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Statistics for Managers 4th Edition
Chapter 4Basic Probability
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Chapter Topics
Basic probability concepts Sample spaces and events, simple
probability, joint probability
Conditional 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 Outcome from a sample space with
one characteristic e.g.: A red card from a deck of cards
Joint event 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
Tree Diagrams
Red 2 24 26
Black 2 24 26
Total 4 48 52
Ace Not Ace Total
Full Deck of Cards
Red Cards
Black Cards
Not an Ace
Ace
Ace
Not an Ace
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Special Events
Impossible evente.g.: Club & diamond on one card
draw Complement of event
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
Null Event
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Contingency TableA Deck of 52 Cards
Ace Not anAce
Total
Red
Black
Total
2 24
2 24
26
26
4 48 52
Sample Space
Red Ace
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Full Deck of Cards
Tree Diagram
Event Possibilities
Red Cards
Black Cards
Ace
Not an Ace
Ace
Not an Ace
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Probability
Probability is the numerical measure of the likelihood that an event will occur
Value is between 0 and 1 Sum of the probabilities of
all mutually exclusive and collective exhaustive events is 1
Certain
Impossible
.5
1
0
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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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(There are 2 ways to get one 6 and the other 4)e.g. P( ) = 2/36
Computing Probabilities
The probability of an event E:
Each of the outcomes in the sample space is equally likely to occur
number of event outcomesP(E)=
total number of possible outcomes in sample space
n(E)
n(S)
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Probability Rules
1 0 ≤ P(E) ≤ 1 Probability of any event must be between 0
and12 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 (Union of Events)
4 Special Rule of AdditionP (AuB) = P(A) + P(B) if and only if A and B are mutually exclusive events
5. General Rule of AdditionP (AuB) = P(A) + P(B) – P(AnB)
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Rules Of Multiplication Intersection of Events)
6 Special Rule of MultiplicationP (AnB) = P(A) x P(B) if and only if A and B are statistically independent events
7 General Rule of MultiplicationP (AnB) = P(A) x P(B/A)
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Conditional Probability Rule
Conditional Probability RuleP(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 = 0.01
What is the probability that Bill filled a prescription that contained a mistake?
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Bayes’s Theorem
1 1
||
| |
and
i ii
k k
i
P A B P BP B A
P A B P B P A B P B
P B A
P A
Adding up the parts of A in all the B’s
Same Event
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Bayes Theorem (cont.)
P(B1 A) =(.20) (.10)
(.20) (.10) + (.80) (.01)===
.02
.028.71 71%
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Bayes Theorem (cont.)
Bill fills prescription
Mary fills prescription
(Prior)Bi
.20
. 80
1.00
(Joint)A Bi
.020
.008
P(A) =.028
(Posterior) Bayes
.02/.028=.71
.008/.028=.29
1.00
(Conditional)
.10
.01
A Bi
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Chapter Summary Discussed basic probability
concepts Sample spaces and events, simple
probability, and joint probability
Defined conditional probability Statistical independence, marginal
probability
Discussed Bayes’s theorem