Be humble in our attribute, be loving and varying in our
attitude, that is the way to live in heaven.
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Probability
Sample
space/event
Independence
Bayes rule
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This lady has lost 10
games in a row on
this slot machine.
Would you play this
slot machine or
another one?
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Random Circumstance
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A random circumstance is one in which the
outcome is unpredictable. The outcome is
unknown until we observe it.
Example 1: Traffic Jam on the Golden
Bridge

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< = 20 mph  traffic jam
How often will a driver encounter traffic jam
on the golden bridge during 7-9 am
weekdays? __ out of 100 times.
Definition of Probability
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Given a random circumstance, the probability
of a specific outcome as the proportion of
times it will occur over the long run, i.e. the
relative frequency of that particular outcome.
Example 2: Coin Flipping
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What is the probability that a flipped coin
shows heads up?
Two Ways to Determine Probability

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Making an assumption about the physical
world and use it to find relative frequencies
Observing outcomes over the long run
Estimating Probability
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Measuring a representative sample and
observing relative frequencies of the sample
that fall into various categories
Example 3: Employment

What is the probability that a CSUEB
graduate gets hired in 3 months after
graduation? Between 3 to 6 months? More
than 6 months?
** Based on a representative sample of 200
graduates, 150, 30, 20 of them got hired in 3,
3-6, and > 6 months, respectively.
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Probability Definitions

Sample space: the collection of all possible
outcomes of a random circumstance

A simple event is one outcome in the sample space.

An event is a collection of one or more simple events
(outcomes) in the sample space.
** what are the random circumstances of the examples?
** what are the sample space/simple event/event of the examples?
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Probability of a Simple Event
Between 0 and 1
 The sum of the probabilities over all possible
simple events is 1
** examples of probability of an “equally likely”
simple event
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Probability of an Event

The sum of the probabilities of simple events
in the event
** Revisit the examples and find the events &
their probabilities
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Complementary Events
One event is the complement of another
event if the two events do not contain any of
the same outcomes, and together they cover
the entire sample space.
** get hired in 3 months or not
** traffic jam or not
** heads or tails
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Mutually Exclusive Events

Two events are mutually exclusive if they do
not contain any of the same outcomes. They
are also called disjoint.
Q: Are an event and its complement mutually
exclusive?
** examples:
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Independent Events
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Two events are independent if the probability
that one event occurs stays the same, no
matter whether or not the other event occurs.
Conditional Probability
The conditional probability of the event B
given the event A, denoted P(B|A), is the
long-run relative frequency with which event
B occurs when circumstances are such that
event A also occurs.
** event A = age 21+; event B = female
** Ask students for age and find P(B|A)
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Basic Rules for Finding Probabilities
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P(A or B) = P(A) + P(B) - P(A and B)
P(A and B) = P(A)P(B|A)
** P(B|A) = P(A and B)/P(A)
** P(A and B) for disjoint events and for
independent events
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Steps for Finding Probabilities
1.
2.
3.
4.
5.
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Identify random circumstance
Identify the sample space
Assign whatever probabilities you know
Specify the event for which the probability is
wanted
Use the probabilities from step 3 and the
probability rules to find the probability of
interest
Tools for Finding Probabilities
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
When conditional or joint probabilities are
known for two events  Two-way tables

For a sequence of events, when conditional
probabilities for events based on previous
events are known  Tree diagrams
Strategies for Finding Complicated
Probabilities
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>2 independent events:
P(A1 and A2 … and Ak) = P(A1)P(A2)…P(Ak)

Bayes Rule:
P( A | B) 
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P( A and B)
P( B | A) P( A)  P( B | A ) P( A )
C
C
Example:
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People are classified into 8 types. For
instance, Type 1 is “Rationalist” and applies
to 15% of men and 8% of women. Type 2 is
“Teacher” and applies to 12% of men and
14% of women. Each person fits one and
only one type.
•What is the probability that a randomly selected
male is “Rationalist”? “Teacher”? Both?
•What is the probability that a randomly selected
female is not a “Teacher”?
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Suppose college roommates have a particularly hard time getting
along with each other if they are both “Rationalists.”
A college randomly assigns roommates of the same sex.
What proportion of male roommate pairs will have this problem?
What proportion of female roommate pairs will have this problem?
Assuming that half of college roommate pairs are male and half are female.
What proportion of all roommate pairs will have this problem?
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A psychologist has noticed that “Teachers” and “Rationalists” get along
particularly well with each other, and she thinks they tend to marry each other.
One of her colleagues disagrees and thinks that the “types” of spouses are
independent of each other.
•
If the “types” are independent, what is the probability that a randomly
selected married couple would consist of one “Rationalist” and one
“Teacher”?
•
In surveys of thousands of randomly selected married couples, she has
found that about 5% of them have one “Rationalist” and one “Teacher.”
Does this contradict her colleague’s theory that the types of spouses are
independent of each other?
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