Question 1: What is an event?
Probability is used to measure the likelihood of something happening. Implicit in the
idea of likelihood is chance. We are uncertain what will happen. An experiment is a
process that generates uncertain occurrences. These occurrences are called the
outcome of the experiment.
For instance, suppose a manufacturer is producing batteries that are sold in a two pack.
If a package of batteries is selected from the production line, the batteries in the
package may be examined to determine whether they work or are defective. The
process of examining whether the batteries in the package are defective is an
experiment. The outcome of the experiment may be listed by indicating whether each
battery is working (W) or defective (D).
First Battery
Second Battery
W
W
W
D
D
W
D
D
We can specify the first outcome of the experiment as (W, W). Other outcomes can be
written in a similar manner. Written this way, this first letter indicates whether the first
battery in the package is working or defective. The second letter indicates whether the
second battery in the package is working or defective. We can refer to these outcomes
collectively as
S  W ,W  , W , D  ,  D,W  ,  D, D  The experiment is carried out many times with each outcome being uncertain. These
repetitions of the experiment are called trials.
2
The collection of all possible outcomes of an experiment is
called the sample space.
The letter S is used to denote the sample space. The outcomes in the sample space are
usually enclosed in brackets.
Example 1
Find the Sample Space
If the battery producer examines the two-pack of batteries and notes the
number of defective batteries, find the sample space.
Solution A two-pack of batteries may have 0, 1, or 2 defective batteries
in it. Since the sample space is the set of all possible outcomes,
S  0,1, 2 A tree diagram is useful for listing all of the outcomes from an experiment. For the
battery packaging, we draw a pair of line segments from a common starting point to
indicate whether the first battery works or does not work.
W
D
These line segments form the first branches of the tree. From each of these
possibilities, another branch is drawn to indicate what might happen when the second
battery in the package is examined.
3
W
W
D
W
D
D
By examining the tree diagram from left to right, we can list all of the outcomes in the
sample space.
W
(W, W)
D
(W, D)
W
(D, W)
D
(D, D)
W
D
A similar strategy can be used for experiments that lead to more complicated branching.
Example 2
Find the Sample Space
A marketing company wishes to survey a group of cell phone customers
regarding their phone usage. On the first question of the survey, they
will ask whether the customer uses a smartphone. On the second
question, they ask whether they are on a family share plan. On a third
question in the survey, they ask whether the customer has a texting
plan. The answers to the questions are recorded as yes (Y) or no (N). If
you consider the administration of the survey to be an experiment, find
the sample space of the experiment.
Solution Construct a tree diagram like the one shown below.
4
smartphone?
family share?
texting?
Y
Y
N
Y
Y
N
N
Y
Y
N
N
Y
N
N
If we corrrespond letters to these branches we can write the sample
space as
S  Y , Y , Y  , Y , Y , N  , Y , N , Y  , Y , N , N  ,  N , Y , Y  ,  N , Y , N  ,  N , N , Y  ,  N , N , N 
Each outcome corresponds to an ordered triple. This is similar to the
ordered pairs we often graph in algebra but with the entries inside the
parentheses matching the answer to the questions. Since the survey
has three questions on it, we need three sets of branches to specify all
possible outcomes and three entries inside of the parentheses..
Often we are interested in a portion of the sample space. An event is any collection of
outcomes from an experiment. We represent events with capital letters.
5
Example 3
Find the Event
The marketing company is interested in several different events.
Specify the outcomes that make up each of the events below.
a. The event A, all three questions are answered yes.
Solution From the tree diagram above, we found the sample space
S  Y , Y , Y  , Y , Y , N  , Y , N , Y  , Y , N , N  ,  N , Y , Y  ,  N , Y , N  ,  N , N , Y  ,  N , N , N 
The event A corresponds to the outcome where each entry in the
ordered triple is Y,
A  Y , Y , Y 
b. The event B, two of the questions are answered no.
Solution We need to find all of the outcomes in the sample space that
contain 2 N’s. This event is
B  Y , N , N  ,  N , Y , N  ,  N , N , Y 
c. The event C, the last question is answered yes.
Solution We need to find all outcomes where the last question was
answered Y. This event is
C  Y , Y , Y  , Y , N , Y  ,  N , Y , Y  ,  N , N , Y 
In each of the parts above, the order in which the outcomes are listed is irrelevant. In
other words, we could also write a collection like B as  N , Y , N  , Y , N , N  ,  N , N , Y  or
Y , N , N  ,  N , N , Y  ,  N , Y , N  . As long as the outcomes are listed inside the brackets,
6
the event is the same. Similarly, the outcomes in the sample space may be listed in any
order.
Example 4
Find the Event
Breweries are classified by the amount of beer they produce in a year.
The American Brewers Association defines a microbrewery as a
brewery that produces less than 15,000 barrels per year. A
nanobrewery is a very small brewery where beer is produced in very
small batches.
One nanobrewery serves only three beers at a time in its tasting room.
The owner conducts an experiment where he keeps track of the first
two beers each customer purchases. He uses the letter b to indicate
brown ale, p for pale ale, and l for lager, and n for no beer ordered. For
instance, bn indicates that the first beer ordered is a brown ale and no
second beer was ordered. List the outcomes in each of the events listed
below.
a. The event A, only one beer is ordered.
Solution For this event, the second letter must be n. If we represent the
event with the letter A,
A  bn, pn, ln
b. The event B, the same beer is ordered for the first and second beer.
Solution List out all of the outcomes where the letters representing a
beer match,
B  bb, pp, ll
Note that nn is not listed since the event assumes a beer is ordered.
7