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Math 1300: Section 8-1 Sample Spaces, Events, and Probability
1. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Math 1300 Finite Mathematics
Section 8-1: Sample Spaces, Events, and Probability
Jason Aubrey
Department of Mathematics
University of Missouri
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Jason Aubrey Math 1300 Finite Mathematics
2. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Probability theory is a branch of mathematics that has been
developed do deal with outcomes of random experiments. A
random experiment (or just experiment) is a situation
involving chance or probability that leads to results called
outcomes.
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Jason Aubrey Math 1300 Finite Mathematics
3. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition
The set S of all possible outcomes of an experiment a way
that in each trial of the experiment one and only one of the
outcomes (events) in the set will occur, we call the set S a
sample space for the experiment. Each element in S is
called a simple outcome, or simple event.
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Jason Aubrey Math 1300 Finite Mathematics
4. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition
The set S of all possible outcomes of an experiment a way
that in each trial of the experiment one and only one of the
outcomes (events) in the set will occur, we call the set S a
sample space for the experiment. Each element in S is
called a simple outcome, or simple event.
An event E is defined to be any subset of S (including the
empty set and the sample space S). Event E is a simple
event if it contains only one element and a compound
event if it contains more than one element.
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Jason Aubrey Math 1300 Finite Mathematics
5. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition
The set S of all possible outcomes of an experiment a way
that in each trial of the experiment one and only one of the
outcomes (events) in the set will occur, we call the set S a
sample space for the experiment. Each element in S is
called a simple outcome, or simple event.
An event E is defined to be any subset of S (including the
empty set and the sample space S). Event E is a simple
event if it contains only one element and a compound
event if it contains more than one element.
We say that an event E occurs if any of the simple events
in E occurs.
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Jason Aubrey Math 1300 Finite Mathematics
6. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose an experiment consists of simultaneously
rolling two fair six-sided dice (say, one red die and one green
die) and recording the values on each die.
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Jason Aubrey Math 1300 Finite Mathematics
7. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose an experiment consists of simultaneously
rolling two fair six-sided dice (say, one red die and one green
die) and recording the values on each die. Some possibilities
are (Red=1, Green=5)
(1, 5)
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Jason Aubrey Math 1300 Finite Mathematics
8. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose an experiment consists of simultaneously
rolling two fair six-sided dice (say, one red die and one green
die) and recording the values on each die. Some possibilities
are (Red=1, Green=5) or (Red=2, Green=2).
(1, 5) (2, 2)
What is the sample space S for this experiment?
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Jason Aubrey Math 1300 Finite Mathematics
10. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
To clarify, the sample space is always a set of objects. In this
case,
(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),
S=
(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)
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Jason Aubrey Math 1300 Finite Mathematics
11. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
To clarify, the sample space is always a set of objects. In this
case,
(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),
S=
(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 the
last chapter to determine the size of a sample space. In this
case, by the multiplication principle:
n(S) = 6 × 6 = 36
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Jason Aubrey Math 1300 Finite Mathematics
12. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Events are subsets of the sample space:
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Jason Aubrey Math 1300 Finite Mathematics
13. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Events are subsets of the sample space:
A simple event is an event (subset) containing only one
outcome. For example,
E = {(3, 2)}
is a simple event.
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Jason Aubrey Math 1300 Finite Mathematics
14. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Events are subsets of the sample space:
A simple event is an event (subset) containing only one
outcome. For example,
E = {(3, 2)}
is a simple event.
A compound event is an event (subset) containing more
than one outcome. For example,
E = {(3, 2), (4, 1), (5, 2)}
is a compound event.
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Jason Aubrey Math 1300 Finite Mathematics
15. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Events will often be described in words, and the first step will
be to determine the correct subset of the sample space.
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Jason Aubrey Math 1300 Finite Mathematics
16. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Events will often be described in words, and the first step will
be to determine the correct subset of the sample space. For
example
“A sum of 11 turns up” corresponds to the event
E = {(5, 6), (6, 5)}.
Notice that n(E) = 2.
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Jason Aubrey Math 1300 Finite Mathematics
17. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Events will often be described in words, and the first step will
be to determine the correct subset of the sample space. For
example
“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 to
the event
F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}.
Here we have n(F ) = 6.
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Jason Aubrey Math 1300 Finite Mathematics
18. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Events will often be described in words, and the first step will
be to determine the correct subset of the sample space. For
example
“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 to
the 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)}
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Here n(G) = 3
Jason Aubrey Math 1300 Finite Mathematics
19. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spun
and allowed to come to rest so that a pointer points within a
numbered sector.
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Jason Aubrey Math 1300 Finite Mathematics
20. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spun
and allowed to come to rest so that a pointer points within a
numbered sector.
(a) What is the sample space for this experiment?
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Jason Aubrey Math 1300 Finite Mathematics
21. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spun
and allowed to come to rest so that a pointer points within a
numbered 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}
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Jason Aubrey Math 1300 Finite Mathematics
22. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spun
and allowed to come to rest so that a pointer points within a
numbered 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 than
15”?
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Jason Aubrey Math 1300 Finite Mathematics
23. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spun
and allowed to come to rest so that a pointer points within a
numbered 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 than
15”?
E = {16, 17, 18}
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Jason Aubrey Math 1300 Finite Mathematics
24. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spun
and allowed to come to rest so that a pointer points within a
numbered 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 than
15”?
E = {16, 17, 18}
(c) Identify the event “the outcome is a number divisible by 12”?
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Jason Aubrey Math 1300 Finite Mathematics
25. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: A wheel with 18 numbers on the perimeter is spun
and allowed to come to rest so that a pointer points within a
numbered 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 than
15”?
E = {16, 17, 18}
(c) Identify the event “the outcome is a number divisible by 12”?
E = {12}
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Jason Aubrey Math 1300 Finite Mathematics
26. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
The first step in defining the probability of an event is to assign
probabilities to each of the outcomes (simple events) in the
sample space.
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Jason Aubrey Math 1300 Finite Mathematics
27. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
The first step in defining the probability of an event is to assign
probabilities to each of the outcomes (simple events) in the
sample space.
Suppose we flip a fair coin twice. The sample space is
S = {HH, HT , TH, TT }
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Jason Aubrey Math 1300 Finite Mathematics
28. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
The first step in defining the probability of an event is to assign
probabilities to each of the outcomes (simple events) in the
sample 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 equally
likely, so P(HH) = P(HT ) = P(TH) = P(TT ) = 1 . 4
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Jason Aubrey Math 1300 Finite Mathematics
29. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Suppose the local meteorologist determines that the
chance of rain is 15%. As an experiment, we go out to
observe the weather. The sample space is
S = {Rain, No Rain}
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Jason Aubrey Math 1300 Finite Mathematics
30. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Suppose the local meteorologist determines that the
chance of rain is 15%. As an experiment, we go out to
observe the weather. The sample space is
S = {Rain, No Rain}
The two outcomes here are not equally likely. We have
P(Rain) = 0.15 and P(No Rain) = 0.85.
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Jason Aubrey Math 1300 Finite Mathematics
31. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Suppose the local meteorologist determines that the
chance of rain is 15%. As an experiment, we go out to
observe the weather. The sample space is
S = {Rain, No Rain}
The two outcomes here are not equally likely. We have
P(Rain) = 0.15 and P(No Rain) = 0.85.
Notice that in both cases, each probability was between zero
and one, and the sum of all of the probabilities was one.
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Jason Aubrey Math 1300 Finite Mathematics
32. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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 real
number, denoted by P(ei ), called the probability of the event
ei .
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Jason Aubrey Math 1300 Finite Mathematics
33. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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 real
number, denoted by P(ei ), called the probability of the event
ei .
1 The probability of a simple event is a number between 0
and 1, inclusive. That is, 0 ≤ P(ei ) ≤ 1.
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Jason Aubrey Math 1300 Finite Mathematics
34. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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 real
number, denoted by P(ei ), called the probability of the event
ei .
1 The probability of a simple event is a number between 0
and 1, inclusive. That is, 0 ≤ P(ei ) ≤ 1.
2 The sum of the probabilities of all simple events in the
sample space is 1. That is,
P(e1 ) + P(e2 ) + · · · + P(en ) = 1.
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Jason Aubrey Math 1300 Finite Mathematics
35. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Two coin flips. . . A possibly rainy day. . .
S = {Rain, No Rain}
S = {HH, HT , TH, TT }
e P(e)
e P(e) Rain 0.15
1
HH 4 No Rain 0.85
1
HT 4
1
TH 4
1
TT 4
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Jason Aubrey Math 1300 Finite Mathematics
36. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose a fair coin is flipped twice. What is the
probability that exactly one head turns up.
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Jason Aubrey Math 1300 Finite Mathematics
37. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose a fair coin is flipped twice. What is the
probability that exactly one head turns up.
The event “exactly one head turns up” is the set
E = {HT , TH}
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Jason Aubrey Math 1300 Finite Mathematics
38. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose a fair coin is flipped twice. What is the
probability that exactly one head turns up.
The event “exactly one head turns up” is the set
E = {HT , TH}
1
We know that P(HT ) = 4 and P(TH) = 1 .
4
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Jason Aubrey Math 1300 Finite Mathematics
39. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose a fair coin is flipped twice. What is the
probability that exactly one head turns up.
The event “exactly one head turns up” is the set
E = {HT , TH}
1
We know that P(HT ) = 4 and P(TH) = 1 .
4
To determine P(E), just add the probabilities of the simple
events in E.
1 1 1
P(E) = P(HT ) + P(TH) = + =
4 4 2
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Jason Aubrey Math 1300 Finite Mathematics
40. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition (Probability of an Event E)
Given a probability assignment for the simple events in a
sample space S, we define the probability of an arbitrary
event E, denoted by P(E), as follows:
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Jason Aubrey Math 1300 Finite Mathematics
41. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition (Probability of an Event E)
Given a probability assignment for the simple events in a
sample space S, we define the probability of an arbitrary
event E, denoted by P(E), as follows:
1 If E is the empty set, then P(E) = 0.
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Jason Aubrey Math 1300 Finite Mathematics
42. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition (Probability of an Event E)
Given a probability assignment for the simple events in a
sample space S, we define the probability of an arbitrary
event 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.
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Jason Aubrey Math 1300 Finite Mathematics
43. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition (Probability of an Event E)
Given a probability assignment for the simple events in a
sample space S, we define the probability of an arbitrary
event 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.
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Jason Aubrey Math 1300 Finite Mathematics
44. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Definition (Probability of an Event E)
Given a probability assignment for the simple events in a
sample space S, we define the probability of an arbitrary
event 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.
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Jason Aubrey Math 1300 Finite Mathematics
45. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: In a family with 3 children, excluding multiple births,
what is the probability of having exactly 2 girls? Assume that a
boy is as likely as a girl at each birth.
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Jason Aubrey Math 1300 Finite Mathematics
46. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: In a family with 3 children, excluding multiple births,
what is the probability of having exactly 2 girls? Assume that a
boy 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}
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Jason Aubrey Math 1300 Finite Mathematics
47. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: In a family with 3 children, excluding multiple births,
what is the probability of having exactly 2 girls? Assume that a
boy 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 8
outcomes in S is equally likely; so each outcome has
1
probability 8 .
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Jason Aubrey Math 1300 Finite Mathematics
48. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Now we identify the event we wish to find the probability of:
E = {GGB, GBG, BGG}
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Jason Aubrey Math 1300 Finite Mathematics
49. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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)
1 1 1 3
= + + =
8 8 8 8
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Jason Aubrey Math 1300 Finite Mathematics
50. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Procedure: Steps for Finding the Probability of an Event E
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Jason Aubrey Math 1300 Finite Mathematics
51. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Procedure: Steps for Finding the Probability of an Event E
1 Set up an appropriate sample space S for the experiment.
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Jason Aubrey Math 1300 Finite Mathematics
52. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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.
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Jason Aubrey Math 1300 Finite Mathematics
53. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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.
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Jason Aubrey Math 1300 Finite Mathematics
54. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Recall two past examples. . .
Two coin flips. . .
S = {HH, HT , TH, TT }
e P(e)
1
HH 4
1
HT 4
1
TH 4
1
TT 4
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Jason Aubrey Math 1300 Finite Mathematics
55. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Recall two past examples. . .
Two coin flips. . . A possibly rainy day. . .
S = {Rain, No Rain}
S = {HH, HT , TH, TT }
e P(e)
e P(e) Rain 0.15
1
HH 4 No Rain 0.85
1
HT 4
1
TH 4
1
TT 4
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Jason Aubrey Math 1300 Finite Mathematics
56. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Recall two past examples. . .
Two coin flips. . . A possibly rainy day. . .
S = {Rain, No Rain}
S = {HH, HT , TH, TT }
e P(e)
e P(e) Rain 0.15
1
HH 4 No Rain 0.85
1
HT 4
1 The outcomes are not equally
TH 4
1 likely.
TT 4
The outcomes are equally
likely. ../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
57. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Sometimes we can assume that all outcomes in a sample
space are equally likely.
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Jason Aubrey Math 1300 Finite Mathematics
58. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Sometimes we can assume that all outcomes in a sample
space are equally likely.
If S = {e1 , e2 , . . . , en } is a sample space in which all
outcomes are equally likely, then we assign the probability
1
n to each outcome. That is
1
P(ei ) =
n
and we have. . .
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Jason Aubrey Math 1300 Finite Mathematics
59. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Theorem (Probability of an Arbitrary Event under an Equally
Likely Assumption)
If we assume that each simple event in a sample space S is as
likely to occur as any other, then the probability of an arbitrary
event E in S is given by
number of elements in E n(E)
P(E) = = .
number of elements in S n(S)
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Jason Aubrey Math 1300 Finite Mathematics
60. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose an experiment consists of simultaneously
rolling two fair six-sided dice (say, one red die and one green
die) and recording the values on each die.
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
61. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose an experiment consists of simultaneously
rolling two fair six-sided dice (say, one red die and one green
die) and recording the values on each die. Some possibilities
are (Red=1, Green=5)
(1, 5)
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Jason Aubrey Math 1300 Finite Mathematics
62. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose an experiment consists of simultaneously
rolling two fair six-sided dice (say, one red die and one green
die) and recording the values on each die. Some possibilities
are (Red=1, Green=5) or (Red=2, Green=2).
(1, 5) (2, 2)
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Jason Aubrey Math 1300 Finite Mathematics
63. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: Suppose an experiment consists of simultaneously
rolling two fair six-sided dice (say, one red die and one green
die) and recording the values on each die. Some possibilities
are (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 comes
out to be 11?
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Jason Aubrey Math 1300 Finite Mathematics
64. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6
are all equally likely to turn up. So each possible pair of
numbers (1, 5), (3, 2), etc, is just as likely as any other. So,
we can make an equally likely assumption.
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Jason Aubrey Math 1300 Finite Mathematics
65. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6
are all equally likely to turn up. So each possible pair of
numbers (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.
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Jason Aubrey Math 1300 Finite Mathematics
66. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6
are all equally likely to turn up. So each possible pair of
numbers (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.
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Jason Aubrey Math 1300 Finite Mathematics
67. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Firstly, since the dice are fair, for each die, the numbers 1-6
are all equally likely to turn up. So each possible pair of
numbers (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
n(E) 2 1
P(E) = = =
n(S) 36 18
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Jason Aubrey Math 1300 Finite Mathematics
68. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(c) What is the probability that the numbers on the dice are
equal?
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Jason Aubrey Math 1300 Finite Mathematics
69. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(c) What is the probability that the numbers on the dice are
equal?
The event here is
F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
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Jason Aubrey Math 1300 Finite Mathematics
70. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(c) What is the probability that the numbers on the dice are
equal?
The event here is
F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
So, n(F ) = 6
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Jason Aubrey Math 1300 Finite Mathematics
71. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(c) What is the probability that the numbers on the dice are
equal?
The event here is
F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
So, n(F ) = 6
Therefore,
n(F ) 6 1
P(F ) = = =
n(S) 36 6
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Jason Aubrey Math 1300 Finite Mathematics
72. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: 5 cards are drawn simultaneously from a standard
deck of 52 cards.
(a) Describe the sample space S. What is n(S)?
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Jason Aubrey Math 1300 Finite Mathematics
73. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: 5 cards are drawn simultaneously from a standard
deck 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 available
cards. So, the sample space S can be described as
S = {all possible 5 card hands}
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Jason Aubrey Math 1300 Finite Mathematics
74. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: 5 cards are drawn simultaneously from a standard
deck 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 available
cards. 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?
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Jason Aubrey Math 1300 Finite Mathematics
75. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: 5 cards are drawn simultaneously from a standard
deck 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 available
cards. 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
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Jason Aubrey Math 1300 Finite Mathematics
76. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
Example: 5 cards are drawn simultaneously from a standard
deck 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 available
cards. 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 any
other, so any five card hand is just as likely as any other. In
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other words, we can make an equally likely assumption.
Jason Aubrey Math 1300 Finite Mathematics
77. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(b)Find the probability that all of the cards are hearts.
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Jason Aubrey Math 1300 Finite Mathematics
78. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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}
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Jason Aubrey Math 1300 Finite Mathematics
79. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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
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Jason Aubrey Math 1300 Finite Mathematics
80. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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
n(E) 1287
P(E) = = ≈ 0.000495
n(S) 2, 598, 960
or about 0.05%. ../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
81. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(c) Find the probability that all the cards are face cards (that is,
jacks, queens or kings).
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Jason Aubrey Math 1300 Finite Mathematics
82. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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}
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Jason Aubrey Math 1300 Finite Mathematics
83. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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
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Jason Aubrey Math 1300 Finite Mathematics
84. Experiments
Sample Spaces and Events
Probability of an Event
Equally 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
n(F ) 792
P(F ) = = ≈ 0.000305
n(S) 2, 598, 960
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or about 0.03%.
Jason Aubrey Math 1300 Finite Mathematics
85. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(d) Find the probability that all the cards are even. (Consider
aces to be 1, jacks to be 11, queens to be 12 and kings to be
13).
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Jason Aubrey Math 1300 Finite Mathematics
86. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(d) Find the probability that all the cards are even. (Consider
aces to be 1, jacks to be 11, queens to be 12 and kings to be
13).
The event “all the cards are even is the set
G = {all 5 card hands consisting of only even cards}
../images/stackedlogo-bw-
Jason Aubrey Math 1300 Finite Mathematics
87. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(d) Find the probability that all the cards are even. (Consider
aces to be 1, jacks to be 11, queens to be 12 and kings to be
13).
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 a
total of 20 even cards in a deck. So,
n(G) = C(20, 6) = 38, 760.
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Jason Aubrey Math 1300 Finite Mathematics
88. Experiments
Sample Spaces and Events
Probability of an Event
Equally Likely Assumption
(d) Find the probability that all the cards are even. (Consider
aces to be 1, jacks to be 11, queens to be 12 and kings to be
13).
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 a
total of 20 even cards in a deck. So,
n(G) = C(20, 6) = 38, 760.
By the qually likely assumption,
n(G) 38, 760
P(G) = = ≈ 0.0149
n(S) 2, 598, 960
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or about 14.9%.
Jason Aubrey Math 1300 Finite Mathematics