Introduction to probability BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12 th edition – some slides are directly.
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Introduction to probability
BSAD 30
Dave Novak
Source: Anderson et al., 2013 Quantitative Methods for Business 12th edition – some slides are directly from J. Loucks © 2013 Cengage Learning
Overview
Experiments and the Sample Space Assigning Probabilities to Experimental
Outcomes Events and Their Probabilities Some Basic Relationships of Probability Bayes’ Theorem Simpson’s Paradox
Uncertainty
Managers often base their decisions on an analysis of uncertainties such as the following:What are the chances that sales will
decrease if we increase prices?What is the likelihood a new assembly
method will increase productivity?What are the odds that a new investment will
be profitable?
Probability
Probability is a numerical measure of the likelihood that an event will occur
Probability values are always assigned on a scale from 0 to 1 You can think of probability in terms of
percentage A probability near zero indicates an event is
quite unlikely to occur A probability near one indicates an event is
almost certain to occur
Probability as a numerical measure of likelihood
0 1.5
Increasing Likelihood of Occurrence
Probability:
The eventis veryunlikelyto occur
The occurrenceof the event is just as likely asit is unlikely
The eventis almostcertainto occur
Statistical experiments
A statistical experiment differs somewhat from an experiment in the physical sciences
In statistical experiments, probability determines outcomes
Even though the experiment is repeated in exactly the same way, an entirely different outcome may occur
For this reason, statistical experiments are often called random experiments
An experiment and its sample space An experiment is any process that
generates well-defined outcomes Flipping a coin 10 times
The sample space for an experiment is the set of all experimental outcomes The exact H or T results from all 10 times
An experimental outcome is also called a sample pointThe result of a particular coin flip
An experiment and its sample space
ExperimentToss a coinInspection a partConduct a sales callRoll a diePlay a football game
Sample SpaceHead, tailDefective, non-defectivePurchase, no purchase1, 2, 3, 4, 5, 6Win, lose, tie
Assigning probabilities
1) Probability assigned to each experimental outcome must be between 0 and 1 inclusive
0 < P(Ei) < 1 for all i
where:
Ei is the ith experimental outcome and P(Ei) is its probability
Assigning probabilities
2) The sum of the probabilities for all experimental outcomes must be equal to 1
P(E1) + P(E2) + . . . + P(En) = 1
where:n is the number of experimental outcomes
Assigning probabilities If we throw two dice together, the possible
outcomes are: 2, 3, 4, … 12 However, each outcome is not equally likely What is the probability that each outcome
will occur?
Assigning probabilities
Total of Dice Specific Outcomes on Pairs of Dice Probability Event Occurs
2 D1=1 + D2=1 (1+1) 1/36 = 3%
3 D1=1 + D2=2 (1+2), D1=2 + D2=1 (2+1) 2/36 = 1/18 = 6%
4 1+3, 2+2, 3+1 3/36 = 1/12 = 8%
5 1+4, 2+3, 3+2, 4+1 4/36 = 1/9 = 11%
6 1+5, 2+4, 3+3, 4+2, 5+1 5/36 = 14%
7 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 6/36 = 1/6 = 17%
8 2+6, 3+5, 4+4, 5+3, 6+2 5/36 = 14%
9 3+6, 4+5, 5+4, 6+3 4/36 = 1/9 = 11%
10 4+6, 5+5, 6+4 3/36 = 1/12 = 8%
11 5+6, 6+5 2/36 = 1/18 = 6%
12 6+6 1/36 = 3%
Assigning probabilities Three ways of assigning probabilities
1) Classical method• Assume equally likely outcomes
2) Relative frequency method• Assign probabilities based on experimentation or
historical data
3) Subjective method• Assign probabilities based on judgment
Classical method Rolling a die
If an experiment has n possible outcomes (where n=6), the classical method would assign a probability of 1/n to each outcome
Sample Space: S = {1, 2, 3, 4, 5, 6}
Probabilities: Each sample point has a 1/6 (or 0.166) chance of occurring
Relative frequency method
Lucas Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records show the following frequencies of daily rentals for the last 40 days
Number ofPolishers Rented
Numberof Days
01234
4 61810 2
Relative frequency methodEach probability assignment is given by dividingthe frequency (number of days) by the total frequency(total number of days)
4/40Probability
Number ofPolishers Rented
Numberof Days
01234
4 61810 240
.10 .15 .45 .25 .051.00
10/40
Subjective method When economic conditions and a
company’s circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data
We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occurWhat is a potentially serious drawback
associated with this approach?
Subjective method Consider the case in which a couple just
made an offer to purchase a house. Two outcomes are possible:
One believes the probability their offer will be accepted is 0.8; thus, P(E1) = 0.8 and P(E2) = 0.2. Two, believes the
probability that their offer will be accepted is 0.6; hence, P(E1) = 0.6 and P(E2) = 0.4. Person two’s probability
estimate for E1 reflects a greater pessimism that their
offer will be accepted
E1 = their offer is accepted
E2 = their offer is rejected
Events and their probabilities An event is a collection of sample points The probability of any event is equal to the
sum of the probabilities of the sample points in the event
If we can identify all the sample points of an experiment and assign a probability to each sample point, we can compute the probability of the event
Events and their probabilities Rolling a die
Event E = Probability of getting an even number when rolling a die
E = {2, 4, 6}
P(E) = P(2) + P(4) + P(6)
= 1/6 + 1/6 + 1/6
= 3/6 = .5
Four relationships of probability These relationships can be used to compute
the probability of an event, without knowing all the sample point probabilities1) Compliment of an event2) Addition law3) Conditional probabilities4) Multiplication law
Example
You invest in two stocks: Markley Oil and Collins Mining, and determine that the possible outcomes of these investments three months from now are:
Investment Gain or Loss in 3 Months (in $000)
Markley Oil Collins Mining
10 5 0-20
8-2
Example Assume an analyst makes the following
probability estimates
Exper. Outcome Net Gain or Loss Probability
(10, 8)(10, -2)(5, 8)(5, -2)(0, 8)(0, -2)(-20, 8)(-20, -2)
$18,000 Gain $8,000 Gain $13,000 Gain $3,000 Gain $8,000 Gain $2,000 Loss $12,000 Loss $22,000 Loss
.20
.08
.16
.26
.10
.12
.02
.06
Example Viewed as a tree diagram
Gain 5Gain 5
Gain 8Gain 8
Gain 8Gain 8
Gain 10Gain 10
Gain 8Gain 8
Gain 8Gain 8
Lose 20Lose 20
Lose 2Lose 2
Lose 2Lose 2
Lose 2Lose 2
Lose 2Lose 2
EvenEven
Markley Oil(Stage 1)
Collins Mining(Stage 2)
ExperimentalOutcomes
(10, 8) Gain $18,000
(10, -2) Gain $8,000
(5, 8) Gain $13,000
(5, -2) Gain $3,000
(0, 8) Gain $8,000
(0, -2) Lose $2,000
(-20, 8) Lose $12,000
(-20, -2) Lose $22,000
Complement of an event The complement of event A is defined to be
the event consisting of all sample points that are not in A
Event A Ac
SampleSpace SSampleSpace S
VennDiagram
Union of two events The union of events A and B is the event
containing all sample points that are in A or B or both
The union of events A and B is denoted by A B
SampleSpace SSampleSpace SEvent A Event B
Intersection of two events The intersection of events A and B is the set
of all sample points that are in both A and B
The intersection of events A and B is denoted by A
SampleSpace SSampleSpace SEvent A Event B
Intersection of A and BIntersection of A and B
Addition law The addition law provides a way to compute
the probability of event A, or B, or both A and B occurring
The addition law is written as:
P(A B) = P(A) + P(B) - P(A B
Mutually exclusive events Two events are said to be mutually
exclusive if the events have no sample points in common
Two events are mutually exclusive if, when one event occurs, the other cannot occur
SampleSpace SSampleSpace SEvent A Event B
Mutually exclusive events If events A and B are mutually exclusive,
P(A B = 0 In this case, the addition law is:
P(A B) = P(A) + P(B)
There is no need toinclude “- P(A B”
Conditional probability The probability of an event occurring given
that another event has already occurred is called a conditional probability
A conditional probability is computed as:
The conditional probability of A given B is denoted by P(A|B)
( )( | )
( )P A B
P A BP B
Multiplication law The multiplication law provides a way to
compute the probability of the intersection of two events
The multiplication law is written as:
P(A B) = P(B) P(A|B)
Joint and marginal conditional probabilities A joint probability gives the probability of an
intersection of two events A marginal probability gives the probability
of each single event separately These probabilities are often shown in a
joint probability table
Joint and marginal conditional probabilities
Collins MiningProfitable (C) Not Profitable (Cc)Markley Oil
Profitable (M) Not Profitable (Mc)
Total .48 .52
Total
.70
.30
1.00
.36 .34
.12 .18
Joint Probabilities(appear in the bodyof the table)
Marginal Probabilities(appear in the marginsof the table)
Independent events If the probability of event A is not changed
by the existence of event B, we would say that events A and B are independent
Two events are independent if:
P(A|B) = P(A) P(B|A) = P(B)or
Multiplication law for independent events The multiplication law can be used to test
whether or not two events are independent of one another
The multiplication law is written as:
P(A B) = P(A) P(B)
Mutual exclusive ≠ independence Two events with nonzero probabilities
cannot be both mutually exclusive and independent
If one mutually exclusive event is known to occur, the other cannot occur; thus, the probability of the other event occurring is reduced to zero (and they are therefore dependent)
Two events that are not mutually exclusive, might or might not be independent
Bayes’ Theorem
When discussing conditional probabilities, it may be possible to revise certain probabilities as new information arisesInitial probabilities are referred to as prior
probabilitiesWhen initial probabilities are revised using
new information, these new probabilities are referred to as posterior probabilities
Bayes’ Theorem
Bayes’ theorem provides the means for revising the prior probabilities
We will not be working through examples of Bayes’ theorem here, but Business Analytics and Finance concentration students would be well served to do so!
NewInformation
Applicationof Bayes’Theorem
PosteriorProbabilities
PriorProbabilities
Simpson’s paradox
A paradox that occurs in probability where a trend that is observed in multiple groups of data disappears or is reversed when those groups are combinedBe careful when aggregating dataLook for meaningful splits in the data, and
consider groups or classes that might have different behavior. If you do not do this, you might obtain odd or incorrect correlations• Different sample sizes, biased samples, etc.
Simpson’s paradox exampleIn 1973, the University of California, Berkeley was sued for discrimination against women who had applied for admission to graduate school
Admission data for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance
Source: http://en.wikipedia.org/wiki/Simpson's_paradox
Applicants Admitted
Men 8442 44%
Women 4321 35%
Simpson’s paradox exampleWhen examining the data from individual departments, it appeared that no department was significantly biased against women. In fact, most departments had a "small but statistically significant bias in favor of women
Source: http://en.wikipedia.org/wiki/Simpson's_paradox
DepartmentMen Women
Applicants Admitted Applicants Admitted
A 825 62% 108 82%
B 560 63% 25 68%
C 325 37% 593 34%
D 417 33% 375 35%
E 191 28% 393 24%
F 373 6% 341 7%
Simpson’s paradox exampleWhat was happening?
Women tended to apply to more competitive departments with low rates of admission even among qualified applicants, whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants
The data from specific departments constitute a proper defense against charges of discrimination
Source: http://en.wikipedia.org/wiki/Simpson's_paradox
Summary
Experiments and the Sample Space Assigning Probabilities to Experimental
OutcomesClassicalRelative frequencySubjective
Events and Their Probabilities
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