probability
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The onset of probability as a useful science is
primarily attributed to Blaise Pascal (1623-1662) and
Pierre de Fermat (1601-1665). While contemplating
a gambling problem posed by Chevalier de Mere in
1654, Blaise Pascal and Pierre de Fermat laid the
fundamental groundwork of probability theory, and
are thereby accredited the fathers of probability.
Chances of a Lifetime
Copyright © 2009 Pearson Education, Inc.
Slide 1- 5
Modeling Probability (cont.)
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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
Copyright © 2009 Pearson Education, Inc.
Slide 1- 6
The First Three Rules of Working with
Probability (MAKE A PICTURE)
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The most common kind of picture to make is
called a Venn diagram.
We will see Venn diagrams in practice shortly…
Copyright © 2009 Pearson Education, Inc.
Slide 1- 7
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.
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.)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 8
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)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 9
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).
Copyright © 2009 Pearson Education, Inc.
Slide 1- 10
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 or B) = P(A) + P(B), provided that A and
B are disjoint.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 11
Formal Probability
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 and B) = P(A) x P(B), provided that A
and B are independent.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 12
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:
Copyright © 2009 Pearson Education, Inc.
Slide 1- 13
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.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 14
Formal Probability - Notation
Notation:
 In this text we use the notation P(A or B) and
P(A and B).
 In other situations, you might see the following:
 P(A  B) instead of P(A or B)
 P(A  B) instead of P(A and B)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 15
Example #1
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A survey of 64 informed voters revealed the
following information: 45 believe that Elvis is still
alive 49 believe that they have been abducted by
space aliens 42 believe both of these things
Copyright © 2009 Pearson Education, Inc.
Slide 1- 16
Example #2
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A survey of 88 faculty and graduate students at the
University of Florida's film school revealed the
following information:
51 admire Moe
49 admire Larry
60 admire Curly
34 admire Moe and Larry
32 admire Larry and Curly
36 admire Moe and Curly
24 admire all three of the Stooges
Copyright © 2009 Pearson Education, Inc.
Slide 1- 17
Chapter 15
Probability Rules!
Copyright © 2009 Pearson Education, Inc.
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 or 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.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 19
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General Addition Rule:
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For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B)
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(On the Formula Sheet)
The following Venn diagram
shows a situation in which we
would use the
general addition rule:
Copyright © 2009 Pearson Education, Inc.
Slide 1- 20
Conditional Probabilities
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To find the probability of the event B given the
event A, we restrict our attention to the outcomes
in A. We then find in what fraction of those
outcomes B also occurred.
P
(A
and
B)
P(B|A) 
P(A)
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(On the Formula Sheet)
Note: P(A) cannot equal 0, since we know that A
has occurred.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 21
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:
P(A and B) = P(A) x P(B)
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However, when our events are not
independent, this earlier multiplication rule
does not work. Thus, we need the General
Multiplication Rule.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 22
The General Multiplication Rule (cont.)
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We encountered the general multiplication rule in
the form of conditional probability.
Rearranging the equation in the definition for
conditional probability, we get the General
Multiplication Rule:
 For any two events A and B,
P(A and B) = P(A) x P(B|A)
or
P(A and B) = P(B) x P(A|B)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 23
Independence
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Independence of two events means that the
outcome of one event does not influence the
probability of the other.
With our new notation for conditional probabilities,
we can now formalize this definition:
 Events A and B are independent whenever
P(B|A) = P(B).
 (Equivalently, events A and B are independent
whenever P(A|B) = P(A).)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 24
Independent ≠ Disjoint
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Disjoint events cannot be independent! Well, why not?
 Since we know that disjoint events have no outcomes in
common, knowing that one occurred means the other
didn’t.
 Thus, the probability of the second occurring changed
based on our knowledge that the first occurred.
 It follows, then, that the two events are not 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.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 25
Drawing Without Replacement
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Sampling without replacement means that once one
object 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.
Drawing without replacement is just another instance of
working with conditional probabilities.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 26
Reversing the Conditioning
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Reversing the conditioning of two events is rarely
intuitive.
Suppose we want to know P(A|B), but we know only
P(A), P(B), and P(B|A).
We also know P(A and B), since
P(A and B) = P(A) x P(B|A)
From this information, we can find P(A|B):
P
(A
and
B)
P(A|B) 
P(B)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 27
Chapter 16
Random Variables
Copyright © 2009 Pearson Education, Inc.
Expected Value: Center
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A random variable assumes a value based on the
outcome of a random event.
 We use a capital letter, like X, to denote a
random variable.
 A particular value of a random variable will be
denoted with a lower case letter, in this case x.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 29
Expected Value: Center (cont.)
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There are two types of random variables:
 Discrete random variables can take one of a
finite number of distinct outcomes.
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Example: Number of credit hours
Continuous random variables can take any
numeric value within a range of values.
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Example: Cost of books this term
Copyright © 2009 Pearson Education, Inc.
Slide 1- 30
Expected Value: Center (cont.)
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A probability model for a random variable
consists of:
 The collection of all possible values of a
random variable, and
 the probabilities that the values occur.
Of particular interest is the value we expect a
random variable to take on, notated μ (for
population mean) or E(X) for expected value.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 31
Expected Value: Center (cont.)
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The expected value of a (discrete) random
variable can be found by summing the products
of each possible value and the probability that it
occurs:
  E  X    x  P  x
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Note: Be sure that every possible outcome is
included in the sum and verify that you have a
valid probability model to start with.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 32
First Center, Now Spread…
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For data, we calculated the standard deviation by
first computing the deviation from the mean and
squaring it. We do that with discrete random
variables as well.
The variance for a random variable is:
  Var  X     x     P  x 
2
2
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The standard deviation for a random variable is:
  SD  X   Var  X 
Copyright © 2009 Pearson Education, Inc.
Slide 1- 33
Continuous Random Variables
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Random variables that can take on any value in a
range of values are called continuous random
variables.
Continuous random variables have means
(expected values) and variances.
We won’t worry about how to calculate these
means and variances in this course, but we can
still work with models for continuous random
variables when we’re given these parameters.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 34