Sections 7.1 & 7.2
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.
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.
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.
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 ∪ F
represents the event that E occurs or F occurs (or both).
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.
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.)
Note: It is impossible for mutually exclusive events to occur simultaneously.
Estimated Probability:
When an experiment is performed a number of times, the estimated probability or relative
frequency of an event E is the fraction of times that the event E occurs. If the experiment
is performed N times and the event E occurs fr(E) times, then the estimated probability is
given by
fr(E)
Fraction of times E occurs.
P(E) 
N
The number fr(E) is called the frequency of E. The number N of times that the
experiment is performed is called the number of trials or the sample size. If E consists of
a single outcome s, then we refer to P(E) as the estimated probability or relative
 frequency of the outcome s, and we write P(s).
Visualizing Estimated Probability
The collection of the estimated probabilities of all the outcomes is the estimated
probability distribution or relative frequency distribution.
Note: We almost always use the term estimated probability rather than relative
frequency.
Some Properties of Estimated Probability Distributions:
Let S  s1,s2 ,...,sn  be a sample space and let P(si ) be the estimated probability of the
event si . Then
1. 0 ≤ P(si ) ≤ 1
 2. P(s1)  P(s2 )   P(sn ) 1

E

e
,e
,...,e
3.
If
,
then
P(E)

P(e
 P(er ) .


1
2
r
1)  P(e2 ) 

In words:

1. The estimated probability of each outcome is a number between 0 and 1 (inclusive).

2. The estimated probabilities of all the outcomes add up to 1.
 3. The estimatedprobability of an event E is the sum of the estimated probabilities of
the individual outcomes in E.
Theoretical Probability: (Intuitive Definition of Theoretical Probability)
The theoretical probability, or probability, P(E), of an event E is the fraction of times we
expect E to occur if we repeat the same experiment over and over.
More Precise Definition of Theoretical Probability:
The theoretical probability, or probability, P(E), of an event E is the limiting value of the
estimated probability as the number of trials gets larger and larger. That is, the estimated
probability approaches the theoretical probability as the number of trials gets larger and
larger. Thus,
Estimated probability is an approximation, or estimate, of theoretical probability. The
larger the number of trials, the more accurate we expect this approximation to be.
If E consists of a single outcome s, we refer to P(E) as the probability of the outcome s,
and write P(s) for P(E). The collection of the probabilities of all the outcomes is the
probability distribution.
Determining Theoretical Probability:
Theoretical probability is determined analytically—that is, by using our knowledge about
the nature of the experiment rather than through actual experimentation. The best we can
obtain through actual experimentation is an estimate of the theoretical probability (hence
the term estimated probability).
Computing Theoretical Probability: Equally Likely Outcomes
In an experiment in which all outcomes are equally likely, the theoretical probability of
an event E is given by
Number of favorable outcomes n(E)
P(E) 

Total number of outcomes
n(S)
Note: Remember that this formula will work only when the outcomes are equally likely.
If, for example, a die is weighted, then the outcomes may not be equally likely, and the
formula above will not apply.
