Question

# Probability of an impossible event is ________.

Use the fundamental theorem of Probability that it represents the extent to which an event is likely to occur, measured by ratio of favorable cases to the whole number of cases possible.

First, we need to understand the fundamental definition that probability is the numerical description of how likely an event is to occur or how likely it is that a proposition is true. It is measured by the ratio of the favorable cases to the whole number of cases possible.
Hence, mathematical formula of probability is given as
P=Number of favorable cases/Total number of cases$\ldots \ldots (1)$
Probability of an event A is written as P(A) and opposite or complement of an event is denoted as P(A’).
Probability of any event will lie in 0 to 1 and cannot be negative. And hence, summation of probability of occurring and non-occurring of any event is zero.
Hence, P(A) +P(A') = 1$\ldots \ldots (2)$
Now, if any event is not occurring, then the number of favorable cases for that event will be zero. Hence, Probability from equation of (1), we get
P=0/Total cases
P=0
Let us relate it with an example given below:-
Ex: One unbiased dice is rolled and now find a probability of any number greater than 7.
Now, one can easily observe that the total number of cases when one dice is rolled is {1, 2, 3, 4, 5, 6} i.e. 6 and where each number is less than 7. Hence, no number is greater than 7. Hence it is an impossible event and the probability of this event is zero as a favorable number of cases is 0. So, P(E) = 0.
Therefore, the probability of an impossible event is 0.

Note: One can get confused about how the probability of any event is zero. So, yes, it’s possible that the probability of any event can be zero if it’s not occurring.