How does an indicator random variable represent whether an event occurs?

In probability and statistics, an indicator random variable—often denoted as $1[A]$—is a real-valued function used to represent the occurrence of a specific event $A$.12

It functions by assigning a numerical value to the outcome of an experiment based on whether the event occurs:
* It equals 1 if the event $A$ occurs.
* It equals 0 if the event $A$ does not occur.12

This tool is particularly useful because it allows mathematicians to represent complex outcomes as sums of simpler events. For instance, the total number of successes in a series of trials can be expressed as the sum of indicator variables for each individual trial. Furthermore, the expected value of an indicator variable is equal to the probability of the event occurring, as $E\{1[A]\}=1\times P(A)+0\times P(A^c)=P(A)$. This property demonstrates that the concept of expectation encompasses probability as a special case.12