1
It’s hard to imagine that anybody has not encountered
probability in some form at some point in their life. That is, we
all have some notion about what probability is. We feel that
probability means the “chance that something happens”. This
of course is not a definition and is really only another way to
say the probability that something happens.
Certainly, most of us have some comfort level with probability
and even feel that we know certain things. But, if we were
asked to write out what it is that we know, our paper might
not contain much writing when we are done. That is, we feel
that we know things, but aren’t exactly sure what it is that we
know. In this chapter, we will use definitions and theorems to
formalize probability mathematically. Additionally, we will provide the student with examples and
conversation in an effort to build an intuitive feel for probability. We feel that solving problems should
be a thought process and not a search for the proper formula.
2.1 A Random Experiment
–
Sample Space, Outcomes and Events
Prior to performing some specific act, experiment, we often cannot say with certainty what will take
place. Some examples would be rolling a die or pair of dice, tossing a single coin or multiple coins, or
measuring the length of time that it will take to drive to work tomorrow. These are just a few of the
endless possibilities of experiments with unknown results. The results will be known after we perform
our experiment.
Definition 2.1:
A
Random Experiment
is an experiment where the result cannot be determined in
advance. We often say there is an element of chance as to what the result will be. Examples of random
experiment were given above.
Definition 2.2:
Given some random experiment, a possible results of the experiment is called an
Outcome
.
Definition 2.3:
The collection of all possible outcomes of a random experiment is the
Sample Space
. We
will denote the sample space of an experiment with the symbol
.
Experiment 1:
A single die is rolled and the number of spots on the top side is noted. While we know we
will get a 1, 2, 3, 4, 5, or 6, we don’t know which one. So, the experiment is random experiment. The set
of all possible outcomes for this experiment yields the sample space
1,2,3,4,5,6
The numbers 1, 2, 3, 4, 5, and 6 are the
Outcomes
of the experiment. We note that there are six
outcomes in the sample space.
Experiment 2:
A coin is to be tossed three times. The sample space can be carefully constructed as
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
There are a total of
3
2
8
outcomes in the sample space.

2
Experiment 3:
Two dice are to be rolled. Can you guess how many outcomes are in the sample space?
Your Guess: ________________
Experiment 4:
A coin is to be tossed. If we get Heads, we are done. If we get Tails, we roll a die.