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7/31/2019 Intro Proba
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Probability Concepts
FOUR
Define the terms: conditional probability and jointprobability.
FIVECalculate probabilities applying the rules of addition
and the rules of multiplication.
Goals
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Definitions continued
There are three definitions of probability: classical,empirical, and subjective.
A PrioriClassical
probabilityapplies
when thereare n
equally likelyoutcomes.
Empirical
Classicalprobabilityapplies whenthe number of
times the eventhappens is
divided by thenumber of
observations.
Subjectiveprobability is
based onwhateverinformation is
available.
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Definitions continued
An Event isthe collection
o f o n e o rm o r eoutcomes ofa nex er iment .
An Outcome is
the part icularr e s u l t o f a ne x p e r i m e n t .
Experiment: A fair die is cast.
Possible outcomes: Thenumbers 1, 2, 3, 4, 5, 6
One possible event: The
occurrence of an evennumber. That is, wecollect the outcomes 2,4 , a n d 6 .
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Collectively ExhaustiveEvents
E v e n t s a r e C o l l e c t i v e l y
Exhaustive if at least one of theevents must occur when ane x p e r i m e n t i s c o n d u c t e d .
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Example 2
155.01200186)( AP
Throughout herteaching careerProfessor Joneshas awarded 186As out of 1,200
students. Whatis the probabilitythat a student inher section this
s e m e s t e r w i l lr e ce i ve a n A?
This is an example of theempirical definition ofprobability.
To find the probability aselected student earned
an A:
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Subjective Probability
Examples of subjective probabilities are:
Estimating the probability theWashington Redskins will winthe Super Bowl this year.
Estimating the probabilitymortgage rates for home
loans wil l top 8 percent.
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Basic Rules of ProbabilityP(A or B) = P(A) + P(B)
If two eventsA and B are mutually
exclusive, the
Special Rule ofAddition states that
theProbability of A or B
occurring equals the sumof their respective
probabilities.
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Example 3
Arrival Frequency
Early 100On Time 800
Late 75
Canceled 25
Total 1000
New England Commuter Airways recentlysupplied the following information on their
commuter flights from Boston to New York:
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Example 3 continued
The probability that a flight is either early orlate is:
P(A or B) = P(A) + P(B) = .10 + .075 =.175.
If A is the event thata flight arrivesearly, then P(A) =100/1000 = .10.
If Bis the event that a
flight arrives late, thenP(B) = 75/1000 = .075.
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The Complement Rule
If P(A) is the probability of event A andP(~A) is the complement of A,
P(A) + P(~A) = 1 or P(A) = 1 - P(~A).
The Complement Rule is used to determinethe probability of an event occurring by subtracting
the probability of the event notoccurring from 1.
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The Complement Rule continued
A~A
A Venn Diagram illustrating the complementrule would appear as:
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Example 4
If Dis the event that
a flight is canceled,then
P(D) = 25/1000 =.025.
Recall example 3. Use thecomplement rule to find the
probability of an early (A) or a late(B) flight
If Cis the event that a
flight arrives on time,then P(C) = 800/1000 =.8.
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Example 4 continued
C.8
D.025
~(C or D) = (A or B).175
P(A or B) = 1 - P(Cor D)= 1 - [.8 +.025]
=.175
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The General Rule ofAddition
If A and Bare two
events that are notmutually exclusive,then P(A or B) isgiven by the
following formula:
P(A or B) = P(A) + P(B) - P(A and B)
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The General Rule of
Addition
A and B
A
B
The Venn Diagram illustrates this rule:
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EXAMPLE 5
Stereo
220
Both100
TV75
In a sample of 500 students, 320 said they had
a stereo, 175 said they had a TV, and 100 saidthey had both. 5 said they had neither.
If d i l d
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Example 5 continued
P(S or TV) = P(S) + P(TV) - P(S and TV)
= 320/500 + 175/500
100/500= .79.
P(S and TV) = 100/500
= .20
If a student is selectedat random, what is theprobab i l i ty tha t the
s t u d e n t h a s o n l y astereo or TV? What isthe probability that thes tudent has bo th a
s t e r e o a n d T V ?
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Joint Probability
A Joint Probability measures thelikelihood that two or more events will
happen concurrently.
An example would
be the event that astudent has botha stereo and TV inhis or her dorm
r o o m .
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Special Rule of Multiplication
Two events A and Bare independentif the
occurrence of one has no effect on theprobability of the occurrence of the other.
This rule is written: P(A and B) = P(A)P(B)
The Special Rule of
Multiplication requires that twoevents A and Bare independent.
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Example 6
5-year stock prices
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5
Year
Stockp
rice
$
IBM
GE
P(IBMand GE) = (.5)(.7) = .35
Chris owns two stocks,IBM and General Electric(GE). The probability that
IBM stock will increase invalue next year is .5 andthe probability that GEstock will increase in
value next year is .7.Assume the two stocksare independent. What isthe probability that bothstocks will increase invalue next year?
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Example 6
continued
What
is theprobability
that at least
one of these stocks
increases in value in
the next year?
This means that
either one can
increase or
both.
P(at least one)
= P(IBM but not GE)
+ P(GE but not IBM)
+ P(IBM and GE)
(.5)(1-.7)+ (.7)(1-.5)
+ (.7)(.5)= .85
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General Multiplication Rule
It states that for two
events A and B, thejoint probability thatboth events willhappen is found by
multiplying theprobability that eventA will happen by theconditional
probability of Bgiventhat A has occurred.
The General
Rule ofMultiplicationis used to find the
joint probability thattwo events willoccur.
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General Multiplication Rule
The joint probability,P(A and B), is given bythe following formula:
P(A and B) = P(A)P(B/A)or
P(A and B) = P(B)P(A/B)
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Example 7
Major Male Female Total
Accounting 170 110 280
Finance 120 100 220
Marketing 160 70 230
Management 150 120 270
Total 600 400 1000
The Dean of the School of Business at OwensUniversity collected the following information
about undergraduate students in her college:
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Example 7
continued
P(A|F) = P(A and F)/P(F)
=[110/1000]/[400/1000] =.275
If a student is selected at random, whatis the probability that the student is afemale (F)accounting major (A)?
P(A and F) =110/1000.
Given that the student is afemale, what is theprobability that she is anaccounting major?