math 1300: section 8-1 sample spaces, events, and probability
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Math 1300 Finite MathematicsSection 8-1: Sample Spaces, Events, and Probability
Jason Aubrey
Department of MathematicsUniversity of Missouri
Jason Aubrey Math 1300 Finite Mathematics
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Probability theory is a branch of mathematics that has beendeveloped do deal with outcomes of random experiments. Arandom experiment (or just experiment) is a situationinvolving chance or probability that leads to results calledoutcomes.
Jason Aubrey Math 1300 Finite Mathematics
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
DefinitionThe set S of all possible outcomes of an experiment a waythat in each trial of the experiment one and only one of theoutcomes (events) in the set will occur, we call the set S asample space for the experiment. Each element in S iscalled a simple outcome, or simple event.
An event E is defined to be any subset of S (including theempty set and the sample space S). Event E is a simpleevent if it contains only one element and a compoundevent if it contains more than one element.We say that an event E occurs if any of the simple eventsin E occurs.
Jason Aubrey Math 1300 Finite Mathematics
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
DefinitionThe set S of all possible outcomes of an experiment a waythat in each trial of the experiment one and only one of theoutcomes (events) in the set will occur, we call the set S asample space for the experiment. Each element in S iscalled a simple outcome, or simple event.An event E is defined to be any subset of S (including theempty set and the sample space S). Event E is a simpleevent if it contains only one element and a compoundevent if it contains more than one element.
We say that an event E occurs if any of the simple eventsin E occurs.
Jason Aubrey Math 1300 Finite Mathematics
![Page 5: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/5.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
DefinitionThe set S of all possible outcomes of an experiment a waythat in each trial of the experiment one and only one of theoutcomes (events) in the set will occur, we call the set S asample space for the experiment. Each element in S iscalled a simple outcome, or simple event.An event E is defined to be any subset of S (including theempty set and the sample space S). Event E is a simpleevent if it contains only one element and a compoundevent if it contains more than one element.We say that an event E occurs if any of the simple eventsin E occurs.
Jason Aubrey Math 1300 Finite Mathematics
![Page 6: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/6.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die.
Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2).
(1,5) (2,2)
What is the sample space S for this experiment?
Jason Aubrey Math 1300 Finite Mathematics
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../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5)
or (Red=2, Green=2).
(1,5)
(2,2)
What is the sample space S for this experiment?
Jason Aubrey Math 1300 Finite Mathematics
![Page 8: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/8.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2).
(1,5) (2,2)
What is the sample space S for this experiment?
Jason Aubrey Math 1300 Finite Mathematics
![Page 9: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/9.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Jason Aubrey Math 1300 Finite Mathematics
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
To clarify, the sample space is always a set of objects. In thiscase,
S =
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
We can often use the counting techniques we learned in thelast chapter to determine the size of a sample space. In thiscase, by the multiplication principle:
n(S) = 6× 6 = 36
Jason Aubrey Math 1300 Finite Mathematics
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../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
To clarify, the sample space is always a set of objects. In thiscase,
S =
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
We can often use the counting techniques we learned in thelast chapter to determine the size of a sample space. In thiscase, by the multiplication principle:
n(S) = 6× 6 = 36
Jason Aubrey Math 1300 Finite Mathematics
![Page 12: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/12.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Events are subsets of the sample space:
A simple event is an event (subset) containing only oneoutcome. For example,
E = {(3,2)}
is a simple event.A compound event is an event (subset) containing morethan one outcome. For example,
E = {(3,2), (4,1), (5,2)}
is a compound event.
Jason Aubrey Math 1300 Finite Mathematics
![Page 13: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/13.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Events are subsets of the sample space:A simple event is an event (subset) containing only oneoutcome. For example,
E = {(3,2)}
is a simple event.
A compound event is an event (subset) containing morethan one outcome. For example,
E = {(3,2), (4,1), (5,2)}
is a compound event.
Jason Aubrey Math 1300 Finite Mathematics
![Page 14: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/14.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Events are subsets of the sample space:A simple event is an event (subset) containing only oneoutcome. For example,
E = {(3,2)}
is a simple event.A compound event is an event (subset) containing morethan one outcome. For example,
E = {(3,2), (4,1), (5,2)}
is a compound event.
Jason Aubrey Math 1300 Finite Mathematics
![Page 15: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/15.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Events will often be described in words, and the first step willbe to determine the correct subset of the sample space.
Forexample
“A sum of 11 turns up” corresponds to the event
E = {(5,6), (6,5)}.
Notice that n(E) = 2.“The numbers on the two dice are equal” corresponds tothe event
F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.
Here we have n(F ) = 6.“A sum less than or equal to 3” corresponds to the event:
G = {(1,1), (1,2), (2,1)}
Here n(G) = 3
Jason Aubrey Math 1300 Finite Mathematics
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../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Events will often be described in words, and the first step willbe to determine the correct subset of the sample space. Forexample
“A sum of 11 turns up” corresponds to the event
E = {(5,6), (6,5)}.
Notice that n(E) = 2.
“The numbers on the two dice are equal” corresponds tothe event
F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.
Here we have n(F ) = 6.“A sum less than or equal to 3” corresponds to the event:
G = {(1,1), (1,2), (2,1)}
Here n(G) = 3
Jason Aubrey Math 1300 Finite Mathematics
![Page 17: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/17.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Events will often be described in words, and the first step willbe to determine the correct subset of the sample space. Forexample
“A sum of 11 turns up” corresponds to the event
E = {(5,6), (6,5)}.
Notice that n(E) = 2.“The numbers on the two dice are equal” corresponds tothe event
F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.
Here we have n(F ) = 6.
“A sum less than or equal to 3” corresponds to the event:
G = {(1,1), (1,2), (2,1)}
Here n(G) = 3
Jason Aubrey Math 1300 Finite Mathematics
![Page 18: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/18.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Events will often be described in words, and the first step willbe to determine the correct subset of the sample space. Forexample
“A sum of 11 turns up” corresponds to the event
E = {(5,6), (6,5)}.
Notice that n(E) = 2.“The numbers on the two dice are equal” corresponds tothe event
F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.
Here we have n(F ) = 6.“A sum less than or equal to 3” corresponds to the event:
G = {(1,1), (1,2), (2,1)}
Here n(G) = 3Jason Aubrey Math 1300 Finite Mathematics
![Page 19: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/19.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.
(a) What is the sample space for this experiment?
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
(b) Identify the event “the outcome is a number greater than15”?
E = {16,17,18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
Jason Aubrey Math 1300 Finite Mathematics
![Page 20: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/20.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.
(a) What is the sample space for this experiment?
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
(b) Identify the event “the outcome is a number greater than15”?
E = {16,17,18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
Jason Aubrey Math 1300 Finite Mathematics
![Page 21: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/21.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.
(a) What is the sample space for this experiment?
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
(b) Identify the event “the outcome is a number greater than15”?
E = {16,17,18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
Jason Aubrey Math 1300 Finite Mathematics
![Page 22: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/22.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.
(a) What is the sample space for this experiment?
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
(b) Identify the event “the outcome is a number greater than15”?
E = {16,17,18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
Jason Aubrey Math 1300 Finite Mathematics
![Page 23: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/23.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.
(a) What is the sample space for this experiment?
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
(b) Identify the event “the outcome is a number greater than15”?
E = {16,17,18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
Jason Aubrey Math 1300 Finite Mathematics
![Page 24: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/24.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.
(a) What is the sample space for this experiment?
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
(b) Identify the event “the outcome is a number greater than15”?
E = {16,17,18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
Jason Aubrey Math 1300 Finite Mathematics
![Page 25: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/25.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.
(a) What is the sample space for this experiment?
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
(b) Identify the event “the outcome is a number greater than15”?
E = {16,17,18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
Jason Aubrey Math 1300 Finite Mathematics
![Page 26: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/26.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
The first step in defining the probability of an event is to assignprobabilities to each of the outcomes (simple events) in thesample space.
Suppose we flip a fair coin twice. The sample space is
S = {HH,HT ,TH,TT}
Since the coin is fair, each of the four outcomes is equallylikely, so P(HH) = P(HT ) = P(TH) = P(TT ) = 1
4 .
Jason Aubrey Math 1300 Finite Mathematics
![Page 27: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/27.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
The first step in defining the probability of an event is to assignprobabilities to each of the outcomes (simple events) in thesample space.
Suppose we flip a fair coin twice. The sample space is
S = {HH,HT ,TH,TT}
Since the coin is fair, each of the four outcomes is equallylikely, so P(HH) = P(HT ) = P(TH) = P(TT ) = 1
4 .
Jason Aubrey Math 1300 Finite Mathematics
![Page 28: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/28.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
The first step in defining the probability of an event is to assignprobabilities to each of the outcomes (simple events) in thesample space.
Suppose we flip a fair coin twice. The sample space is
S = {HH,HT ,TH,TT}
Since the coin is fair, each of the four outcomes is equallylikely, so P(HH) = P(HT ) = P(TH) = P(TT ) = 1
4 .
Jason Aubrey Math 1300 Finite Mathematics
![Page 29: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/29.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Suppose the local meteorologist determines that thechance of rain is 15%. As an experiment, we go out toobserve the weather. The sample space is
S = {Rain,No Rain}
The two outcomes here are not equally likely. We haveP(Rain) = 0.15 and P(No Rain) = 0.85.
Notice that in both cases, each probability was between zeroand one, and the sum of all of the probabilities was one.
Jason Aubrey Math 1300 Finite Mathematics
![Page 30: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/30.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Suppose the local meteorologist determines that thechance of rain is 15%. As an experiment, we go out toobserve the weather. The sample space is
S = {Rain,No Rain}
The two outcomes here are not equally likely. We haveP(Rain) = 0.15 and P(No Rain) = 0.85.
Notice that in both cases, each probability was between zeroand one, and the sum of all of the probabilities was one.
Jason Aubrey Math 1300 Finite Mathematics
![Page 31: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/31.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Suppose the local meteorologist determines that thechance of rain is 15%. As an experiment, we go out toobserve the weather. The sample space is
S = {Rain,No Rain}
The two outcomes here are not equally likely. We haveP(Rain) = 0.15 and P(No Rain) = 0.85.
Notice that in both cases, each probability was between zeroand one, and the sum of all of the probabilities was one.
Jason Aubrey Math 1300 Finite Mathematics
![Page 32: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/32.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probabilities of Simple Events)Given a sample space
S = {e1,e2, . . . ,en}
with n simple events, to each simple event ei we assign a realnumber, denoted by P(ei), called the probability of the eventei .
1 The probability of a simple event is a number between 0and 1, inclusive. That is, 0 ≤ P(ei) ≤ 1.
2 The sum of the probabilities of all simple events in thesample space is 1. That is,
P(e1) + P(e2) + · · ·+ P(en) = 1.
Jason Aubrey Math 1300 Finite Mathematics
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probabilities of Simple Events)Given a sample space
S = {e1,e2, . . . ,en}
with n simple events, to each simple event ei we assign a realnumber, denoted by P(ei), called the probability of the eventei .
1 The probability of a simple event is a number between 0and 1, inclusive. That is, 0 ≤ P(ei) ≤ 1.
2 The sum of the probabilities of all simple events in thesample space is 1. That is,
P(e1) + P(e2) + · · ·+ P(en) = 1.
Jason Aubrey Math 1300 Finite Mathematics
![Page 34: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/34.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probabilities of Simple Events)Given a sample space
S = {e1,e2, . . . ,en}
with n simple events, to each simple event ei we assign a realnumber, denoted by P(ei), called the probability of the eventei .
1 The probability of a simple event is a number between 0and 1, inclusive. That is, 0 ≤ P(ei) ≤ 1.
2 The sum of the probabilities of all simple events in thesample space is 1. That is,
P(e1) + P(e2) + · · ·+ P(en) = 1.
Jason Aubrey Math 1300 Finite Mathematics
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../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Two coin flips. . .
S = {HH,HT ,TH,TT}
e P(e)HH 1
4HT 1
4TH 1
4TT 1
4
A possibly rainy day. . .
S = {Rain,No Rain}
e P(e)Rain 0.15
No Rain 0.85
Jason Aubrey Math 1300 Finite Mathematics
![Page 36: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/36.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose a fair coin is flipped twice. What is theprobability that exactly one head turns up.
The event “exactly one head turns up” is the set
E = {HT ,TH}
We know that P(HT ) = 14 and P(TH) = 1
4 .
To determine P(E), just add the probabilities of the simpleevents in E .
P(E) = P(HT ) + P(TH) =14+
14=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 37: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/37.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose a fair coin is flipped twice. What is theprobability that exactly one head turns up.
The event “exactly one head turns up” is the set
E = {HT ,TH}
We know that P(HT ) = 14 and P(TH) = 1
4 .
To determine P(E), just add the probabilities of the simpleevents in E .
P(E) = P(HT ) + P(TH) =14+
14=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 38: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/38.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose a fair coin is flipped twice. What is theprobability that exactly one head turns up.
The event “exactly one head turns up” is the set
E = {HT ,TH}
We know that P(HT ) = 14 and P(TH) = 1
4 .
To determine P(E), just add the probabilities of the simpleevents in E .
P(E) = P(HT ) + P(TH) =14+
14=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 39: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/39.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose a fair coin is flipped twice. What is theprobability that exactly one head turns up.
The event “exactly one head turns up” is the set
E = {HT ,TH}
We know that P(HT ) = 14 and P(TH) = 1
4 .
To determine P(E), just add the probabilities of the simpleevents in E .
P(E) = P(HT ) + P(TH) =14+
14=
12
Jason Aubrey Math 1300 Finite Mathematics
![Page 40: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/40.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we define the probability of an arbitraryevent E , denoted by P(E), as follows:
1 If E is the empty set, then P(E) = 0.2 If E is a simple event, i.e. E = {ei}, then P(E) = P(ei) as
defined previously.3 If E is a compound event, then P(E) is the sum of the
probabilities of all the simple events in E .4 If E is the sample space S, then P(E) = P(S) = 1.
Jason Aubrey Math 1300 Finite Mathematics
![Page 41: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/41.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we define the probability of an arbitraryevent E , denoted by P(E), as follows:
1 If E is the empty set, then P(E) = 0.
2 If E is a simple event, i.e. E = {ei}, then P(E) = P(ei) asdefined previously.
3 If E is a compound event, then P(E) is the sum of theprobabilities of all the simple events in E .
4 If E is the sample space S, then P(E) = P(S) = 1.
Jason Aubrey Math 1300 Finite Mathematics
![Page 42: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/42.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we define the probability of an arbitraryevent E , denoted by P(E), as follows:
1 If E is the empty set, then P(E) = 0.2 If E is a simple event, i.e. E = {ei}, then P(E) = P(ei) as
defined previously.
3 If E is a compound event, then P(E) is the sum of theprobabilities of all the simple events in E .
4 If E is the sample space S, then P(E) = P(S) = 1.
Jason Aubrey Math 1300 Finite Mathematics
![Page 43: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/43.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we define the probability of an arbitraryevent E , denoted by P(E), as follows:
1 If E is the empty set, then P(E) = 0.2 If E is a simple event, i.e. E = {ei}, then P(E) = P(ei) as
defined previously.3 If E is a compound event, then P(E) is the sum of the
probabilities of all the simple events in E .
4 If E is the sample space S, then P(E) = P(S) = 1.
Jason Aubrey Math 1300 Finite Mathematics
![Page 44: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/44.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Definition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we define the probability of an arbitraryevent E , denoted by P(E), as follows:
1 If E is the empty set, then P(E) = 0.2 If E is a simple event, i.e. E = {ei}, then P(E) = P(ei) as
defined previously.3 If E is a compound event, then P(E) is the sum of the
probabilities of all the simple events in E .4 If E is the sample space S, then P(E) = P(S) = 1.
Jason Aubrey Math 1300 Finite Mathematics
![Page 45: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/45.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: In a family with 3 children, excluding multiple births,what is the probability of having exactly 2 girls? Assume that aboy is as likely as a girl at each birth.
First we determine the sample space S:
S = {GGG,GGB,GBG,BGG,GBB,BGB,BBG,BBB}
Since a boy is as likely as a girl at each birth, each of the 8outcomes in S is equally likely; so each outcome hasprobability 1
8 .
Jason Aubrey Math 1300 Finite Mathematics
![Page 46: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/46.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: In a family with 3 children, excluding multiple births,what is the probability of having exactly 2 girls? Assume that aboy is as likely as a girl at each birth.
First we determine the sample space S:
S = {GGG,GGB,GBG,BGG,GBB,BGB,BBG,BBB}
Since a boy is as likely as a girl at each birth, each of the 8outcomes in S is equally likely; so each outcome hasprobability 1
8 .
Jason Aubrey Math 1300 Finite Mathematics
![Page 47: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/47.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: In a family with 3 children, excluding multiple births,what is the probability of having exactly 2 girls? Assume that aboy is as likely as a girl at each birth.
First we determine the sample space S:
S = {GGG,GGB,GBG,BGG,GBB,BGB,BBG,BBB}
Since a boy is as likely as a girl at each birth, each of the 8outcomes in S is equally likely; so each outcome hasprobability 1
8 .
Jason Aubrey Math 1300 Finite Mathematics
![Page 48: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/48.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Now we identify the event we wish to find the probability of:
E = {GGB,GBG,BGG}
Therefore,
P(E) = P(GGB) + P(GBG) + P(BGG)
=18+
18+
18=
38
Jason Aubrey Math 1300 Finite Mathematics
![Page 49: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/49.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Now we identify the event we wish to find the probability of:
E = {GGB,GBG,BGG}
Therefore,
P(E) = P(GGB) + P(GBG) + P(BGG)
=18+
18+
18=
38
Jason Aubrey Math 1300 Finite Mathematics
![Page 50: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/50.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Procedure: Steps for Finding the Probability of an Event E
1 Set up an appropriate sample space S for the experiment.2 Assign acceptable probabilities to the simple events in S.3 To obtain the probability of an arbitrary event E , add the
probabilities of the simple events in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 51: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/51.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Procedure: Steps for Finding the Probability of an Event E1 Set up an appropriate sample space S for the experiment.
2 Assign acceptable probabilities to the simple events in S.3 To obtain the probability of an arbitrary event E , add the
probabilities of the simple events in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 52: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/52.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Procedure: Steps for Finding the Probability of an Event E1 Set up an appropriate sample space S for the experiment.2 Assign acceptable probabilities to the simple events in S.
3 To obtain the probability of an arbitrary event E , add theprobabilities of the simple events in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 53: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/53.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Procedure: Steps for Finding the Probability of an Event E1 Set up an appropriate sample space S for the experiment.2 Assign acceptable probabilities to the simple events in S.3 To obtain the probability of an arbitrary event E , add the
probabilities of the simple events in E .
Jason Aubrey Math 1300 Finite Mathematics
![Page 54: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/54.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Recall two past examples. . .
Two coin flips. . .
S = {HH,HT ,TH,TT}
e P(e)HH 1
4HT 1
4TH 1
4TT 1
4
The outcomes are equallylikely.
A possibly rainy day. . .
S = {Rain,No Rain}
e P(e)Rain 0.15
No Rain 0.85
The outcomes are not equallylikely.
Jason Aubrey Math 1300 Finite Mathematics
![Page 55: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/55.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Recall two past examples. . .
Two coin flips. . .
S = {HH,HT ,TH,TT}
e P(e)HH 1
4HT 1
4TH 1
4TT 1
4
The outcomes are equallylikely.
A possibly rainy day. . .
S = {Rain,No Rain}
e P(e)Rain 0.15
No Rain 0.85
The outcomes are not equallylikely.
Jason Aubrey Math 1300 Finite Mathematics
![Page 56: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/56.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Recall two past examples. . .
Two coin flips. . .
S = {HH,HT ,TH,TT}
e P(e)HH 1
4HT 1
4TH 1
4TT 1
4
The outcomes are equallylikely.
A possibly rainy day. . .
S = {Rain,No Rain}
e P(e)Rain 0.15
No Rain 0.85
The outcomes are not equallylikely.
Jason Aubrey Math 1300 Finite Mathematics
![Page 57: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/57.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Sometimes we can assume that all outcomes in a samplespace are equally likely.
If S = {e1,e2, . . . ,en} is a sample space in which alloutcomes are equally likely, then we assign the probability1n to each outcome. That is
P(ei) =1n
and we have. . .
Jason Aubrey Math 1300 Finite Mathematics
![Page 58: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/58.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Sometimes we can assume that all outcomes in a samplespace are equally likely.If S = {e1,e2, . . . ,en} is a sample space in which alloutcomes are equally likely, then we assign the probability1n to each outcome. That is
P(ei) =1n
and we have. . .
Jason Aubrey Math 1300 Finite Mathematics
![Page 59: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/59.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Theorem (Probability of an Arbitrary Event under an EquallyLikely Assumption)If we assume that each simple event in a sample space S is aslikely to occur as any other, then the probability of an arbitraryevent E in S is given by
P(E) =number of elements in Enumber of elements in S
=n(E)
n(S).
Jason Aubrey Math 1300 Finite Mathematics
![Page 60: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/60.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die.
Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2).
(1,5) (2,2)
(a) What is the probability that the sum on the two dice comesout to be 11?
Jason Aubrey Math 1300 Finite Mathematics
![Page 61: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/61.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5)
or (Red=2, Green=2).
(1,5)
(2,2)
(a) What is the probability that the sum on the two dice comesout to be 11?
Jason Aubrey Math 1300 Finite Mathematics
![Page 62: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/62.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2).
(1,5) (2,2)
(a) What is the probability that the sum on the two dice comesout to be 11?
Jason Aubrey Math 1300 Finite Mathematics
![Page 63: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/63.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2).
(1,5) (2,2)
(a) What is the probability that the sum on the two dice comesout to be 11?
Jason Aubrey Math 1300 Finite Mathematics
![Page 64: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/64.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1,5), (3,2), etc, is just as likely as any other. So,we can make an equally likely assumption.
We know from earlier that n(S) = 36.E = {(5,6), (6,5)}, so n(E) = 2.Therefore
P(E) =n(E)
n(S)=
236
=1
18
Jason Aubrey Math 1300 Finite Mathematics
![Page 65: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/65.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1,5), (3,2), etc, is just as likely as any other. So,we can make an equally likely assumption.We know from earlier that n(S) = 36.
E = {(5,6), (6,5)}, so n(E) = 2.Therefore
P(E) =n(E)
n(S)=
236
=1
18
Jason Aubrey Math 1300 Finite Mathematics
![Page 66: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/66.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1,5), (3,2), etc, is just as likely as any other. So,we can make an equally likely assumption.We know from earlier that n(S) = 36.E = {(5,6), (6,5)}, so n(E) = 2.
ThereforeP(E) =
n(E)
n(S)=
236
=1
18
Jason Aubrey Math 1300 Finite Mathematics
![Page 67: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/67.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1,5), (3,2), etc, is just as likely as any other. So,we can make an equally likely assumption.We know from earlier that n(S) = 36.E = {(5,6), (6,5)}, so n(E) = 2.Therefore
P(E) =n(E)
n(S)=
236
=1
18
Jason Aubrey Math 1300 Finite Mathematics
![Page 68: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/68.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) What is the probability that the numbers on the dice areequal?
The event here isF = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}So, n(F ) = 6Therefore,
P(F ) =n(F )
n(S)=
636
=16
Jason Aubrey Math 1300 Finite Mathematics
![Page 69: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/69.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) What is the probability that the numbers on the dice areequal?
The event here isF = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
So, n(F ) = 6Therefore,
P(F ) =n(F )
n(S)=
636
=16
Jason Aubrey Math 1300 Finite Mathematics
![Page 70: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/70.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) What is the probability that the numbers on the dice areequal?
The event here isF = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}So, n(F ) = 6
Therefore,
P(F ) =n(F )
n(S)=
636
=16
Jason Aubrey Math 1300 Finite Mathematics
![Page 71: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/71.jpg)
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ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) What is the probability that the numbers on the dice areequal?
The event here isF = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}So, n(F ) = 6Therefore,
P(F ) =n(F )
n(S)=
636
=16
Jason Aubrey Math 1300 Finite Mathematics
![Page 72: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/72.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: 5 cards are drawn simultaneously from a standarddeck of 52 cards.
(a) Describe the sample space S. What is n(S)?
Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as
S = {all possible 5 card hands}
How many 5-card hands can be drawn from a 52-card deck?From the previous chapter, we know this is
n(S) = C(52,5) = 2,598,960
When the cards are dealt, each card is just as likely as anyother, so any five card hand is just as likely as any other. Inother words, we can make an equally likely assumption.
Jason Aubrey Math 1300 Finite Mathematics
![Page 73: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/73.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: 5 cards are drawn simultaneously from a standarddeck of 52 cards.
(a) Describe the sample space S. What is n(S)?
Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as
S = {all possible 5 card hands}
How many 5-card hands can be drawn from a 52-card deck?From the previous chapter, we know this is
n(S) = C(52,5) = 2,598,960
When the cards are dealt, each card is just as likely as anyother, so any five card hand is just as likely as any other. Inother words, we can make an equally likely assumption.
Jason Aubrey Math 1300 Finite Mathematics
![Page 74: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/74.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: 5 cards are drawn simultaneously from a standarddeck of 52 cards.
(a) Describe the sample space S. What is n(S)?
Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as
S = {all possible 5 card hands}
How many 5-card hands can be drawn from a 52-card deck?
From the previous chapter, we know this is
n(S) = C(52,5) = 2,598,960
When the cards are dealt, each card is just as likely as anyother, so any five card hand is just as likely as any other. Inother words, we can make an equally likely assumption.
Jason Aubrey Math 1300 Finite Mathematics
![Page 75: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/75.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: 5 cards are drawn simultaneously from a standarddeck of 52 cards.
(a) Describe the sample space S. What is n(S)?
Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as
S = {all possible 5 card hands}
How many 5-card hands can be drawn from a 52-card deck?From the previous chapter, we know this is
n(S) = C(52,5) = 2,598,960
When the cards are dealt, each card is just as likely as anyother, so any five card hand is just as likely as any other. Inother words, we can make an equally likely assumption.
Jason Aubrey Math 1300 Finite Mathematics
![Page 76: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/76.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
Example: 5 cards are drawn simultaneously from a standarddeck of 52 cards.
(a) Describe the sample space S. What is n(S)?
Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as
S = {all possible 5 card hands}
How many 5-card hands can be drawn from a 52-card deck?From the previous chapter, we know this is
n(S) = C(52,5) = 2,598,960
When the cards are dealt, each card is just as likely as anyother, so any five card hand is just as likely as any other. Inother words, we can make an equally likely assumption.
Jason Aubrey Math 1300 Finite Mathematics
![Page 77: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/77.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(b)Find the probability that all of the cards are hearts.
The event “all of the cards are hearts” is the set
E = {all 5 card hands with only hearts}
Since there are 13 hearts in a standard deck of cards, we have
n(E) = C(13,5) = 1287
By the equally likely assumption
P(E) =n(E)
n(S)=
12872,598,960
≈ 0.000495
or about 0.05%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 78: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/78.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(b)Find the probability that all of the cards are hearts.
The event “all of the cards are hearts” is the set
E = {all 5 card hands with only hearts}
Since there are 13 hearts in a standard deck of cards, we have
n(E) = C(13,5) = 1287
By the equally likely assumption
P(E) =n(E)
n(S)=
12872,598,960
≈ 0.000495
or about 0.05%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 79: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/79.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(b)Find the probability that all of the cards are hearts.
The event “all of the cards are hearts” is the set
E = {all 5 card hands with only hearts}
Since there are 13 hearts in a standard deck of cards, we have
n(E) = C(13,5) = 1287
By the equally likely assumption
P(E) =n(E)
n(S)=
12872,598,960
≈ 0.000495
or about 0.05%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 80: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/80.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(b)Find the probability that all of the cards are hearts.
The event “all of the cards are hearts” is the set
E = {all 5 card hands with only hearts}
Since there are 13 hearts in a standard deck of cards, we have
n(E) = C(13,5) = 1287
By the equally likely assumption
P(E) =n(E)
n(S)=
12872,598,960
≈ 0.000495
or about 0.05%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 81: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/81.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings).
The event “all the cards are face cards” is the set
F = {all 5 card hands consisting only of face cards}
There are a total of 4× 3 = 12 face cards. So,
n(F ) = C(12,5) = 792
By the equally likely assumption
P(F ) =n(F )
n(S)=
7922,598,960
≈ 0.000305
or about 0.03%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 82: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/82.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings).
The event “all the cards are face cards” is the set
F = {all 5 card hands consisting only of face cards}
There are a total of 4× 3 = 12 face cards. So,
n(F ) = C(12,5) = 792
By the equally likely assumption
P(F ) =n(F )
n(S)=
7922,598,960
≈ 0.000305
or about 0.03%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 83: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/83.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings).
The event “all the cards are face cards” is the set
F = {all 5 card hands consisting only of face cards}
There are a total of 4× 3 = 12 face cards. So,
n(F ) = C(12,5) = 792
By the equally likely assumption
P(F ) =n(F )
n(S)=
7922,598,960
≈ 0.000305
or about 0.03%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 84: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/84.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings).
The event “all the cards are face cards” is the set
F = {all 5 card hands consisting only of face cards}
There are a total of 4× 3 = 12 face cards. So,
n(F ) = C(12,5) = 792
By the equally likely assumption
P(F ) =n(F )
n(S)=
7922,598,960
≈ 0.000305
or about 0.03%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 85: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/85.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13).
The event “all the cards are even is the set
G = {all 5 card hands consisting of only even cards}
There are 6 even cards per suit (2,4,6,8,10,Q); so there are atotal of 20 even cards in a deck. So,
n(G) = C(20,6) = 38,760.
By the qually likely assumption,
P(G) =n(G)
n(S)=
38,7602,598,960
≈ 0.0149
or about 14.9%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 86: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/86.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13).
The event “all the cards are even is the set
G = {all 5 card hands consisting of only even cards}
There are 6 even cards per suit (2,4,6,8,10,Q); so there are atotal of 20 even cards in a deck. So,
n(G) = C(20,6) = 38,760.
By the qually likely assumption,
P(G) =n(G)
n(S)=
38,7602,598,960
≈ 0.0149
or about 14.9%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 87: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/87.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13).
The event “all the cards are even is the set
G = {all 5 card hands consisting of only even cards}
There are 6 even cards per suit (2,4,6,8,10,Q); so there are atotal of 20 even cards in a deck. So,
n(G) = C(20,6) = 38,760.
By the qually likely assumption,
P(G) =n(G)
n(S)=
38,7602,598,960
≈ 0.0149
or about 14.9%.
Jason Aubrey Math 1300 Finite Mathematics
![Page 88: Math 1300: Section 8-1 Sample Spaces, Events, and Probability](https://reader038.vdocument.in/reader038/viewer/2022110115/5482a8c7b47959f60c8b4857/html5/thumbnails/88.jpg)
../images/stackedlogo-bw-medium.png
ExperimentsSample Spaces and Events
Probability of an EventEqually Likely Assumption
(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13).
The event “all the cards are even is the set
G = {all 5 card hands consisting of only even cards}
There are 6 even cards per suit (2,4,6,8,10,Q); so there are atotal of 20 even cards in a deck. So,
n(G) = C(20,6) = 38,760.
By the qually likely assumption,
P(G) =n(G)
n(S)=
38,7602,598,960
≈ 0.0149
or about 14.9%.Jason Aubrey Math 1300 Finite Mathematics