Transcript Chapter 4, part 1: Probability

Learn to let go. That is the key to happiness.
~Jack Kornfield
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Probability
Section
4.1-4.5
Basic terms and rules
Conditional probability and
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
A random circumstance is one in which the
outcome is unpredictable. The outcome is
unknown until we observe it.
Eg. Toss a die

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Interpretations of Probability

Given a random circumstance, the probability of a
specific outcome can be interpreted as
1.
(classical interpretation) a % arising from the nature
of the circumstance.
(relative frequency interpretation) the proportion of
times this outcome will occur over a large number
of the same circumstances.
(personal interpretation) what a person believes.
2.
3.
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Two Ways to Determine Probability
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
Making an assumption about the physical
world and use it to find the probabilities.

Repeating the same circumstances many
times and calculating the relative
frequencies.

Based on the person’s experiences.
Example 1: Coin Flipping
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
What is the probability that a flipped coin
shows heads up?

Simulation in Minitab (random digit table and
binomial way)
Example 2: Traffic Jam on the I-880

< = 30 mph  traffic jam
How often will a driver encounter traffic jam
on the I-880 during 7-9 am in weekdays?
Ans: __ out of 100 times.

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Terms

Sample space SS: the collection of all
possible outcomes of a random circumstance

An event is a collection of one or more
outcomes in the sample space.

A simple event is an event of one outcome.
** what are the random circumstances of the examples?
** what are the outcomes/sample space/event of the examples?
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Probability Models
A probability model assigns a value to each
outcome which satisfies the following properties:
The probability of an outcome must be
between 0 and 1
The
sum of the probabilities over all possible
outcomes must be 1 (i.e. P(SS) =1)
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Probability of an Event

The sum of the probabilities of outcomes in
the event
** Revisit the examples: 1) build up probability
models and 2) find probabilities of events
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Probability Models
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
A probability model with a finite sample space is
called discrete

A continuous probability model assigns probabilities
as areas under a density curve. The area under the
curve and above any range of values is the
probability of an outcome in that range
Example: randomly pick a number between 0 and 1
Equally Likely Probability Model

If the sample space S is finite in number and
the outcomes have the same likelihood of
occurrence, then each outcome has
probability equal to 1 divided by the number
of possible outcomes and so
# in A
P ( A) 
.
# in S
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Random Variables

A random variable is a function assigning a real value
to an outcome of a RC
The distribution of a random variable X tells us what
values X can take and how to assign probabilities to
those values
Example:
1. # of dots (RC: rolling a die)
2. height of a student (RC: randomly pick from the class)

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Basic Event Relations
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
Mutually exclusive events
Two events A and B are called mutually
exclusive if the occurrence of one excludes
the occurrence of the other.

Complement events
The complement of an event A is the event
that A does not occur, denoted as A.
Basic Probability Laws



The union of 2 events A and B, A  B , is
the event when either A or B or both
occur.
The intersection of A and B, A  B, is the
event when A and B both occur.
Venn diagram shows us that
P( A  B)  P( A)  P( B)  P( A  B)
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Operation Laws
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Conditional Probability
Base on a survey of 1000 government employees:
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Conditional Probability
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1.
If an employee is selected randomly (out of
the 1000 surveyed), what is the probability
that the selected one is a male employee?
2.
If a male employee is selected, what is the
probability that he is also married? (called the
conditional probability of selecting a married
employee given that the selected one is male.)
Conditional Probability
P(married| male)=
# of married male employees
# of male employees
i.e. proportion of married male employees
proportion of male employees
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Conditional Probability

In general, the conditional probability of
event B given event A is
P( A  B)
P( A | B) 
P( B)
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Independent Events
Event A is called independent of event B if
the knowledge that B has occurred
DOES NOT
change the probability of the occurrence of A,
i.e. P(A|B) = P(A).


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Does P(B|A) = P(B)?
Independent Events


Events A and B are independent events if and
only if, P(A|B) = P(A) or P(B|A) = P(B).
Otherwise, A and B are dependent.
A and B are independent if and only if,
P( A  B)  P( A) P( B).

A1, A2, …, Ak are independent if and only if,
P( A1  A2  ... Ak )  P( A1) P( A2)...P( Ak ).
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Independent Events


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An event cannot be both mutually exclusive
and independent (unless it is trivial i.e.
probability 0):
If events are independent, then they cannot
be mutually exclusive.
If events are mutually exclusive, they cannot
be independent.
Independent Events
Example 1:
A red die and a white die are rolled. Define the
events:
A= 4 on red die; B= sum of two dice is odd.
Show that A and B are independent.
Example 2:
Given that P(grade A in 6204)= .60; P(grade A in
6304)= .60; P(grade A in both) =.36. Are A, B
independent?
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Diagnostic Tests




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A diagnostic testing or screening is the
application of a test to individuals who have not
yet exhibited any clinical symptoms in order to
classify them with respect to their probability of
having a particular disease.
“sensitivity” is the true + rate
“specificity” is the true - rate
“prevalence” is the proportion of subjects with
the disease in a population
Diagnostic Tests
Consider a common pregnancy test
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Truth\Test result Positive
Negative
Pregnancy
Correct
False Negative
No pregnancy
False Positive
Correct
Diagnostic Tests
Eg. Pregnancy tests
Sensitivity
False positive rate
Specificity
False negative rate
=
=
=
=
P( + | pregnancy)
P( + | non-preg)
P( - | non-preg)
P( - | pregnancy)
Q: What is the probability that a woman with a positive
result is actually NOT pregnant?
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Bayes’ Rule
P( A | B) 
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P( A  B)
P( B)
P( A  B)
P( B | A) P ( A)  P ( B | A) P( A)
Example: Pap Smear
1,000,000
women
Cervical
cancer
83
Test +
70
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No cervical
cancer
999,917
Test –
13
Test +
186,385
Test –
813,532
Bayes’ Theorem
P( Ai  B)
P( Ai | B) 
P( B | A1) P( A1)  ...  P( B | Ak ) P( Ak )
Think of the events A1, A2,…, Ak as representing all possible
conditions that can produce the observable “effect” B. In this
context, the probabilities P(Ai)’s are called prior probabilities. Now
suppose that the effect B is observed to occur. Bayes’ theorem
gives a way to calculate the probability that B was produced or
caused by the particular condition Ai than by any of the other
conditions. The conditional probability P(Ai|B) is called the posterior
probability of Ai .
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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
(building a probability model if possible)
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
Example 4.3:
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A book club classifies members as heavy,
medium, or light purchasers, and separate
mailings are prepared for each of these
groups. Overall, 20% of the members are
heavy purchasers, 30% medium, and 50%
light.
Example 4.3:

The following % are obtained from existing
records of individuals classified as heavy,
medium, or light purchasers:
1st 3 months’
purchases
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Group (%)
Heavy
Medium
Light
0
5
15
60
1
10
30
20
2
30
40
15
3+
55
15
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Example 4.3:


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If a member is a heavy purchaser, what is
the probability he/she will buy 2 books in the
first 3 months?
What percent of members will buy 2 books in
the first 3 months?
If a member purchases 2 books in the first 3
months, what is the probability that he/she is
a light purchaser?
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
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.
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?