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7
Probability
Copyright © Cengage Learning. All rights reserved.
7.1
Sample Spaces and Events
Copyright © Cengage Learning. All rights reserved.
Sample Spaces
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Sample Spaces
At the beginning of a football game, to ensure fairness, the
referee tosses a coin to decide who will get the ball first.
When the ref tosses the coin and observes which side
faces up, there are two possible results: heads (H) and
tails (T).
These are the only possible results, ignoring the (remote)
possibility that the coin lands on its edge.
The act of tossing the coin is an example of an
experiment.
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Sample Spaces
The two possible results, H and T, are possible outcomes
of the experiment, and the set S = {H, T} of all possible
outcomes is the sample space for the experiment.
Experiments, Outcomes, and Sample Spaces
An experiment is an occurrence with a result, or outcome,
that is uncertain before the experiment takes place.
The set of all possible outcomes is called the sample
space for the experiment.
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Sample Spaces
Quick Examples
1. Experiment: Flip a coin and observe the side facing up.
Outcomes: H, T
Sample Space: S = {H, T}
2. Experiment: Select a student in your class.
Outcomes: The students in your class
Sample Space: The set of students in your class
3. Experiment: Cast a die and observe the number facing
up.
Outcomes: 1, 2, 3, 4, 5, 6
Sample Space: S = {1, 2, 3, 4, 5, 6}
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Sample Spaces
4. Experiment: Cast two distinguishable dice and observe
the numbers facing up.
Outcomes: (1, 1), (1, 2), . . . , (6, 6) (36 outcomes)
Sample
S=
Space:
(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)
n(S) = the number of outcomes in S = 36
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Sample Spaces
5. Experiment: Choose 2 cars (without regard to order) at
random from a fleet of 10.
Outcomes: Collections of 2 cars chosen from 10
Sample Space: The set of all collections of 2 cars
chosen from 10
n(S) = C(10, 2) = 45
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Example 1 – School and Work
In a survey conducted by the Bureau of Labor Statistics,
the high school graduating class of 2010 was divided into
those who went on to college and those who did not. Those
who went on to college were further divided into those who
went to 2-year colleges and those who went to 4-year
colleges. All graduates were also asked whether they were
working or not.
Find the sample space for the experiment “Select a
member of the high school graduating class of 2010 and
classify his or her subsequent school and work activity.”
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Example 1 – Solution
The tree in Figure 1 shows the various possibilities.
Figure 1
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Example 1 – Solution
cont’d
The sample space is
S = {2-year college & working, 2-year college & not
working, 4-year college & working, 4-year
college & not working, no college & working, no
college & not working}.
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Events
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Events
In Example 1, suppose we are interested in the event that a
2010 high school graduate was working.
In mathematical language, we are interested in the subset
of the sample space consisting of all outcomes in which the
graduate was working.
Events
Given a sample space S, an event E is a subset of S. The
outcomes in E are called the favorable outcomes.
We say that E occurs in a particular experiment if the
outcome of that experiment is one of the elements of
E—that is, if the outcome of the experiment is favorable.
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Events
Visualizing an Event
In the following figure, the favorable outcomes (events in E)
are shown in green.
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Events
Quick Examples
1. Experiment: Roll a die and observe the number facing
up.
S = {1, 2, 3, 4, 5, 6}
Event: E: The number observed is odd.
E = {1, 3, 5}
2. Experiment: Roll two distinguishable dice and observe
the numbers facing up.
S = {(1, 1), (1, 2), . . . , (6, 6)}
Event: F: The dice show the same number.
F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
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Events
3. Experiment: Roll two distinguishable dice and observe
the numbers facing up.
S = {(1, 1), (1, 2), . . . , (6, 6)}
Event: G: The sum of the numbers is 1.
There are no favorable outcomes.
G=∅
4. Experiment: Select a city beginning with “J.”
Event: E: The city is Johannesburg.
E = {Johannesburg} An event can consist of a single
outcome.
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Events
5. Experiment: Draw a hand of 2 cards from a deck of 52.
Event: H: Both cards are diamonds.
H is the set of all hands of 2 cards chosen from 52
such that both cards are diamonds.
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Example 2 – Dice
We roll a red die and a green die and observe the numbers
facing up. Describe the following events as subsets of the
sample space.
a. E: The sum of the numbers showing is 6.
b. F: The sum of the numbers showing is 2.
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Example 2 – Solution
Here (again) is the sample space for the experiment of
throwing two dice.
(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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Example 2 – Solution
cont’d
a. In mathematical language, E is the subset of S that
consists of all those outcomes in which the sum of the
numbers showing is 6. Here is the sample space once
again, with the outcomes in question shown in color:
(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)
Thus, E = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}.
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Example 2 – Solution
cont’d
b. The only outcome in which the numbers showing add to
2 is (1, 1).
Thus,
F = {(1, 1)}.
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Complement, Union, and
Intersection of Events
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Complement, Union, and Intersection of Events
Events may often be described in terms of other events,
using set operations such as complement, union, and
intersection.
Complement of an Event
The complement of an event E is the set of outcomes not
in E. Thus, the complement of E represents the event that
E does not occur.
Visualizing the Complement
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Complement, Union, and Intersection of Events
Quick Examples
1. You take four shots at the goal during a soccer game
and record the number of times you score.
Describe the event that you score at least twice, and
also its complement.
S = {0, 1, 2, 3, 4}
Set of outcomes
E = {2, 3, 4}
Event that you score at least twice
E = {0, 1}
Event that you do not score at least twice
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Complement, Union, and Intersection of Events
2. You roll a red die and a green die and observe the two
numbers facing up. Describe the event that the sum of
the numbers is not 6.
S = {(1, 1), (1, 2), . . . , (6, 6)}
F = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}
F =
(1, 1), (1, 2), (1, 3),
(2, 1), (2, 2), (2, 3),
(3, 1), (3, 2),
(4, 1),
(4, 3),
(5, 2), (5, 3),
(6, 1), (6, 2), (6, 3),
Sum of numbers is 6.
(1, 4),
(3, 4),
(4, 4),
(5, 4),
(6, 4),
(2, 5),
(3, 5),
(4, 5),
(5, 5),
(6, 5),
(1, 6),
(2, 6),
(3, 6),
(4, 6),
(5, 6),
(6, 6)
Sum of numbers is not 6.
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Complement, Union, and Intersection of Events
Union of Events
The union of the events E and F is the set of all outcomes
in E or F (or both). Thus, E U F represents the event that E
occurs or F occurs (or both).
Quick Example
Roll a die.
E: The outcome is a 5; E = {5}.
F: The outcome is an even number; F = {2, 4, 6}.
E U F: The outcome is either a 5 or an even number;
E U F = {2, 4, 5, 6}.
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Complement, Union, and Intersection of Events
Intersection of Events
The intersection of the events E and F is the set of all
outcomes common to E and F. Thus, E ∩ F represents the
event that both E and F occur.
Quick Example
Roll two dice; one red and one green.
E: The red die is 2.
F: The green die is odd.
E ∩ F: The red die is 2 and the green die is odd;
E ∩ F = {(2, 1), (2, 3), (2, 5)}.
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Example 4 – Weather
Let R be the event that it will rain tomorrow, let P be the
event that it will be pleasant, let C be the event that it will
be cold, and let H be the event that it will be hot.
a. Express in words: R ∩ P, R U (P ∩ C).
b. Express in symbols: Tomorrow will be either a pleasant
day or a cold and rainy day; it will not, however, be hot.
Solution:
The key here is to remember that intersection corresponds
to and and union to or.
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Example 4 – Solution
cont’d
a. R ∩ P is the event that it will rain and it will not be
pleasant. R U (P ∩ C) is the event that either it will rain,
or it will be pleasant and cold.
b. If we rephrase the given statement using and and or we
get “Tomorrow will be either a pleasant day or a cold and
rainy day, and it will not be hot.”
[P U (C ∩ R)] ∩ H
Pleasant, or cold and
rainy, and not hot.
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Example 4 – Solution
cont’d
The nuances of the English language play an important
role in this formulation. For instance, the effect of the pause
(comma) after “rainy day” suggests placing the preceding
clause P U (C ∩ R) in parentheses.
In addition, the phrase “cold and rainy” suggests that C and
R should be grouped together in their own parentheses.
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Complement, Union, and Intersection of Events
The case where E ∩ F is empty is interesting, and we give
it a name.
Mutually Exclusive Events
If E and F are events, then E and F are said to be disjoint
or mutually exclusive if E ∩ F is empty. (Hence, they have
no outcomes in common.)
Visualizing Mutually Exclusive Events
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Complement, Union, and Intersection of Events
Interpretation
It is impossible for mutually exclusive events to occur
simultaneously.
Quick Examples
In each of the following examples, E and F are mutually
exclusive events.
1. Roll a die and observe the number facing up. E: The
outcome is even; F: The outcome is odd.
E = {2, 4, 6}, F = {1, 3, 5}
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Complement, Union, and Intersection of Events
2. Toss a coin three times and record the sequence of
heads and tails. E: All three tosses land the same way
up, F: One toss shows heads and the other two show
tails.
E = {HHH, TTT}, F = {HTT, THT, TTH}
3. Observe tomorrow’s weather. E: It is raining; F: There is
not a cloud in the sky.
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