CHAPTER
7
Probability Basics
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Overview of Chapter 7
In this lecture, we will review the basics of probability
theory. Since uncertainty is a typical aspect of
problems, rigorous and accurate problem solving
requires using probability theory (i.e., math and logic).
 Specifically, we want you to:
•
•
•
•
Understand probability concepts
Use probability to model simple situations
Interpret probability statements
Manipulate and analyze models.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
A Quick Note on Terminology
 Please don’t confuse the term “chance event” (or
just “event”) with the term “outcome.” We use the
term chance event to refer to something about
which a decision maker is uncertain. In turn, a
chance event has more than one possible outcome.
 Symbology:
• Chance Events are designated with boldface letters (A)
• Outcomes are designated with lightface letters (A) and
sometimes letters with subscripts (Ai)
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
A Little Probability Theory
 Probability theory uses math and logic to deal with
chance events, events that we are uncertain as to
their outcome.
 Chance events have more than one possible
outcome and we use probability theory to rigorously
deal with the chances associated with the different
outcomes.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
A Little Probability Theory
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
A Little Probability Theory
3. Total Probability Must Equal 1
• If we have a set of outcomes such that one (and only
one) of them has to occur, then the probability of the set
must sum to 100% and there is a 100% chance that one
(and only one) of the outcomes will occur.
• Thus, the set of outcomes is called collectively
exhaustive and mutually exclusive.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Venn Diagrams
Venn diagrams graphically represent probability.
The entire Venn diagram
is the set of all possible
outcomes (A1, A2, and C),
and the entire area of the
diagram is 1 or 100%.
There is a 100% chance
one the outcomes in the
diagram will occur.
The area of A1, e.g., is
the probability of A1
occurring, say, 10%. That
is, just as A1 is 10% of the
diagram’s area, so it has
a 10% chance of
happening.
A1 and A2 do not overlap since they are mutually
exclusive. They cannot both happen. Thus, their
probabilities are additive, just as A1, A2 and C must
add up to 100%.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
4. Conditional Probability
•
•
•
The probability of A happening when B has also
occurred
Stated as “the probability of A given B”
“The probability of A given B is the joint probability of
A and B divided by the probability of B.”
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
Conditional Probability
as a Venn Diagram—
the probability of a certain
stock price going up
given that the Dow Jones
went up (the diagonally
shaded area)
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
5. Independence
• The probability of outcome A occurring stays the same
no matter which outcome of B has occurred.
where
• Note that independence is a special instance of
conditional probability.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Some Additional Probability Rules
 Symmetry in independence: independence in one
direction implies independence in the other direction
If P(Ai Bj) = P(Ai), then P(Bj Ai) = P(Bj)
• This is also a special case of conditional probability.
 Independent chance events are not the same as
mutually exclusive outcomes.
 Two chance events being probabilistically
dependent does not imply a causal relationship.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
6. Complements
• One event is said to be the complement of another if the
two together are the case of all that is.
• “Every object in the universe is either a butterfly (B) or not a
butterfly (B). Thus, the probability of not being a butterfly
(p=0.9999999) is the complement of the probability of being a
butterfly (p=0.0000001).”
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
7. Total Probability of an Event
• A convenient way to calculate the probability of A: P(A)
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
Total Probability of Outcome A
The probability of
outcome A is
made up of the
probability of
outcome “A and
B” and the
probability of
outcome “A and
B.”
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
8. Bayes’ Theorem
• Lets you simultaneously relate P(A|B), the chance that
an event A happened given B, and P(B|A), the chance B
happened given that event A occurred
• Via the symmetry principle of conditional probability
• Lets you factor in false positives (concluding things don’t
exist when in fact they do) and false negatives
(concluding things do exist when they don’t)
• Type I error—incorrectly rejecting a true null hypothesis
• Type II error—incorrectly not rejecting a false null hypothesis
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas
Deriving Bayes’ theorem
Starting with the
symmetry formula
Rearranging it
algebraically
And replacing P(A) with the formula for total probability
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas - Example

P(A)
Total
P(B)
Total
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
More Probability Formulas – Example Continue
P(A)
P(B)
Total
Total
0.204
0.0064
0.2104
0.476
0.3136
0.7896
0.68
0.32
1.00
Uncertain Quantities (Random Variables)
 Probability distribution
• The set of probabilities associated with all possible
outcomes of an uncertain quantity
• The probabilities must add to 1 because the events are
collectively exhaustive.
 Two types:
• Discrete uncertain events and probability distributions
• Continuous uncertain events and probability
distributions
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Discrete Probability Distributions
A discrete uncertain quantity is one that can assume a
finite or countable number of possible values, e.g., the
numbers of cars in a parking lot.
Graph of Probability Mass Function
Graph of Cumulative Distribution Function
Expected Value: Discrete Case
 The probability-weighted average of an uncertain
quantity’s possible values
• Denoted as E(X) or µx
E(X) = x1p1 + x2p2 + x3p3 + . . . + xnpn.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Expected Value: Discrete Case

© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Variance and Standard Deviation
 Are two ways to describe how far numbers in a
probability distribution lie from the expected value
(e.g., the average or mean)
• That is, how far a set of numbers are spread out
• Variance is denoted as Var(X) or σ2x
Var(X) = E[(X-E(X))2]
• Standard deviation is denoted as σx and is the square
root of variance
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Variance and Standard Deviation

© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Continuous Probability Distributions
 A continuous uncertain quantity is one that can take
any value within a range, e.g., time, speed.
• The probability of any specific value of a continuous
variable equals zero.
Graph of Cumulative Distribution Function
Probability Density Functions
 A function for a continuous variable in which the
area under the curve within a specific interval
represents the probability that the uncertain quantity
will fall in that interval.
 Corresponds to the probability mass function of
discrete uncertain quantities
• Whereas height represents probability for discrete
distributions, it is area that represents probability for
continuous distributions.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Stochastic Dominance
In the continuous case, the same rules for stochastic
dominance applied as discussed in Chapter 4.
Investment B
stochastically
dominates
Investment A
because B is always
to the right of A.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Yearly Profit Probability
Probability Density Functions
Yearly Profit
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Expected Value, Variance, Standard Deviation: The
Continuous Case
 Continuous probability distributions also have
expected values, variances, and standard
deviations.
• Characteristics are defined by calculus (integral sign ∫
replaces the summation sign ∑ )
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example
 An oil company is considering two sites for an exploratory well.
Because of budget constraints, only one well can be drilled. Site 1 is
fairly risky, with substantial uncertainty about the amount of oil that
can be found. On the other hand, site 2 is fairly certain to produce a
low level of oil. The characteristics of the two sites are as follows:
Site 1: Cost to drill $ 100,000
Site 2: Cost to drill $ 200,000
Outcome
Payoff
Outcome
Payoff
Probability
Dry
-$100,000
Dry
-$200,000
0.2
Low producer
$150,000
Low producer
$50,000
0.8
High producer
$ 500,000
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)
 If the rock strata underlying Site 1 are characterized by having a “dome” structure,
the chances for finding oil are somewhat greater than if no dome structure exists.
The probability of dome structure is P(Dome)=0.6. the ,conditional probabilities of
finding oil at Site 1 are shown in the table . The decision tree is also shown .
Estimate site 1 probabilities, and determine the expected values and the variances
for each option
If Dome Structure Exists
Outcome
P(outcome\ Dome)
Dry
0.6
Low
0.25
High
0.15
If No Dome Structure Exists
Outcome
P(outcome\ No Dome)
Dry
0.85
Low
0.125
High
0.025
Oil Wildcatting Example (continue)

0.7
0.2
0.1
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)

© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)
 Risk Profiles for both sites
Although site 1 has higher EMV but it has higher risk; i.e., there is a 70%
chance of losing $100,000 if site one is selected, whereas site 2 has a chance
of only 20% to lose $200,000
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)
The probabilities can be presented in the following table:
Dome
No Dome
Total
Dry
0.36
0.34
0.7
Low
0.15
0.05
0.20
High
0.09
0.01
0.10
Sum
0.60
0.40
1.00
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)

© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Another Example
S=schizophrenia
A=atrophy (brain) by
CAT scans
(1) Can you do the sensitivity analysis for variable P(S) (= 0 to 1) with 0.1
interval?
(2) Plot the sensitivity graph.
(3) Make a comment on your finding.
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Oil Wildcatting Example (continue)
A small
change has
a high
impact on
the result up
to 40%
cases
(the curve is
steep)
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.
Summary
In this lecture, we have reviewed the basics of
probability theory. We covered:
 Eight of its fundamental rules and aspects
 Venn diagrams
 Discrete and continuous probability distributions
 Expected value, variance and standard deviation
Update project progress??
© 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website for classroom use.