Math 141: Business Mathematics I
Fall 2015
§7.1 Experiments, Sample Spaces, and Events
Instructor: Yeong-Chyuan Chung
In this section, we will mainly be introducing terminology that we need in order to discuss
probability.
An experiment is an activity with observable results.
The results of the experiment are called outcomes of the experiment.
An outcome of an experiment may also be called a sample point.
The sample space is the set consisting of all possible sample points of an experiment. In
other words, it is the set of all possible outcomes of an experiment. We think of the sample
space as a universal set (recall §6.1).
An event is a subset of the sample space of an experiment. In other words, it is a collection
of some outcomes of an experiment.
An event is said to occur if it contains the observed outcome.
Since we are dealing with sets, everything that we discussed in §6.1 applies here as well.
In particular, we may talk about the union or intersection of two events, and also the
complement of an event.
Recall that in §6.1, we said that two sets E and F are disjoint if their intersection is empty
(E ∩ F = ∅). If E and F are events, and E ∩ F = ∅, then we say that E and F are mutually
exclusive events.
Recall that the empty set, ∅, is a set with no elements. It is also a subset of every other set.
In particular, in our current context, it is a subset of the sample space of any experiment.
In other words, it is an event. However, it has no elements, which means that it does not
contain any of the outcomes of the experiment. Thus this event never occurs, and we say
that it is an impossible event.
On the other hand, the universal set, which is the sample space in our current context, is
also an event, and it contains all possible outcomes of the experiment. Thus this event must
occur, and we say that is a certain event.
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§7.1 Experiments, Sample Spaces, and Events
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Example. Consider the experiment of rolling a fair six-sided die and observing the number
on the top face.
(a) What is the sample space?
(b) Write the event E = {x|x is an even number} is roster notation.
(c) If F = {3, 4, 5, 6}, what is the event E ∩ F ?
(d) What is the event E ∪ F c ?
(e) Are the events E and F mutually exclusive?
(f ) Are the events E and F complementary?
Example. Consider the experiment of tossing a fair coin twice and observing the resulting
sequence of heads and tails.
(a) What is the sample space?
(b) List all events of this experiment.
Example (Exercises 16,18,20 in the text). Let S be any sample space, and let E, F , and G
be any three events associated with the experiment. Describe the following events using ∪, ∩,
and/or c .
(a) The event that both E and F occur.
(b) The event that E but not F occurs.
(c) The event that E occurs but neither of the events F or G occurs.
§7.1 Experiments, Sample Spaces, and Events
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Example (Exercise 26 in the text). An experiment consists of selecting a letter at random
from the letters in the word MASSACHUSETTS and observing the outcomes.
(a) What is an appropriate sample space for this experiment?
(b) Describe the event “the letter selected is a vowel.”
Example (Exercise 34 in the text). Human blood is classified by the presence or absence of
three main antigens (A, B, and Rh). When a blood specimen is typed, the presence of the
A and/or B antigen is indicated by listing the letter A and/or the letter B. If neither the A
nor the B antigen is present, the letter O is used. The presence or absence of the Rh antigen
is indicated by the symbols + or −, respectively. Thus, if a blood specimen is classified as
AB+ , it contains the A and the B antigens as well as the Rh antigen. Similarly, O− blood
contains none of the three antigens. Using this information, determine the sample space
corresponding to the different blood groups.
Example (Exercise 36 in the text). The manager of a local bank observes how long it takes
a customer to complete his transactions at the automatic bank teller machine (ATM).
(a) Describe an appropriate sample space for this experiment.
(b) Describe the event that it takes a customer between 2 and 3 min, inclusive, to complete
his transactions at the ATM.
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§7.1 Experiments, Sample Spaces, and Events
Example (Exercise 42 in the text). Am opinion poll is conducted among a state’s electorate
to determine the relationship between their income levels and their stands on a proposition
aimed at reducing state income taxes. Voters are classified as belonging to either the low-,
middle-, or upper-income group. They were also asked to indicate their political affiliations:
Democrat, Republican, or Independent. Let the letters L, M, and U represent the low-,
middle-, and upper-income groups, respectively, and let the letters D, R, and I represent
Democrat, Republican, and Independent, respectively.
(a) Describe a sample space corresponding to this poll.
(b) Describe the event that a respondent belongs to the middle-income group and is not a
Democrat.