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Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 41
Chapter 4
Basic Probability
Statistics for ManagersUsing Microsoft® Excel
4th Edition
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 42
Chapter Goals
After completing this chapter, you should be able to:
Explain basic probability concepts and definitions Use contingency tables to view a sample space Apply common rules of probability Compute conditional probabilities Determine whether events are statistically
independent Use Bayes’ Theorem for conditional probabilities
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 43
Important Terms
Probability – the chance that an uncertain event will occur (always between 0 and 1)
Event – Each possible type of occurrence or outcome
Simple Event – an event that can be described by a single characteristic
Sample Space – the collection of all possible events
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 44
Assessing Probability
There are three approaches to assessing the probability of un uncertain event:
1. a priori classical probability
2. empirical classical probability
3. subjective probability an individual judgment or opinion about the probability of occurrence
outcomeselementaryofnumbertotal
occurcaneventthewaysofnumber
T
Xoccurrenceofyprobabilit
observedoutcomesofnumbertotal
observedoutcomesfavorableofnumberoccurrenceofyprobabilit
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 45
Sample Space
The Sample Space is the collection of all possible events
e.g. All 6 faces of a die:
e.g. All 52 cards of a bridge deck:
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 46
Events
Simple event An outcome from a sample space with one
characteristic e.g., A red card from a deck of cards
Complement of an event A (denoted A’) All outcomes that are not part of event A e.g., All cards that are not diamonds
Joint event Involves two or more characteristics simultaneously e.g., An ace that is also red from a deck of cards
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 47
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 52 Cards
Red Card
Black Card
Not an Ace
Ace
Ace
Not an Ace
Sample Space
Sample Space2
24
2
24
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 48
Mutually Exclusive Events
Mutually exclusive events Events that cannot occur together
example:
A = queen of diamonds; B = queen of clubs
Events A and B are mutually exclusive
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 49
Collectively Exhaustive Events
Collectively exhaustive events One of the events must occur The set of events covers the entire sample space
example: A = aces; B = black cards;
C = diamonds; D = hearts
Events A, B, C and D are collectively exhaustive (but not mutually exclusive – an ace may also be a heart)
Events B, C and D are collectively exhaustive and also mutually exclusive
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 410
Probability
Probability is the numerical measure of the likelihood that an event will occur
The probability of any event must be between 0 and 1, inclusively
The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1
Certain
Impossible
.5
1
0
0 ≤ P(A) ≤ 1 For any event A
1P(C)P(B)P(A) If A, B, and C are mutually exclusive and collectively exhaustive
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 411
Computing Joint and Marginal Probabilities
The probability of a joint event, A and B:
Computing a marginal (or simple) probability:
Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events
outcomeselementaryofnumbertotal
BandAsatisfyingoutcomesofnumber)BandA(P
)BdanP(A)BandP(A)BandP(AP(A) k21
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 412
Joint Probability Example
P(Red and Ace)
BlackColor
Type Red Total
Ace 2 2 4
NonAce 24 24 48
Total 26 26 52
52
2
cards of number total
ace and red are that cards of number
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 413
Marginal Probability Example
P(Ace)
BlackColor
Type Red Total
Ace 2 2 4
NonAce 24 24 48
Total 26 26 52
52
4
52
2
52
2)BlackandAce(P)dReandAce(P
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 414
P(A1 and B2) P(A1)
TotalEvent
Joint Probabilities Using Contingency Table
P(A2 and B1)
P(A1 and B1)
Event
Total 1
Joint Probabilities Marginal (Simple) Probabilities
A1
A2
B1 B2
P(B1) P(B2)
P(A2 and B2) P(A2)
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 415
General Addition Rule
P(A or B) = P(A) + P(B)  P(A and B)
General Addition Rule:
If A and B are mutually exclusive, then
P(A and B) = 0, so the rule can be simplified:
P(A or B) = P(A) + P(B)
For mutually exclusive events A and B
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 416
General Addition Rule Example
P(Red or Ace) = P(Red) +P(Ace)  P(Red and Ace)
= 26/52 + 4/52  2/52 = 28/52Don’t count the two red aces twice!
BlackColor
Type Red Total
Ace 2 2 4
NonAce 24 24 48
Total 26 26 52
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 417
Computing Conditional Probabilities
A conditional probability is the probability of one event, given that another event has occurred:
P(B)
B)andP(AB)P(A
P(A)
B)andP(AA)P(B
Where P(A and B) = joint probability of A and B
P(A) = marginal probability of A
P(B) = marginal probability of B
The conditional probability of A given that B has occurred
The conditional probability of B given that A has occurred
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 418
What is the probability that a car has a CD player, given that it has AC ?
i.e., we want to find P(CD  AC)
Conditional Probability Example
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 419
Conditional Probability Example
No CDCD Total
AC .2 .5 .7
No AC .2 .1 .3
Total .4 .6 1.0
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.
.2857.7
.2
P(AC)
AC)andP(CDAC)P(CD
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 420
Conditional Probability Example
No CDCD Total
AC .2 .5 .7
No AC .2 .1 .3
Total .4 .6 1.0
Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is about 28.57%.
.2857.7
.2
P(AC)
AC)andP(CDAC)P(CD
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 421
Using Decision Trees
Has AC
Does not have AC
Has CD
Does not have CD
Has CD
Does not have CD
P(AC)= .7
P(AC’)= .3
P(AC and CD) = .2
P(AC and CD’) = .5
P(AC’ and CD’) = .1
P(AC’ and CD) = .2
7.
5.
3.
2.
3.
1.
AllCars
7.
2.
Given AC or no AC:
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 422
Using Decision Trees
Has CD
Does not have CD
Has AC
Does not have AC
Has AC
Does not have AC
P(CD)= .4
P(CD’)= .6
P(CD and AC) = .2
P(CD and AC’) = .2
P(CD’ and AC’) = .1
P(CD’ and AC) = .5
4.
2.
6.
5.
6.
1.
AllCars
4.
2.
Given CD or no CD:
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 423
Statistical Independence
Two events are independent if and only if:
Events A and B are independent when the probability of one event is not affected by the other event
P(A)B)P(A
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 424
Multiplication Rules
Multiplication rule for two events A and B:
P(B)B)P(AB)andP(A
P(A)B)P(A Note: If A and B are independent, thenand the multiplication rule simplifies to
P(B)P(A)B)andP(A
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 425
Marginal Probability
Marginal probability for event A:
Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events
)P(B)BP(A)P(B)BP(A)P(B)BP(A P(A) kk2211
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 426
Bayes’ Theorem
where:
Bi = ith event of k mutually exclusive and collectively
exhaustive events
A = new event that might impact P(Bi)
))P(BBP(A))P(BBP(A))P(BBP(A
))P(BBP(AA)P(B
kk2211
iii
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 427
Bayes’ Theorem Example
A drilling company has estimated a 40% chance of striking oil for their new well.
A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests.
Given that this well has been scheduled for a detailed test, what is the probability
that the well will be successful?
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 428
Let S = successful well
U = unsuccessful well P(S) = .4 , P(U) = .6 (prior probabilities)
Define the detailed test event as D
Conditional probabilities:
P(DS) = .6 P(DU) = .2
Goal is to find P(SD)
Bayes’ Theorem Example(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 429
667.12.24.
24.
)6)(.2(.)4)(.6(.
)4)(.6(.
U)P(U)P(DS)P(S)P(D
S)P(S)P(DD)P(S
Bayes’ Theorem Example(continued)
Apply Bayes’ Theorem:
So the revised probability of success, given that this well has been scheduled for a detailed test, is .667
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 430
Given the detailed test, the revised probability of a successful well has risen to .667 from the original estimate of .4
Bayes’ Theorem Example
EventPrior
Prob.
Conditional Prob.
Joint
Prob.
Revised
Prob.
S (successful) .4 .6 .4*.6 = .24 .24/.36 = .667
U (unsuccessful) .6 .2 .6*.2 = .12 .12/.36 = .333
Sum = .36
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 PrenticeHall, Inc. Chap 431
Chapter Summary
Discussed basic probability concepts Sample spaces and events, contingency tables, simple
probability, and joint probability
Examined basic probability rules General addition rule, addition rule for mutually exclusive events,
rule for collectively exhaustive events
Defined conditional probability Statistical independence, marginal probability, decision trees,
and the multiplication rule
Discussed Bayes’ theorem