![Page 1: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/1.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Math 1300 Finite MathematicsSection 8-2: Union, Intersection, and Complement of Events;
Odds
Jason Aubrey
Department of MathematicsUniversity of Missouri
Jason Aubrey Math 1300 Finite Mathematics
![Page 2: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/2.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
In this section, we will develop the rules of probability forcompound events (more than one simple event) and willdiscuss probabilities involving the union of events as well asintersection of two events.
Jason Aubrey Math 1300 Finite Mathematics
![Page 3: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/3.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
If A and B are two events in a sample space S, then the unionof A and B, denoted by A ∪ B, and the intersection of A and B,denoted by A ∩ B, are defined as follows:
Definition (Union: A ∪ B)
A B
A ∪ B = {e ∈ S|e ∈ A or e ∈ B}
The event “A or B” is defined as the set A ∪ B.
Jason Aubrey Math 1300 Finite Mathematics
![Page 4: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/4.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
If A and B are two events in a sample space S, then the unionof A and B, denoted by A ∪ B, and the intersection of A and B,denoted by A ∩ B, are defined as follows:
Definition (Union: A ∪ B)
A B
A ∪ B = {e ∈ S|e ∈ A or e ∈ B}
The event “A or B” is defined as the set A ∪ B.Jason Aubrey Math 1300 Finite Mathematics
![Page 5: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/5.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Definition (Intersection: A ∩ B)
A B
A ∩ B = {e ∈ S|e ∈ A and e ∈ B}
The event “A and B” is defined as the set A ∩ B.
Jason Aubrey Math 1300 Finite Mathematics
![Page 6: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/6.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Definition (Intersection: A ∩ B)
A B
A ∩ B = {e ∈ S|e ∈ A and e ∈ B}
The event “A and B” is defined as the set A ∩ B.
Jason Aubrey Math 1300 Finite Mathematics
![Page 7: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/7.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Consider the sample space of equally likely eventsfor the rolling of a single fair die: S = {1,2,3,4,5,6}
(a) What is the probability of rolling an odd number and a primenumber?
Let A be the event “an odd number is rolled”. Let B be the event“a prime number is rolled”.
Since only the outcomes 3 and 5 are both odd and prime,A ∩ B = {3,5}.By the equally likely assumption,
P(A ∩ B) =n(A ∩ B)
n(S)=
26=
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 8: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/8.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Consider the sample space of equally likely eventsfor the rolling of a single fair die: S = {1,2,3,4,5,6}
(a) What is the probability of rolling an odd number and a primenumber?
Let A be the event “an odd number is rolled”. Let B be the event“a prime number is rolled”.
Since only the outcomes 3 and 5 are both odd and prime,A ∩ B = {3,5}.By the equally likely assumption,
P(A ∩ B) =n(A ∩ B)
n(S)=
26=
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 9: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/9.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Consider the sample space of equally likely eventsfor the rolling of a single fair die: S = {1,2,3,4,5,6}
(a) What is the probability of rolling an odd number and a primenumber?
Let A be the event “an odd number is rolled”. Let B be the event“a prime number is rolled”.
Since only the outcomes 3 and 5 are both odd and prime,A ∩ B = {3,5}.By the equally likely assumption,
P(A ∩ B) =n(A ∩ B)
n(S)=
26=
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 10: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/10.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Consider the sample space of equally likely eventsfor the rolling of a single fair die: S = {1,2,3,4,5,6}
(a) What is the probability of rolling an odd number and a primenumber?
Let A be the event “an odd number is rolled”. Let B be the event“a prime number is rolled”.
Since only the outcomes 3 and 5 are both odd and prime,A ∩ B = {3,5}.
By the equally likely assumption,
P(A ∩ B) =n(A ∩ B)
n(S)=
26=
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 11: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/11.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Consider the sample space of equally likely eventsfor the rolling of a single fair die: S = {1,2,3,4,5,6}
(a) What is the probability of rolling an odd number and a primenumber?
Let A be the event “an odd number is rolled”. Let B be the event“a prime number is rolled”.
Since only the outcomes 3 and 5 are both odd and prime,A ∩ B = {3,5}.By the equally likely assumption,
P(A ∩ B) =n(A ∩ B)
n(S)=
26=
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 12: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/12.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
(b) What is the probability of rolling an odd number or a primenumber?
Again, A = {1,3,5} and B = {2,3,5} so
A ∪ B = {1,2,3,5}
By the equally likely assumption
P(A ∪ B) =n(A ∪ B)
n(S)=
46=
23
Jason Aubrey Math 1300 Finite Mathematics
![Page 13: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/13.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
(b) What is the probability of rolling an odd number or a primenumber?
Again, A = {1,3,5} and B = {2,3,5} so
A ∪ B = {1,2,3,5}
By the equally likely assumption
P(A ∪ B) =n(A ∪ B)
n(S)=
46=
23
Jason Aubrey Math 1300 Finite Mathematics
![Page 14: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/14.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
(b) What is the probability of rolling an odd number or a primenumber?
Again, A = {1,3,5} and B = {2,3,5} so
A ∪ B = {1,2,3,5}
By the equally likely assumption
P(A ∪ B) =n(A ∪ B)
n(S)=
46=
23
Jason Aubrey Math 1300 Finite Mathematics
![Page 15: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/15.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
(b) What is the probability of rolling an odd number or a primenumber?
Again, A = {1,3,5} and B = {2,3,5} so
A ∪ B = {1,2,3,5}
By the equally likely assumption
P(A ∪ B) =n(A ∪ B)
n(S)=
46=
23
Jason Aubrey Math 1300 Finite Mathematics
![Page 16: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/16.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
Is P(E) = P(A) + P(B)?Only if A and B are mutually exclusive (disjoint), that is,if A ∩ B = ∅.In this case, P(A ∪ B) is the sum of the probabilities of allof the simple events in A plus the sum of the probabilitiesof all of the simple events in B.
Jason Aubrey Math 1300 Finite Mathematics
![Page 17: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/17.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
Is P(E) = P(A) + P(B)?
Only if A and B are mutually exclusive (disjoint), that is,if A ∩ B = ∅.In this case, P(A ∪ B) is the sum of the probabilities of allof the simple events in A plus the sum of the probabilitiesof all of the simple events in B.
Jason Aubrey Math 1300 Finite Mathematics
![Page 18: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/18.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
Is P(E) = P(A) + P(B)?Only if A and B are mutually exclusive (disjoint), that is,if A ∩ B = ∅.
In this case, P(A ∪ B) is the sum of the probabilities of allof the simple events in A plus the sum of the probabilitiesof all of the simple events in B.
Jason Aubrey Math 1300 Finite Mathematics
![Page 19: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/19.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that an event E is
E = A ∪ B.
Is P(E) = P(A) + P(B)?Only if A and B are mutually exclusive (disjoint), that is,if A ∩ B = ∅.In this case, P(A ∪ B) is the sum of the probabilities of allof the simple events in A plus the sum of the probabilitiesof all of the simple events in B.
Jason Aubrey Math 1300 Finite Mathematics
![Page 20: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/20.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
But what happens if A and B are not mutually exclusive;that is, what if A ∩ B 6= ∅?
In this case, we must use a version of the addition principlefor probability.
Jason Aubrey Math 1300 Finite Mathematics
![Page 21: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/21.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
But what happens if A and B are not mutually exclusive;that is, what if A ∩ B 6= ∅?In this case, we must use a version of the addition principlefor probability.
Jason Aubrey Math 1300 Finite Mathematics
![Page 22: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/22.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Definition (Probability of a Union of Two Events)For any events A and B,
P(A ∪ B) = P(A) + P(B)− P(A ∩ B)
If A and B are mutually exclusive, then
P(A ∪ B) = P(A) + P(B)
Jason Aubrey Math 1300 Finite Mathematics
![Page 23: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/23.jpg)
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Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example Let E and F be events in a sample space S. IfP(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what isP(E ∩ F )?
Notice here that P(E ∪ F ) 6= P(E) + P(F ), so we know that theevents E and F are not mutually exclusive; that is,P(E ∩ F ) 6= 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
0.55 = 0.35 + 0.25− P(E ∩ F )
P(E ∩ F ) = 0.35 + 0.25− 0.55 = 0.05
Jason Aubrey Math 1300 Finite Mathematics
![Page 24: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/24.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example Let E and F be events in a sample space S. IfP(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what isP(E ∩ F )?
Notice here that P(E ∪ F ) 6= P(E) + P(F ), so we know that theevents E and F are not mutually exclusive; that is,P(E ∩ F ) 6= 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
0.55 = 0.35 + 0.25− P(E ∩ F )
P(E ∩ F ) = 0.35 + 0.25− 0.55 = 0.05
Jason Aubrey Math 1300 Finite Mathematics
![Page 25: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/25.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example Let E and F be events in a sample space S. IfP(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what isP(E ∩ F )?
Notice here that P(E ∪ F ) 6= P(E) + P(F ), so we know that theevents E and F are not mutually exclusive; that is,P(E ∩ F ) 6= 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
0.55 = 0.35 + 0.25− P(E ∩ F )
P(E ∩ F ) = 0.35 + 0.25− 0.55 = 0.05
Jason Aubrey Math 1300 Finite Mathematics
![Page 26: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/26.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example Let E and F be events in a sample space S. IfP(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what isP(E ∩ F )?
Notice here that P(E ∪ F ) 6= P(E) + P(F ), so we know that theevents E and F are not mutually exclusive; that is,P(E ∩ F ) 6= 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
0.55 = 0.35 + 0.25− P(E ∩ F )
P(E ∩ F ) = 0.35 + 0.25− 0.55 = 0.05
Jason Aubrey Math 1300 Finite Mathematics
![Page 27: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/27.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example Let E and F be events in a sample space S. IfP(E) = 0.35, P(F ) = 0.25 and P(E ∪ F ) = 0.55 then what isP(E ∩ F )?
Notice here that P(E ∪ F ) 6= P(E) + P(F ), so we know that theevents E and F are not mutually exclusive; that is,P(E ∩ F ) 6= 0.
To find P(E ∩ F ) we use the addition principle:
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
0.55 = 0.35 + 0.25− P(E ∩ F )
P(E ∩ F ) = 0.35 + 0.25− 0.55 = 0.05
Jason Aubrey Math 1300 Finite Mathematics
![Page 28: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/28.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find theprobability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are Eand F mutually exclusive? No because a card can be both ajack and a club?
Now, n(E) = 4 - there are four jacks; by the equally likelyassumption,
P(E) =n(E)
n(S)=
452
.
Also, n(F ) = 13 - there are thirteen clubs; by the equallylikely assumption,
P(F ) =n(F )
n(S)=
1352
.
Jason Aubrey Math 1300 Finite Mathematics
![Page 29: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/29.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find theprobability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are Eand F mutually exclusive?
No because a card can be both ajack and a club?
Now, n(E) = 4 - there are four jacks; by the equally likelyassumption,
P(E) =n(E)
n(S)=
452
.
Also, n(F ) = 13 - there are thirteen clubs; by the equallylikely assumption,
P(F ) =n(F )
n(S)=
1352
.
Jason Aubrey Math 1300 Finite Mathematics
![Page 30: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/30.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find theprobability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are Eand F mutually exclusive? No because a card can be both ajack and a club?
Now, n(E) = 4 - there are four jacks; by the equally likelyassumption,
P(E) =n(E)
n(S)=
452
.
Also, n(F ) = 13 - there are thirteen clubs; by the equallylikely assumption,
P(F ) =n(F )
n(S)=
1352
.
Jason Aubrey Math 1300 Finite Mathematics
![Page 31: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/31.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find theprobability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are Eand F mutually exclusive? No because a card can be both ajack and a club?
Now, n(E) = 4 - there are four jacks; by the equally likelyassumption,
P(E) =n(E)
n(S)=
452
.
Also, n(F ) = 13 - there are thirteen clubs; by the equallylikely assumption,
P(F ) =n(F )
n(S)=
1352
.
Jason Aubrey Math 1300 Finite Mathematics
![Page 32: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/32.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A single card is drawn from a deck of cards. Find theprobability that the card is a jack or club.
Let E = “the set of jacks”, and F = “the set of clubs”. Are Eand F mutually exclusive? No because a card can be both ajack and a club?
Now, n(E) = 4 - there are four jacks; by the equally likelyassumption,
P(E) =n(E)
n(S)=
452
.
Also, n(F ) = 13 - there are thirteen clubs; by the equallylikely assumption,
P(F ) =n(F )
n(S)=
1352
.
Jason Aubrey Math 1300 Finite Mathematics
![Page 33: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/33.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;by the equally likely assumption,
P(E ∩ F ) =1
52.
Now we can use the addition principle to get
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
P(E ∪ F ) =4
52+
1352− 1
52
=1652
=4
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 34: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/34.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;by the equally likely assumption,
P(E ∩ F ) =1
52.
Now we can use the addition principle to get
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
P(E ∪ F ) =4
52+
1352− 1
52
=1652
=4
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 35: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/35.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;by the equally likely assumption,
P(E ∩ F ) =1
52.
Now we can use the addition principle to get
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
P(E ∪ F ) =4
52+
1352− 1
52
=1652
=4
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 36: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/36.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Finally, n(E ∩ F ) = 1 - there is one jack that is also a club;by the equally likely assumption,
P(E ∩ F ) =1
52.
Now we can use the addition principle to get
P(E ∪ F ) = P(E) + P(F )− P(E ∩ F )
P(E ∪ F ) =4
52+
1352− 1
52
=1652
=4
13
Jason Aubrey Math 1300 Finite Mathematics
![Page 37: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/37.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.(Tossing three coins is the same experiment as tossing onecoin three times.)
(a) Use the multiplication principle to calculate the total numberof outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.Let A be the event that exactly two tails are flipped.Let B be the event that exactly three tails are flipped.
Jason Aubrey Math 1300 Finite Mathematics
![Page 38: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/38.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.(Tossing three coins is the same experiment as tossing onecoin three times.)
(a) Use the multiplication principle to calculate the total numberof outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.Let A be the event that exactly two tails are flipped.Let B be the event that exactly three tails are flipped.
Jason Aubrey Math 1300 Finite Mathematics
![Page 39: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/39.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.(Tossing three coins is the same experiment as tossing onecoin three times.)
(a) Use the multiplication principle to calculate the total numberof outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.Let A be the event that exactly two tails are flipped.Let B be the event that exactly three tails are flipped.
Jason Aubrey Math 1300 Finite Mathematics
![Page 40: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/40.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.(Tossing three coins is the same experiment as tossing onecoin three times.)
(a) Use the multiplication principle to calculate the total numberof outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.Let A be the event that exactly two tails are flipped.Let B be the event that exactly three tails are flipped.
Jason Aubrey Math 1300 Finite Mathematics
![Page 41: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/41.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.(Tossing three coins is the same experiment as tossing onecoin three times.)
(a) Use the multiplication principle to calculate the total numberof outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.
Let A be the event that exactly two tails are flipped.Let B be the event that exactly three tails are flipped.
Jason Aubrey Math 1300 Finite Mathematics
![Page 42: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/42.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.(Tossing three coins is the same experiment as tossing onecoin three times.)
(a) Use the multiplication principle to calculate the total numberof outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.Let A be the event that exactly two tails are flipped.
Let B be the event that exactly three tails are flipped.
Jason Aubrey Math 1300 Finite Mathematics
![Page 43: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/43.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Three coins are tossed. Assume they are fair coins.(Tossing three coins is the same experiment as tossing onecoin three times.)
(a) Use the multiplication principle to calculate the total numberof outcomes in the sample space.
n(S) = (2)(2)(2) = 8
(b) Find the probability of flipping at least two tails.
Let E be the event of flipping at least two tails.Let A be the event that exactly two tails are flipped.Let B be the event that exactly three tails are flipped.
Jason Aubrey Math 1300 Finite Mathematics
![Page 44: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/44.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutuallyexclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
P(A) =n({HTT ,THT ,TTH})
n(S)=
38
P(B) =n({TTT})
n(S)=
18
Therefore,
P(E) = P(A) + P(B) =38+
18=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 45: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/45.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutuallyexclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
P(A) =n({HTT ,THT ,TTH})
n(S)=
38
P(B) =n({TTT})
n(S)=
18
Therefore,
P(E) = P(A) + P(B) =38+
18=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 46: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/46.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutuallyexclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
P(A) =n({HTT ,THT ,TTH})
n(S)=
38
P(B) =n({TTT})
n(S)=
18
Therefore,
P(E) = P(A) + P(B) =38+
18=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 47: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/47.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutuallyexclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
P(A) =n({HTT ,THT ,TTH})
n(S)=
38
P(B) =n({TTT})
n(S)=
18
Therefore,
P(E) = P(A) + P(B) =38+
18=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 48: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/48.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Notice that E = A ∪ B and that A and B are mutuallyexclusive; that is,
E = A ∪ B and A ∩ B = ∅
So P(E) = P(A ∪ B) = P(A) + P(B)
P(A) =n({HTT ,THT ,TTH})
n(S)=
38
P(B) =n({TTT})
n(S)=
18
Therefore,
P(E) = P(A) + P(B) =38+
18=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 49: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/49.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1, . . . ,en}
into two subsets E and E ′ such that
E ∩ E ′ = ∅.
That is, E and E ′ are mutually exclusive, and
E ∪ E ′ = S.
Then E ′ is called the complement of E relative to S. The setE ′ contains all the elements of S that are not in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 50: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/50.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1, . . . ,en}
into two subsets E and E ′ such that
E ∩ E ′ = ∅.
That is, E and E ′ are mutually exclusive, and
E ∪ E ′ = S.
Then E ′ is called the complement of E relative to S. The setE ′ contains all the elements of S that are not in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 51: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/51.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1, . . . ,en}
into two subsets E and E ′ such that
E ∩ E ′ = ∅.
That is, E and E ′ are mutually exclusive, and
E ∪ E ′ = S.
Then E ′ is called the complement of E relative to S.
The setE ′ contains all the elements of S that are not in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 52: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/52.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Suppose that we divide a finite sample space
S = {e1, . . . ,en}
into two subsets E and E ′ such that
E ∩ E ′ = ∅.
That is, E and E ′ are mutually exclusive, and
E ∪ E ′ = S.
Then E ′ is called the complement of E relative to S. The setE ′ contains all the elements of S that are not in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 53: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/53.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Furthermore,
P(S) = P(E ∪ E ′)
= P(E) + P(E ′) = 1
Therefore,
P(E) = 1− P(E ′) P(E ′) = 1− P(E)
Many times it is easier to first compute the probability that andevent won’t occur, and then use that to find the probability thatthe event will occur.
Jason Aubrey Math 1300 Finite Mathematics
![Page 54: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/54.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Furthermore,
P(S) = P(E ∪ E ′)
= P(E) + P(E ′) = 1
Therefore,
P(E) = 1− P(E ′) P(E ′) = 1− P(E)
Many times it is easier to first compute the probability that andevent won’t occur, and then use that to find the probability thatthe event will occur.
Jason Aubrey Math 1300 Finite Mathematics
![Page 55: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/55.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Furthermore,
P(S) = P(E ∪ E ′)
= P(E) + P(E ′) = 1
Therefore,
P(E) = 1− P(E ′) P(E ′) = 1− P(E)
Many times it is easier to first compute the probability that andevent won’t occur, and then use that to find the probability thatthe event will occur.
Jason Aubrey Math 1300 Finite Mathematics
![Page 56: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/56.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 57: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/57.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)
P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 58: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/58.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 59: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/59.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 60: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/60.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 61: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/61.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)
P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 62: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/62.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 63: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/63.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 64: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/64.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If A and B are mutually exclusive events withP(A) = 0.6 and P(B) = 0.3, find the following probabilities:
(a) P(A ∩ B)P(A ∩ B) = 0
(b) P(A ∩ B)
P(A ∩ B) = P(A) + P(B)− P(A ∩ B) = 0.6 + 0.3− 0 = 0.9
(c) P(A′)P(A′) = 1− P(A) = 1− 0.6 = 0.4
(d) P(A ∩ B′)
Since A ∩ B = ∅, A ⊆ B′ so A ∩ B′ = A and
P(A ∩ B′) = P(A) = 0.6
Jason Aubrey Math 1300 Finite Mathematics
![Page 65: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/65.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: What is the probability that when two dice aretossed, the number of points on each die will not be the same?
This is the same as saying that doubles will not occur. Forexample,
E be the set of all rolls of two dice which do not result indoubles. Mathematically we can represent this as
E = {(n,m)|1 ≤ n,m ≤ 6 and n 6= m}We wish to find P(E). Let S be be the sample space for thisexperiment.
S = {(n,m)|1 ≤ n,m ≤ 6} and n(S) = 36
Jason Aubrey Math 1300 Finite Mathematics
![Page 66: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/66.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: What is the probability that when two dice aretossed, the number of points on each die will not be the same?
This is the same as saying that doubles will not occur. Forexample,
E be the set of all rolls of two dice which do not result indoubles. Mathematically we can represent this as
E = {(n,m)|1 ≤ n,m ≤ 6 and n 6= m}We wish to find P(E). Let S be be the sample space for thisexperiment.
S = {(n,m)|1 ≤ n,m ≤ 6} and n(S) = 36
Jason Aubrey Math 1300 Finite Mathematics
![Page 67: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/67.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: What is the probability that when two dice aretossed, the number of points on each die will not be the same?
This is the same as saying that doubles will not occur. Forexample,
E be the set of all rolls of two dice which do not result indoubles. Mathematically we can represent this as
E = {(n,m)|1 ≤ n,m ≤ 6 and n 6= m}We wish to find P(E).
Let S be be the sample space for thisexperiment.
S = {(n,m)|1 ≤ n,m ≤ 6} and n(S) = 36
Jason Aubrey Math 1300 Finite Mathematics
![Page 68: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/68.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: What is the probability that when two dice aretossed, the number of points on each die will not be the same?
This is the same as saying that doubles will not occur. Forexample,
E be the set of all rolls of two dice which do not result indoubles. Mathematically we can represent this as
E = {(n,m)|1 ≤ n,m ≤ 6 and n 6= m}We wish to find P(E). Let S be be the sample space for thisexperiment.
S = {(n,m)|1 ≤ n,m ≤ 6} and n(S) = 36
Jason Aubrey Math 1300 Finite Mathematics
![Page 69: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/69.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Here we have
E ′ = {(n,m)|1 ≤ n,m ≤ 6 and n = m}= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
Since
P(E ′) =n(E ′)
n(S)=
636
=16
we haveP(E) = 1− P(E ′) = 1− 1
6=
56
Jason Aubrey Math 1300 Finite Mathematics
![Page 70: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/70.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Here we have
E ′ = {(n,m)|1 ≤ n,m ≤ 6 and n = m}
= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
Since
P(E ′) =n(E ′)
n(S)=
636
=16
we haveP(E) = 1− P(E ′) = 1− 1
6=
56
Jason Aubrey Math 1300 Finite Mathematics
![Page 71: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/71.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Here we have
E ′ = {(n,m)|1 ≤ n,m ≤ 6 and n = m}= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
Since
P(E ′) =n(E ′)
n(S)=
636
=16
we haveP(E) = 1− P(E ′) = 1− 1
6=
56
Jason Aubrey Math 1300 Finite Mathematics
![Page 72: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/72.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Here we have
E ′ = {(n,m)|1 ≤ n,m ≤ 6 and n = m}= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
Since
P(E ′) =n(E ′)
n(S)=
636
=16
we haveP(E) = 1− P(E ′) = 1− 1
6=
56
Jason Aubrey Math 1300 Finite Mathematics
![Page 73: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/73.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Here we have
E ′ = {(n,m)|1 ≤ n,m ≤ 6 and n = m}= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
Since
P(E ′) =n(E ′)
n(S)=
636
=16
we haveP(E) = 1− P(E ′) = 1− 1
6=
56
Jason Aubrey Math 1300 Finite Mathematics
![Page 74: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/74.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability thatheads turn up at least once?
Let E represent the event “heads turns up at least once” and letS represent the sample space. Notice first that
S = {HTTHT ,HHHTT ,THHHT ,HTHTH,TTTTT ,HHHHT , . . .}
We assume the coin is fair so that we may also assume that allof the outcomes in the sample space are equally likely. What isn(S)?
n(S) = (2)(2)(2)(2)(2) = 32
E contains all outcomes that have at least one H. E.g. HTTHT ,HHHTT , etc.
Jason Aubrey Math 1300 Finite Mathematics
![Page 75: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/75.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability thatheads turn up at least once?
Let E represent the event “heads turns up at least once” and letS represent the sample space. Notice first that
S = {HTTHT ,HHHTT ,THHHT ,HTHTH,TTTTT ,HHHHT , . . .}
We assume the coin is fair so that we may also assume that allof the outcomes in the sample space are equally likely. What isn(S)?
n(S) = (2)(2)(2)(2)(2) = 32
E contains all outcomes that have at least one H. E.g. HTTHT ,HHHTT , etc.
Jason Aubrey Math 1300 Finite Mathematics
![Page 76: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/76.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability thatheads turn up at least once?
Let E represent the event “heads turns up at least once” and letS represent the sample space. Notice first that
S = {HTTHT ,HHHTT ,THHHT ,HTHTH,TTTTT ,HHHHT , . . .}
We assume the coin is fair so that we may also assume that allof the outcomes in the sample space are equally likely. What isn(S)?
n(S) = (2)(2)(2)(2)(2) = 32
E contains all outcomes that have at least one H. E.g. HTTHT ,HHHTT , etc.
Jason Aubrey Math 1300 Finite Mathematics
![Page 77: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/77.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability thatheads turn up at least once?
Let E represent the event “heads turns up at least once” and letS represent the sample space. Notice first that
S = {HTTHT ,HHHTT ,THHHT ,HTHTH,TTTTT ,HHHHT , . . .}
We assume the coin is fair so that we may also assume that allof the outcomes in the sample space are equally likely. What isn(S)?
n(S) = (2)(2)(2)(2)(2) = 32
E contains all outcomes that have at least one H. E.g. HTTHT ,HHHTT , etc.
Jason Aubrey Math 1300 Finite Mathematics
![Page 78: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/78.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A coin is tossed 5 times. What is the probability thatheads turn up at least once?
Let E represent the event “heads turns up at least once” and letS represent the sample space. Notice first that
S = {HTTHT ,HHHTT ,THHHT ,HTHTH,TTTTT ,HHHHT , . . .}
We assume the coin is fair so that we may also assume that allof the outcomes in the sample space are equally likely. What isn(S)?
n(S) = (2)(2)(2)(2)(2) = 32
E contains all outcomes that have at least one H. E.g. HTTHT ,HHHTT , etc.
Jason Aubrey Math 1300 Finite Mathematics
![Page 79: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/79.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
What is in the set E ′?
The opposite of “heads turn up at leastonce” is “heads do not turn up at all.” So,
E ′ = {TTTTT} and P(E ′) =1
32
Therefore,
P(E) = 1− P(E ′) = 1− 132
=3132
Tip: Consider using complements whenever you encounter aprobability (or even counting problems) that contains the phrase“at least once”.
Jason Aubrey Math 1300 Finite Mathematics
![Page 80: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/80.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
What is in the set E ′? The opposite of “heads turn up at leastonce” is “heads do not turn up at all.” So,
E ′ = {TTTTT} and P(E ′) =1
32
Therefore,
P(E) = 1− P(E ′) = 1− 132
=3132
Tip: Consider using complements whenever you encounter aprobability (or even counting problems) that contains the phrase“at least once”.
Jason Aubrey Math 1300 Finite Mathematics
![Page 81: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/81.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
What is in the set E ′? The opposite of “heads turn up at leastonce” is “heads do not turn up at all.” So,
E ′ = {TTTTT} and P(E ′) =1
32
Therefore,
P(E) = 1− P(E ′) = 1− 132
=3132
Tip: Consider using complements whenever you encounter aprobability (or even counting problems) that contains the phrase“at least once”.
Jason Aubrey Math 1300 Finite Mathematics
![Page 82: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/82.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
What is in the set E ′? The opposite of “heads turn up at leastonce” is “heads do not turn up at all.” So,
E ′ = {TTTTT} and P(E ′) =1
32
Therefore,
P(E) = 1− P(E ′) = 1− 132
=3132
Tip: Consider using complements whenever you encounter aprobability (or even counting problems) that contains the phrase“at least once”.
Jason Aubrey Math 1300 Finite Mathematics
![Page 83: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/83.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A shipment of 40 precision parts, including 8 that aredefective, is sent to an assembly plant. The quality controldivision selects 10 at random for testing and rejects theshipment if 1 or more in the sample are found defective. Whatis the probability that the shipment will be rejected?
Notice first that the question
What is the probability that the shipment will berejected?
is really asking
What is the probability that the 10 parts selected fortesting contain at least one defective part?
Jason Aubrey Math 1300 Finite Mathematics
![Page 84: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/84.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: A shipment of 40 precision parts, including 8 that aredefective, is sent to an assembly plant. The quality controldivision selects 10 at random for testing and rejects theshipment if 1 or more in the sample are found defective. Whatis the probability that the shipment will be rejected?
Notice first that the question
What is the probability that the shipment will berejected?
is really asking
What is the probability that the 10 parts selected fortesting contain at least one defective part?
Jason Aubrey Math 1300 Finite Mathematics
![Page 85: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/85.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of40.
n(S) = C(40,10) = 847,660,528
Let E be the set of all selections of 10 parts that contain at leastone defective part. We want to find P(E).
Notice that every set of 10 parts eithercontains at least one defective part (so is in E), orcontains no defective parts
Thus E ′ is the set of all selections of 10 parts that contain nodefective parts.
Jason Aubrey Math 1300 Finite Mathematics
![Page 86: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/86.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of40.
n(S) = C(40,10) = 847,660,528
Let E be the set of all selections of 10 parts that contain at leastone defective part. We want to find P(E).
Notice that every set of 10 parts eithercontains at least one defective part (so is in E), orcontains no defective parts
Thus E ′ is the set of all selections of 10 parts that contain nodefective parts.
Jason Aubrey Math 1300 Finite Mathematics
![Page 87: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/87.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of40.
n(S) = C(40,10) = 847,660,528
Let E be the set of all selections of 10 parts that contain at leastone defective part. We want to find P(E).
Notice that every set of 10 parts eithercontains at least one defective part (so is in E), orcontains no defective parts
Thus E ′ is the set of all selections of 10 parts that contain nodefective parts.
Jason Aubrey Math 1300 Finite Mathematics
![Page 88: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/88.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of40.
n(S) = C(40,10) = 847,660,528
Let E be the set of all selections of 10 parts that contain at leastone defective part. We want to find P(E).
Notice that every set of 10 parts either
contains at least one defective part (so is in E), orcontains no defective parts
Thus E ′ is the set of all selections of 10 parts that contain nodefective parts.
Jason Aubrey Math 1300 Finite Mathematics
![Page 89: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/89.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of40.
n(S) = C(40,10) = 847,660,528
Let E be the set of all selections of 10 parts that contain at leastone defective part. We want to find P(E).
Notice that every set of 10 parts eithercontains at least one defective part (so is in E), or
contains no defective partsThus E ′ is the set of all selections of 10 parts that contain nodefective parts.
Jason Aubrey Math 1300 Finite Mathematics
![Page 90: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/90.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of40.
n(S) = C(40,10) = 847,660,528
Let E be the set of all selections of 10 parts that contain at leastone defective part. We want to find P(E).
Notice that every set of 10 parts eithercontains at least one defective part (so is in E), orcontains no defective parts
Thus E ′ is the set of all selections of 10 parts that contain nodefective parts.
Jason Aubrey Math 1300 Finite Mathematics
![Page 91: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/91.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Let S be all possible selections of 10 parts from the shipment of40.
n(S) = C(40,10) = 847,660,528
Let E be the set of all selections of 10 parts that contain at leastone defective part. We want to find P(E).
Notice that every set of 10 parts eithercontains at least one defective part (so is in E), orcontains no defective parts
Thus E ′ is the set of all selections of 10 parts that contain nodefective parts.
Jason Aubrey Math 1300 Finite Mathematics
![Page 92: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/92.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick10 that have no defective parts, we choose from the 32 that arenot defective, so
n(E ′) = C(32,10) = 64,512,240
Therefore,
P(E) = 1− P(E ′) = 1− n(E ′)
n(S)
= 1− 64,512,240847,660,528
≈ 1− 0.0761
≈ 0.9239
So there is about a 92.4% chance that the shipment will berejected.
Jason Aubrey Math 1300 Finite Mathematics
![Page 93: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/93.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick10 that have no defective parts, we choose from the 32 that arenot defective, so
n(E ′) = C(32,10) = 64,512,240
Therefore,
P(E) = 1− P(E ′) = 1− n(E ′)
n(S)
= 1− 64,512,240847,660,528
≈ 1− 0.0761
≈ 0.9239
So there is about a 92.4% chance that the shipment will berejected.
Jason Aubrey Math 1300 Finite Mathematics
![Page 94: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/94.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick10 that have no defective parts, we choose from the 32 that arenot defective, so
n(E ′) = C(32,10) = 64,512,240
Therefore,
P(E) = 1− P(E ′) = 1− n(E ′)
n(S)
= 1− 64,512,240847,660,528
≈ 1− 0.0761
≈ 0.9239
So there is about a 92.4% chance that the shipment will berejected.
Jason Aubrey Math 1300 Finite Mathematics
![Page 95: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/95.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick10 that have no defective parts, we choose from the 32 that arenot defective, so
n(E ′) = C(32,10) = 64,512,240
Therefore,
P(E) = 1− P(E ′) = 1− n(E ′)
n(S)
= 1− 64,512,240847,660,528
≈ 1− 0.0761
≈ 0.9239
So there is about a 92.4% chance that the shipment will berejected.
Jason Aubrey Math 1300 Finite Mathematics
![Page 96: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/96.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick10 that have no defective parts, we choose from the 32 that arenot defective, so
n(E ′) = C(32,10) = 64,512,240
Therefore,
P(E) = 1− P(E ′) = 1− n(E ′)
n(S)
= 1− 64,512,240847,660,528
≈ 1− 0.0761
≈ 0.9239
So there is about a 92.4% chance that the shipment will berejected.
Jason Aubrey Math 1300 Finite Mathematics
![Page 97: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/97.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
The shipment of 40 parts contains 8 that are defective. To pick10 that have no defective parts, we choose from the 32 that arenot defective, so
n(E ′) = C(32,10) = 64,512,240
Therefore,
P(E) = 1− P(E ′) = 1− n(E ′)
n(S)
= 1− 64,512,240847,660,528
≈ 1− 0.0761
≈ 0.9239
So there is about a 92.4% chance that the shipment will berejected.
Jason Aubrey Math 1300 Finite Mathematics
![Page 98: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/98.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Definition (From Probabilities to Odds)
If P(E) is the probability of the event E , then1 the odds for E are given by
P(E)
1− P(E)=
P(E)
P(E ′),P(E) 6= 1.
2 the odds against E are given by
1− P(E)
P(E)=
P(E ′)
P(E),P(E) 6= 0
Note: When possible, odds are to be expressed as ratios ofwhole numbers.
Jason Aubrey Math 1300 Finite Mathematics
![Page 99: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/99.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Definition (From Probabilities to Odds)
If P(E) is the probability of the event E , then1 the odds for E are given by
P(E)
1− P(E)=
P(E)
P(E ′),P(E) 6= 1.
2 the odds against E are given by
1− P(E)
P(E)=
P(E ′)
P(E),P(E) 6= 0
Note: When possible, odds are to be expressed as ratios ofwhole numbers.
Jason Aubrey Math 1300 Finite Mathematics
![Page 100: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/100.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Definition (From Probabilities to Odds)
If P(E) is the probability of the event E , then1 the odds for E are given by
P(E)
1− P(E)=
P(E)
P(E ′),P(E) 6= 1.
2 the odds against E are given by
1− P(E)
P(E)=
P(E ′)
P(E),P(E) 6= 0
Note: When possible, odds are to be expressed as ratios ofwhole numbers.
Jason Aubrey Math 1300 Finite Mathematics
![Page 101: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/101.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Definition (From Probabilities to Odds)
If P(E) is the probability of the event E , then1 the odds for E are given by
P(E)
1− P(E)=
P(E)
P(E ′),P(E) 6= 1.
2 the odds against E are given by
1− P(E)
P(E)=
P(E ′)
P(E),P(E) 6= 0
Note: When possible, odds are to be expressed as ratios ofwhole numbers.
Jason Aubrey Math 1300 Finite Mathematics
![Page 102: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/102.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Given the following probabilities for an event E , findthe odds for and against E :
P(E) = 35
Since P(E) = 3/5 we know P(E ′) = 1− P(E) = 2/5.Then the odds for E are
P(E)
P(E ′)=
3/52/5
=32
And the odds against E are
P(E ′)
P(E)=
2/53/5
=23
Jason Aubrey Math 1300 Finite Mathematics
![Page 103: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/103.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Given the following probabilities for an event E , findthe odds for and against E :
P(E) = 35
Since P(E) = 3/5 we know P(E ′) = 1− P(E) = 2/5.Then the odds for E are
P(E)
P(E ′)=
3/52/5
=32
And the odds against E are
P(E ′)
P(E)=
2/53/5
=23
Jason Aubrey Math 1300 Finite Mathematics
![Page 104: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/104.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Given the following probabilities for an event E , findthe odds for and against E :
P(E) = 35
Since P(E) = 3/5 we know P(E ′) = 1− P(E) = 2/5.
Then the odds for E are
P(E)
P(E ′)=
3/52/5
=32
And the odds against E are
P(E ′)
P(E)=
2/53/5
=23
Jason Aubrey Math 1300 Finite Mathematics
![Page 105: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/105.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Given the following probabilities for an event E , findthe odds for and against E :
P(E) = 35
Since P(E) = 3/5 we know P(E ′) = 1− P(E) = 2/5.Then the odds for E are
P(E)
P(E ′)=
3/52/5
=32
And the odds against E are
P(E ′)
P(E)=
2/53/5
=23
Jason Aubrey Math 1300 Finite Mathematics
![Page 106: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/106.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Given the following probabilities for an event E , findthe odds for and against E :
P(E) = 35
Since P(E) = 3/5 we know P(E ′) = 1− P(E) = 2/5.Then the odds for E are
P(E)
P(E ′)=
3/52/5
=32
And the odds against E are
P(E ′)
P(E)=
2/53/5
=23
Jason Aubrey Math 1300 Finite Mathematics
![Page 107: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/107.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
P(E) = 0.35
Since P(E) = 0.35 we know P(E ′) = 1− 0.35 = 0.65Then the odds for E are
P(E)
P(E ′)=
0.350.65
=7
13
And the odds against E are
P(E ′)
P(E)=
0.650.35
=137
Jason Aubrey Math 1300 Finite Mathematics
![Page 108: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/108.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
P(E) = 0.35Since P(E) = 0.35 we know P(E ′) = 1− 0.35 = 0.65
Then the odds for E are
P(E)
P(E ′)=
0.350.65
=7
13
And the odds against E are
P(E ′)
P(E)=
0.650.35
=137
Jason Aubrey Math 1300 Finite Mathematics
![Page 109: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/109.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
P(E) = 0.35Since P(E) = 0.35 we know P(E ′) = 1− 0.35 = 0.65Then the odds for E are
P(E)
P(E ′)=
0.350.65
=7
13
And the odds against E are
P(E ′)
P(E)=
0.650.35
=137
Jason Aubrey Math 1300 Finite Mathematics
![Page 110: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/110.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
P(E) = 0.35Since P(E) = 0.35 we know P(E ′) = 1− 0.35 = 0.65Then the odds for E are
P(E)
P(E ′)=
0.350.65
=7
13
And the odds against E are
P(E ′)
P(E)=
0.650.35
=137
Jason Aubrey Math 1300 Finite Mathematics
![Page 111: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/111.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven whentwo dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
n(E) = 6, n(S) = 36,
P(E) =6
36and P(E ′) =
3036
ThereforeP(E)
P(E ′)=
6/3630/36
=6
30=
15
Jason Aubrey Math 1300 Finite Mathematics
![Page 112: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/112.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven whentwo dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
n(E) = 6, n(S) = 36,
P(E) =6
36and P(E ′) =
3036
ThereforeP(E)
P(E ′)=
6/3630/36
=6
30=
15
Jason Aubrey Math 1300 Finite Mathematics
![Page 113: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/113.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven whentwo dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
n(E) = 6, n(S) = 36,
P(E) =6
36and P(E ′) =
3036
ThereforeP(E)
P(E ′)=
6/3630/36
=6
30=
15
Jason Aubrey Math 1300 Finite Mathematics
![Page 114: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/114.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven whentwo dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
n(E) = 6, n(S) = 36,
P(E) =6
36and P(E ′) =
3036
ThereforeP(E)
P(E ′)=
6/3630/36
=6
30=
15
Jason Aubrey Math 1300 Finite Mathematics
![Page 115: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/115.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: Find the odds in favor of rolling a total of seven whentwo dice are tossed.
Let E be the event that the sum of the two dice is seven. So,
E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
n(E) = 6, n(S) = 36,
P(E) =6
36and P(E ′) =
3036
ThereforeP(E)
P(E ′)=
6/3630/36
=6
30=
15
Jason Aubrey Math 1300 Finite Mathematics
![Page 116: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/116.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
If the odds for an event E areab
, then the probability of Eis,
P(E) =a
a + b
If the odds against an event E areab
then the probability ofE is
P(E) =b
a + b
Jason Aubrey Math 1300 Finite Mathematics
![Page 117: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/117.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
If the odds for an event E areab
, then the probability of Eis,
P(E) =a
a + b
If the odds against an event E areab
then the probability ofE is
P(E) =b
a + b
Jason Aubrey Math 1300 Finite Mathematics
![Page 118: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/118.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds againstrolling a 6 before rolling a 7 are 6:5, what is the probability ofrolling a 6 before rolling a 7?
Let E be the event “a 6 is rolled before a 7 is rolled”.odds against E are 6:5Therefore,
P(E) =5
6 + 5=
511
Jason Aubrey Math 1300 Finite Mathematics
![Page 119: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/119.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds againstrolling a 6 before rolling a 7 are 6:5, what is the probability ofrolling a 6 before rolling a 7?
Let E be the event “a 6 is rolled before a 7 is rolled”.
odds against E are 6:5Therefore,
P(E) =5
6 + 5=
511
Jason Aubrey Math 1300 Finite Mathematics
![Page 120: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/120.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds againstrolling a 6 before rolling a 7 are 6:5, what is the probability ofrolling a 6 before rolling a 7?
Let E be the event “a 6 is rolled before a 7 is rolled”.odds against E are 6:5
Therefore,
P(E) =5
6 + 5=
511
Jason Aubrey Math 1300 Finite Mathematics
![Page 121: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/121.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: If in repeated rolls of two fair dice the odds againstrolling a 6 before rolling a 7 are 6:5, what is the probability ofrolling a 6 before rolling a 7?
Let E be the event “a 6 is rolled before a 7 is rolled”.odds against E are 6:5Therefore,
P(E) =5
6 + 5=
511
Jason Aubrey Math 1300 Finite Mathematics
![Page 122: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/122.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: The data below was obtained from a random surveyof 1,000 residents of a state. The participants were asked theirpolitical affiliations and their preferences in an upcominggubernatorial election (D = Democrat, R = Republican, U =Unaffiliated. )
D R U TotalsCandidate A 200 100 85 385Candidate B 250 230 50 530No Preference 50 20 15 85Totals 500 350 150 1,000
If a resident of the state is selected at random, what is theempirical probability that the resident is not affiliated with apolitical party or has no preference? What are the odds for thisevent?
Jason Aubrey Math 1300 Finite Mathematics
![Page 123: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/123.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
Example: The data below was obtained from a random surveyof 1,000 residents of a state. The participants were asked theirpolitical affiliations and their preferences in an upcominggubernatorial election (D = Democrat, R = Republican, U =Unaffiliated. )
D R U TotalsCandidate A 200 100 85 385Candidate B 250 230 50 530No Preference 50 20 15 85Totals 500 350 150 1,000
If a resident of the state is selected at random, what is theempirical probability that the resident is not affiliated with apolitical party or has no preference? What are the odds for thisevent?
Jason Aubrey Math 1300 Finite Mathematics
![Page 124: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/124.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
D R U TotalsCandidate A 200 100 85 385Candidate B 250 230 50 530No Preference 50 20 15 85Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N)− P(U ∩ N)
=150
1000+
851000
− 151000
=220
1000= 0.22 or 22%
Then the odds for this event are22
100− 22=
2278
=1139
Jason Aubrey Math 1300 Finite Mathematics
![Page 125: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/125.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
D R U TotalsCandidate A 200 100 85 385Candidate B 250 230 50 530No Preference 50 20 15 85Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N)− P(U ∩ N)
=150
1000+
851000
− 151000
=220
1000= 0.22 or 22%
Then the odds for this event are22
100− 22=
2278
=1139
Jason Aubrey Math 1300 Finite Mathematics
![Page 126: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/126.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
D R U TotalsCandidate A 200 100 85 385Candidate B 250 230 50 530No Preference 50 20 15 85Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N)− P(U ∩ N)
=150
1000+
851000
− 151000
=220
1000= 0.22 or 22%
Then the odds for this event are22
100− 22=
2278
=1139
Jason Aubrey Math 1300 Finite Mathematics
![Page 127: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/127.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
D R U TotalsCandidate A 200 100 85 385Candidate B 250 230 50 530No Preference 50 20 15 85Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N)− P(U ∩ N)
=150
1000+
851000
− 151000
=220
1000= 0.22 or 22%
Then the odds for this event are22
100− 22=
2278
=1139
Jason Aubrey Math 1300 Finite Mathematics
![Page 128: Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; Odds](https://reader038.vdocument.in/reader038/viewer/2022102823/5482f1105806b51a058b4770/html5/thumbnails/128.jpg)
../images/stackedlogo-bw-medium.png
Union and IntersectionComplement of an Event
OddsApplications to Empirical Probability
D R U TotalsCandidate A 200 100 85 385Candidate B 250 230 50 530No Preference 50 20 15 85Totals 500 350 150 1,000
We are looking for P(U ∪ N):
P(U ∪ N) = P(U) + P(N)− P(U ∩ N)
=150
1000+
851000
− 151000
=220
1000= 0.22 or 22%
Then the odds for this event are22
100− 22=
2278
=1139
Jason Aubrey Math 1300 Finite Mathematics