chapter 5 probability 5.2a addition rule. addition rule (general rule) if we have two events a and...
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Chapter 5 Probability5.2A Addition Rule
Addition Rule
(General Rule) If we have two events A and B, then:
P(A or B) = P(A) + P(B) – P(A & B)
Why subtract P(A and B)
P(A or B)
A BA and B
Why subtract P(A and B)
P(A)
A
B
Why subtract P(A and B)
P(B)
A
B
Why subtract P(A and B)
P(A and B) is in there twice!!!We subtract so it is only in there once!
A BA and B
Mutually Exclusive:If events A and B cannot occur at the same time, then they are mutually exclusive or disjoint events.
*If two events A and B are disjoint or mutually exclusive, then
P(A and B)=0
Venn diagram – Mutually ExclusiveEvents
A B
A B
In this case there is no intersection. Since P(A and B) = 0, we do not have to subtract.
Example #1
Events that are mutually exclusive when we roll a single die:1.Rolling a 5 and a 22.Rolling an even number and an odd
number3. Getting a number greater than 4
and a number less than 2
Events that are not mutuallyexclusive when we roll a single die:1.Rolling a 3 and an odd number2.Rolling an even number and a
number greater than 4.3. Rolling a multiple of 3 and an
even number.
Try These Example #2
Determine whether the events listed are mutually exclusive or not mutually exclusive when selecting cars from the Dorman High School Parking Lot:a.Selecting a white car and an SUV.b. Selecting a car with a V6 engine
and a car with a V8 engine.
Try These
Determine whether the events listed are mutually exclusive or not mutually exclusive when selecting cars from the Dorman High School Parking Lot:c.Selecting a Chevrolet and a Fordd. Selecting a car that is a 2002 model
and a truck
a. Selecting a white car and an SUV. NME b. Selecting a car with a V6 engine and a
car with a V8 engine. MEc. Selecting a Chevrolet and a Ford MEd. Selecting a car that is a 2002 model
and a truck NME
Addition Rule (Reminder)
(General Rule) If we have two events A and B , then:
P(A or B) = P(A) + P(B) – P(A & B)
Now we will learn how to use the Addition Rule!!
Example #3
Suppose there are 23 professors at a small college. Of these, 8 teach Math, 8 teach English, 7 teach Science and 6 teach social sciences. We also know that 4 teach Math and Science and 2 teach English and Social Science. We randomly select a professor. Find the following probabilities:
a. The professor teaches Math or Social Science
b. The professor teaches English or Science
ME?
ME?
)()()( SSPMPMorSP 609.23
14
23
6
23
8
)()()( SPEPEorSP652.
23
15
23
7
23
8
c. The professor teaches Math or Science
d. The professor teaches English or Social Science
ME?
ME?
)()()()( MandSPSPMPMorSP 478.23
11
23
4
23
7
23
8
)()()()( EandSSPSSPEPEorSSP522.
23
12
23
2
23
6
23
8
Example #4
A pizza buffet is almost out of pizza and only has 27 pieces of pizza left. 16 of these are hamburger, 7 are pepperoni and 4 are cheese. 17 are thin crust and 10 are thick crust. 10 of the Hamburger are thin crust and 6 are thick crust. All 7 pepperoni are thin crust and all 4 cheese are thick crust. If a piece of pizza is selected at random, find the probability that it is:
a. Pepperoni or Hamburger
b. Cheese or Thick Crust
ME?
ME?
)()()( HPPPPorHP
)()()()( CandTkPTkPCPCorTkP
852.27
23
27
16
27
7
370.27
10
27
4
27
10
27
4
c. Hamburger or Thin Crust
d. Pepperoni or Thick Crust
ME?
ME?
)()()()( HandTnPTnPHPHorTnP
852.27
10
27
17
27
16
)()()( TkPPPPorTkP
630.27
17
27
10
27
7
Example #5
Musical styles other than rock and pop are becoming more popular. A survey of college students finds that the probability they like country music is .40. The probability that they liked jazz is .30 and that they liked both is .10. What is the probability that they like country or jazz?
P(C or J) = .4 + .3 -.1 = .6
Probabilities from two way tablesExample #6
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
a) What is the probability that the driver is a student?
)(StudentP 543.359
195
Probabilities from two way tables
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
b) What is the probability that the driver drives an American or Asian car?
ME?875.
359
314
359
102
359
212
)()()( AsPAmPAsorAmP
Probabilities from two way tables
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
c) What is the probability that the driver is staff or drives an Asian car?
ME? )()()()( AsandStPAsPStPAsorStP
610.359
219
359
47
359
102
359
164
Probabilities from two way tables
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
d) What is the probability that the driver is student or drives an American car?
ME? )()()()( AmandStuPAmPStuPAmorStuP
836.359
300
359
107
359
195
359
212
Example #7: A certain ophthalmic trait is associated with eye color. See table below:
Blue Brown Other TotalTrait Present 70 30 20 120Not Present 20 110 50 180Total 90 140 70 300
a) What is the probability that the trait is present?
4.300
120)( TPP
ME?
Example #7: A certain ophthalmic trait is associated with eye color. See table below:
Blue Brown Other TotalTrait Present 70 30 20 120Not Present 20 110 50 180Total 90 140 70 300
b) What is the probability that the trait is not present or the person has blue eyes?
)()()()( BlandTNPPBlPTNPPBlorTNPP ME?833.
300
250
300
20
300
90
300
180
Example #7: A certain ophthalmic trait is associated with eye color. See table below:
Blue Brown Other TotalTrait Present 70 30 20 120Not Present 20 110 50 180Total 90 140 70 300
c) What is the probability that the trait is present or the person has brown eyes?
)()()()( BrandTPPBrPTPPBrorTPP ME?767.
300
30
300
140
300
120
Example #7: A certain ophthalmic trait is associated with eye color. See table below:
Blue Brown Other TotalTrait Present 70 30 20 120Not Present 20 110 50 180Total 90 140 70 300
d) What is the probability that the person has blue or brown eyes?
)()()( BrPBlPBrorBlP ME?767.
300
230
300
140
300
90
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