What is the definition of an impossible event in probability theory?
If you search on the web, you will find many sources that define it as an event whose probability is equal to zero. This definition is wrong.
Although the probability of an impossible event is zero, in many probabilistic models there are events that are not impossible but have zero probability.
The right definition might be a bit difficult to grasp: "the impossible event is the empty set".
What does this definition mean? In order to fully understand it, we need to recall how events are defined in probability theory.
Remember that an event is a subset of the sample space, which is the set of all possible outcomes of a probabilistic experiment.
Example
If the probabilistic experiment we are considering is the toss of a die, the
sample space
is![[eq1]](/images/impossible-event__1.png){kind=small}
The
possible outcomes are the numbers from
to
.
An event is a subset of the sample space
.
So, for example,
![[eq2]](/images/impossible-event__5.png){kind=small}
and
![[eq3]](/images/impossible-event__6.png){kind=small}
and![[eq4]](/images/impossible-event__7.png){kind=small}
are
events.
In set theory, the empty set
is the set that contains no elements. Given any set
,
the empty set
is a subset of
.
In
symbols:![[eq5]](/images/impossible-event__12.png){kind=small}
So, given a sample space
,
the empty set is one of its
subsets:![[eq6]](/images/impossible-event__14.png){kind=small}
It is an event and it is called the impossible event.
In other words, the impossible event is the event which does not contain any
of the possible outcomes (remember that the possible outcomes are the elements
of
).
Let us see why this definition of impossible event is reasonable with an example.
Example
Let us continue with the toss of a die experiment introduced in the previous
example. Consider the event "the number that will appear face up is both
greater than 4 and smaller than 2". This is obviously impossible because there
is no number that satisfies these two conditions simultaneously. Let us see
why the event is impossible also according to the formal definition. We are
talking about two events. The event "greater than 4"
is![[eq7]](/images/impossible-event__16.png){kind=small}
and
the event "smaller than 2"
is![[eq8]](/images/impossible-event__17.png){kind=small}
The
event "both greater than 4 and smaller than 2" is
![[eq9]](/images/impossible-event__18.png){kind=small}
Thus,
it is impossible also according to the definition.
One of the basic properties of probability is that the empty set must have zero probability (see the lecture on Probability for a formal proof). Therefore, by definition, impossible events have zero probability.
Although the impossible event has zero probability, not all zero-probability events are impossible. As a matter of fact, there are common probabilistic settings where the sample space is uncountable and each of the possible outcomes has zero probability. In other words, there are non-empty sets (events) that have zero probability.
If you are puzzled by the last statement (there are possible events with zero probability), you are advised to read the lecture on Zero-probability events.
If you want to know more about sample spaces and events, you can read the introductory lecture on Probability.
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Please cite as:
Taboga, Marco (2021). "Impossible event", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/impossible-event.
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