Download Probability

3-1
Chapter Three
Probability
3-2
McGraw-Hill/Irwin
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Probability
3.1
3.2
3.3
3.4
*3.5
3-3
The Concept of Probability
Sample Spaces and Events
Some Elementary Probability Rules
Conditional Probability and Independence
Bayes’ Theorem
3.1 Probability Concepts
An experiment is any process of observation with
an uncertain outcome.
The possible outcomes for an experiment are called
the experimental outcomes.
Probability is a measure of the chance that an
experimental outcome will occur when an
experiment is carried out
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Probability
If E is an experimental outcome, then P(E) denotes the
probability that E will occur and
Conditions
0  P( E )  1
If E can never occur, then P(E) = 0
If E is certain to occur, then P(E) = 1
The probabilities of all the experimental outcomes must
sum to 1.
Interpretation: long-run relative frequency or
subjective
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3.2 The Sample Space
The sample space of an experiment is the set of all
experimental outcomes.
Example 3.2: Genders of Two Children
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Computing Probabilities of Events
An event is a set (or collection) of experimental
outcomes.
The probability of an event is the sum of the
probabilities of the experimental outcomes that
belong to the event.
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Example: Computing Probabilities
Example 3.4: Genders of Two Children
Events
P(one boy and one girl) =
P(BG) + P(GB) = ¼ + ¼ = ½
P(at least one girl) =
P(BG) + P(GB) + P(GG) = ¼ + ¼ + ¼ = ¾
Note: Experimental Outcomes: BB, BG, GB, GG
All outcomes equally likely: P(BB) = … = P(GG) = ¼
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Probabilities: Equally Likely Outcomes
If the sample space outcomes (or experimental
outcomes) are all equally likely, then the
probability that an event will occur is equal to the
ratio
the number of sample space outcomes that correspond to the event
The total number of sample space outcomes
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Example: AccuRatings Case
Of 5528 residents
sampled, 445 prefer
KPWR.
Estimated Share:
P(KPWR) = 445/5528
= 0.0805
Assuming 8,300,000 Los Angeles residents aged 12 or older:
Listeners = Population x Share = 8,300,000 x 0.08 = 668,100
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3.3 Event Relations
The complement A of an event A is the set of all
sample space outcomes not in A.
Further, P(A) = 1 - P(A)
Union of A and B, A  B
Elementary events that belong to
either A or B (or both.)
Intersection of A and B, A  B
Elementary events that belong to
both A and B.
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The Addition Rule for Unions
The probability that A or B (the union of A and B) will
occur is
P(A  B) = P(A) + P(B) - P(A  B)
A and B are mutually exclusive if they have no sample
space outcomes in common, or equivalently if
P(A  B) = 0
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3.4 Conditional Probability
The probability of an event A, given that the event B has
occurred is called the “conditional probability of A
given B” and is denoted as P(A | B). Further,
P(A | B) =
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P(A  B)
P(B)
Independence of Events
Two events A and B are said to be independent if
and only if:
P(A|B) = P(A) or, equivalently,
P(B|A) = P(B)
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The Multiplication Rule for Intersections
The probability that A and B (the intersection of
A and B) will occur is
P(A  B) = P(A) P(B | A)
= P(B) P(A | B)
If A and B are independent, then the probability
that A and B (the intersection of A and B) will
occur is P(A  B) = P(A) P(B)  P(B) P(A)
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Contingency Tables
P(R1 )
P(R1  C1 )
R1
R2
Total
P(R 2  C2 )
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C1
.4
.1
.5
C2
.2
.3
.5
Total
.6
.4
1.00
P(C 2 )
Example: AccuRatings Case
Example 3.16: Estimating Radio Station Share by
Daypart
5528 L.A. residents sampled.
2827 of residents sampled
listen during some portion of
the 6-10 a.m. daypart.
Of those, 201 prefer KIIS.
KIIS Share for 6-10 a.m. daypart:
P(KIIS|6-10 a.m.) = P(KIIS  6-10 a.m.) / P(6-10 a.m.)
= (201/5528)  (2827/5528) = 201/2827 = 0.0711
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Bayes’ Theorem
S1, S2, …, Sk represents k mutually exclusive possible states of nature,
one of which must be true.
P(S1), P(S2), …, P(Sk) represents the prior probabilities of the k
possible states of nature.
If E is a particular outcome of an experiment designed to determine
which is the true state of nature, then the posterior (or revised)
probability of a state Si, given the experimental outcome E, is:
P(Si  E)
P(Si|E) =
P(E)
P(Si )P(E|S i )

P(E)
P(Si )P(E|S i )

P(S1 )P(E|S1 )+P(S 2 )P(E|S 2 )+ ...+P(Sk )P(E|S k )
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3.5 Bayes’ Theorem:
An Example, AIDS Testing
Question:
Answer:
Suppose that a person selected randomly for testing, tests
positive for AIDS. The test is known to be highly accurate
(99.9% for people who have AIDS, 99% for people who
don’t.) What is the probability that the person actually has
AIDS?
Surprisingly, much lower than most of us would guess!
The Facts :
AIDS Incidence Rate : 6 cases per 1000 Americans
P(AIDS)  0.006
P( AIDS )  0.994
Testing Accuracy :
P(POS|AIDS )  0.999
P(POS|AIDS )  0.01
Solution : P(AIDS|POS )
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An Example, AIDS Testing (Continued)
P(AIDS|POS ) 
P(AIDS  POS)
P(AIDS  POS)  P( AIDS  POS)

P(AIDS)P(P OS|AIDS)
P(AIDS)P(P OS|AIDS)  P( AIDS )P(POS| AIDS )

( 0.006 )( 0.999 )
0.005994

( 0.006 )( 0.999 )  ( 0.994 )( 0.01 ) 0.005994  0.00994

0.005994
0.015934
 0.3762
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P(AIDS  POS)

P(POS)
( Bayes ' Theorem)
Probability
Summary:
3.1
3.2
3.3
3.4
*3.5
3-21
The Concept of Probability
Sample Spaces and Events
Some Elementary Probability Rules
Conditional Probability and Independence
Bayes’ Theorem