Probability: An Introduction
Statistical Experiment
An statistical experiment is a repeatable process that allows observations of the results of the process.
Outcome of an Experiment
Each observation of an experiment is called an outcome.
Sample Space of an Experiment
The sample space S of an experiment is the set of all possible outcomes. There should be at least as many outcomes in S as times the experiment process is repeated. In some cases, one repitition of the experiment may produce more than one outcome.
Sample Size of an Experiment
The sample size n of an experiment is the number of elements in its sample space S.
Event of an Experiment
An event of interest E consists of one or more outcomes of an experiment – that is, one or more of the elements of the associated experiment's sample space S.
Complement of an Event of an Experiment
The dictionary definition of "complement" is something that completes to make a perfect whole. The complement of an event E consists of all the outcomes in the associated experiment that are not in E. This means the union of E together with its complement comprises the experiment's entire sample space S (the perfect whole in this case).
In this tutorial, we will use EC to denote the complement of an event E.
Probability of an Event
The probability of an event E occurring in an experiment is, in loose terms, the likelihood or chance that the event will occur some time again in repetitions of the experiment.
In Statistics, the probability of an event, denoted by P(E), is defined to be a real number between 0 and 1 which is a mathematical measure of the likelihood or chance that the event will occur again during such an experiment. If the probability of an event is measured to be 1, that event is certain to occur. On the other hand, if its probability is 0, the event is certain never to occur. We can extrapolate to the numbers between 0 and 1. For example, if P(E) = .75, the event E can be expected to happen 75% of the time.
Remembering that an event E is just a subset of the sample space S of its associated experiment, we can formulate a definition of P(E) as follows, noting that # of E-outcomes is always less than or equal to the sample size.
# of ways E can happen # of E-outcomesP(E) = _______________________ = _________________ or
total # outcomes sample size
# of E-outcomes
P(E) = _________________ .
n
If the event E (that is, one of the outcomes in E) occurs every time the experiment is repeated, then:
# of E-outcomes in the experiment = total # of outcomes = sample size = n.
In this case, substituting these values into the above formula yields P(E) = n/n = 1. In other words, the event E is certain to occur in every repetition of that experiment.
If the event E never occurs each time the experiment is repeated, then:
# of E-outcomes in the experiment = 0.
In this case, substituting these values into the above formula yields P(E) = 0/n = 0. In other words, the event E is certain never to occur no matter how often that experiment is repeated.
Last Updated by SAS October 07, 2026, 16:00:54
