Transcript Statistics for Managers Using Microsoft Excel, 3/e

Business Statistics:
A First Course
4th Edition
Chapter 4
Basic Probability
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Chapter Topics

Basic probability concepts


Conditional probability


Sample spaces and events, simple
probability, joint probability
Statistical independence, marginal
probability
Bayes’s Theorem
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Terminology




Experiment- Process of Observation
Outcome-Result of an Experiment
Sample Space- All Possible Outcomes
of a Given Experiment
Event- A Subset of a Sample Space
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Sample Spaces

Collection of all possible outcomes

e.g.: All six faces of a die:

e.g.: All 52 cards in a deck:
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Events

Simple event
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Outcome from a sample space with
one characteristic
e.g.: A red card from a deck of cards
Joint event
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Involves two outcomes
simultaneously
e.g.: An ace that is also red from a
deck of cards
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Visualizing Events

Contingency Tables
Ace

Total
Black
Red
2
2
24
24
26
26
Total
4
48
52
Tree Diagrams
Full
Deck
of Cards
Not Ace
Ace
Red
Cards
Black
Cards
Not an Ace
Ace
Not an Ace
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Special Events

Null Event
Impossible event

e.g.: Club & diamond on one card 
draw

Complement of event
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For event A, all events not in A
Denoted as A’
e.g.: A: queen of diamonds
A’: all cards in a deck that are
not queen of diamonds
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Contingency Table
A Deck of 52 Cards
Red Ace
Ace
Not an
Ace
Total
Red
2
24
26
Black
2
24
26
Total
4
48
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Sample Space
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Tree Diagram
Event Possibilities
Full
Deck
of Cards
Red
Cards
Ace
Not an Ace
Ace
Black
Cards
Not an Ace
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Probability
Certain



Probability is the numerical 1
measure of the likelihood
that an event will occur
Value is between 0 and 1
.5
Sum of the probabilities of
all mutually exclusive and
collective exhaustive events
is 1
0
Impossible
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Types of Probability
•Classical (a priori) Probability
P (Jack) = 4/52
•Empirical (Relative Frequency) Probability
Probability it will rain today = 60%
•Subjective Probability
Probability that new product will be successful
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Computing Probabilities

The probability of an event E:
number of event outcomes
P(E)=
total number of possible outcomes in sample space
n(E)
=
e.g. P(
) = 2/36
n(S)
(There are 2 ways to get one 6 and the other 4)

Each of the outcomes in the sample
space is equally likely to occur
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Probability Rules
1
0 ≤ P(E) ≤ 1
Probability of any event must be between 0
and1
2
P(S) = 1 ; P(Ǿ) = 0
Probability that an event in the sample space
will occur is 1; the probability that an event
that is not in the sample space will occur is 0
3
P (E) = 1 – P(E)
Probability that event E will not occur is 1
minus the probability that it will occur
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Rules of Addition
4
5.
Special Rule of Addition
P (AuB) = P(A) + P(B) if and only if A
and B are mutually exclusive events
General Rule of Addition
P (AuB) = P(A) + P(B) – P(AnB)
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Rules Of Multiplication
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Special Rule of Multiplication
P (AnB) = P(A) x P(B) if and only if A
and B are statistically independent
events
General Rule of Multiplication
P (AnB) = P(A) x P(B/A)
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Conditional Probability Rule
Conditional Probability Rule
P(B/A) = P (AnB)/ P(A)
This is a rewrite of the formula for the
general rule of multiplication.
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Bayes Theorem
P(B1) = probability that Bill fills prescription = .20
P(B2) = probability that Mary fills prescription = .80
P(A B1) = probability mistake Bill fills prescription = 0.10
P(A B2) = probability mistake Mary fills prescription = 0.0
What is the probability that Bill filled a
prescription that contained a mistake?
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Bayes’s Theorem
P  A | Bi  P  Bi 
P  Bi | A  =
P  A | B1  P  B1       P  A | Bk  P  Bk 
P  Bi and A 
=
P  A
Adding up
Same
Event
the parts
of A in all
the B’s
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Bayes Theorem (cont.)
(.20) (.10)
P(B1 A) =
(.20) (.10) + (.80) (.01)
.02
=
= .71 = 71%
.028
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Bayes Theorem (cont.)
(Prior)
Bi
Bill fills prescription
(Conditional)
A Bi
(Joint)
A  Bi
.20
.10
.020
Mary fills prescription. 80
.01
.008
1.00
P(A) =.028
(Posterior)
Bayes
.02/.028=.71
.008/.028=.29
1.00
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Chapter Summary

Discussed basic probability concepts


Defined conditional probability


Sample spaces and events, simple
probability, and joint probability
Statistical independence, marginal
probability
Discussed Bayes’s theorem
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