Question

What is an impossible event?

Answer 1

Hint: We will first define what are events and then we will give an example to it. Further we will first define the impossible events in general and then give a technical definition from the probability theory and define the sample space and the empty sets, and we will assign those empty sets as impossible events.

Complete step-by-step answer:
First let’s understand what is meant by the events in probability, so the outcomes of a random experiment are called events connected with the experiment. For example; head and tails are the outcomes of the random experiment of throwing a coin and they are called events of probability.
Now, an impossible event really means an event that cannot happen, 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 technical definition is that the impossible event is the empty set.
Now let’s define how events are defined in probability theory, an event is basically a subset of the sample space, which is the set of all possible outcomes of an experiment. For example, we consider the experiment of tossing a dice so the sample space is: $\Omega =\left\{ 1,2,3,4,5,6 \right\}$ , which means that the possible outcomes are the numbers from $1\text{ to }6$. Now, let’s say we want the event of an odd number coming out then as we said that an event is a subset of the sample space, in this case \[\Omega \] . So, $E=\left\{ 1,2,3 \right\}$ and if the event is for an even number showing up then $F=\left\{ 2,4,6 \right\}$ .
Now as we know that in set theory, the empty set $\phi $ is the set that contains no elements. So, given a sample space $\Omega $ , the empty set is one of its subsets therefore: $\phi \subseteq \Omega $ . 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.

Note: 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.