These considerations show that our probability model must deal with
- A
trial which results in (selects) an
outcome from a
set of conceptually
possible outcomes. The trial is not successfully completed until one of the outcomes isrealized.
- Associated with each outcome is a certain characteristic (or combination of characteristics)
pertinent to the problem at hand. In polling for political opinions, it is a person who isselected. That person has many features and characteristics (race, age, gender, occupation,
religious preference, preferences for food, etc.). But the primary feature, which characterizesthe outcome, is the political opinion on the question asked. Of course, some of the
other features may be of interest for analysis of the poll.
Inherent in informal thought, as well as in precise analysis, is the notion of an
event to which a
probability may be assigned as a measure of the
likelihood the event
will
occur on any trial. A successful mathematical model must formulate these
notions with precision.An
event is identified in terms of the characteristic of the outcome observed.
The event “a favorable response” to a polling question
occurs if the outcome observed
has that characteristic; i.e., iff (if and only if) the respondent replies in the affirmative. A hand of five cardsis drawn. The event “one or more aces”
occurs iff the hand actually drawn has at least
one ace. If that same hand has two cards of the suit of clubs, then the event “twoclubs” has
occurred . These considerations lead to the following definition.
Definition. The
event determined by some characteristic of the possible outcomes
is the set of those outcomes having this characteristic. The event
occurs iff
the outcome of the trial is a member of that set (i.e., has the characteristic determiningthe event).
- The event of throwing a “seven” with a pair of dice (which we call the event
SEVEN) consists of the set of those possible outcomes with a total of seven spots turned up.The event SEVEN occurs iff the outcome is one of those combinations with a total of seven spots
(i.e., belongs to the event SEVEN). This could be represented as follows. Supposethe two dice are distinguished (say by color) and a picture is taken of each of the thirty
six possible combinations. On the back of each picture, write the number of spots. Now theevent SEVEN consists of the set of all those pictures with seven on the back. Throwing
the dice is equivalent to selecting randomly one of the thirty six pictures. The event SEVENoccurs iff the picture selected is one of the set of those pictures with seven on the back.
- Observing for a very long (theoretically infinite) time the signal passing through
a communication channel is equivalent to selecting one of the conceptually possible signals.Now such signals have many characteristics: the maximum peak value, the frequency spectrum,
the degree of differentibility, the average value over a given time period, etc. If thesignal has a peak absolute value less than ten volts, a frequency spectrum essentially
limited from 60 herz to 10,000 herz, with peak rate of change 10,000 voltsper second, then
it is
one of the
set of signals with those characteristics. The event "the signal
has these characteristics" has occured. This set (event) consists of an uncountable infinityof such signals.
