Probability
We will spend several classes discussing the concept of probability. A
probability is a number which measures the likelihood of an event. Given an event E, we
will write the probability of E as
P(E) or P{E}.
We can imagine an event as occurring as an outcome of an experiment or trial,
where we use these words in their broadest possible sense. If there is only one possible
outcome to the experiment, then the experiment is said to be deterministic. If there are
many possible outcomes to the experiment, then the experiment is probabilistic or
stochastic.
What is an example of a deterministic experiment?
Of a probabilistic experiment?
Suppose we could list the outcomes or events of an experiment, say E1, E2, …, En.
Assume that the Ei are mutually exclusive and collectively exhaustive. Then we require
that
(1)
0  P( Ei )  1
for i  1, 2, , n.
n
(2)
 P( E )  1
i 1
i
Three Views of Probability
(1) Relative Frequency Perspective
One of the earliest notions of probability came from the observation of the
relative frequencies of events which occurred from the replication of an experiment.
Specifically, if we perform an experiment N times and event E is observed nE times, then
nE
N  N
P ( E )  lim
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Example: If we flip a fair coin a large number of times, how frequently will heads occur?
(2) The Outcome Space Perspective
Suppose that an experiment will result in exactly one of n events, E1, E2, …, En.
We refer to this set of mutually exclusive and collectively exhaustive events as an
outcome space.
Example: Suppose that you randomly select a family with two children and record the
gender of each child, F (female) or M (male). What is the outcome space?
If each event E1, E2, …, En is equally likely to occur, then the outcome space is said to be
symmetric. In this case
P(E1) = P(E2) = … = P(En) = 1/n
n
since  P( Ei )  1.
i 1
Example: What is the outcome space for the throw of a die?
What is P{throw a 3}?
An event is said to be simple or elementary if the event cannot be decomposed into other
events. Events which can be decomposed are called compound events.
Example: Is the event {throw an odd number on a die} simple or compound?
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After some thought, you will see that two simple events cannot occur at the same
time; that is, they cannot both be realized as the result of a single experiment. This fact
helps us compute probabilities of compound events.
The probability of a compound event is the sum of the probabilities of the
simple events which cause the compound event to occur.
Example 1:
P{throwing an odd number on a die}
= P{1} + P{3} + P{5}
= 1/6 + 1/6 + 1/6
=½
Example 2:
P{having one or more girls in a family with two children}
=
Important idea: For a symmetric outcome space, if you can figure out the number of
simple events in the outcome space, n, and the number of simple events which causes the
event E to occur, then
P{E} 
Number of ways which cause E to occur
n
Example: In the die throw example, the number of simple events in the outcome space is
6, while an odd number is caused by 3 of those outcomes. Therefore,
P{odd number} = 3/6
(3) Subjective Probabilities
Subjective probabilities reflect an assessment of an individual as to his or her
confidence that a particular event will occur. For example, you might look at a new
product prototype and say, “there is an 80% chance that it will be successful.” Clearly
the relative frequency or outcome space approach will not be applicable since the new
product prototype will only occur once, not over and over again. I might say the chance
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of success is 90%. The fact is that it will either be successful or not. Thus, neither of us
is right or wrong.
One time decisions occur frequently in business environments, so you should not
be surprised if you encounter subjective probabilities. A large body of work exists on
this subject know as Bayesian Decision Theory.
Homework on probabilities and outcome spaces
An experiment is performed by rolling two dice and noting the values of the top surfaces.
(1) Write down the outcome space associated with this experiment.
(2) What is the probability that the sum of the two values equals 7?
(3) What is the probability that the absolute difference in the values on the dice is one?
Three? Five? Odd?
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