The probability of an event not occurring is calculated as one minus the probability of the event occurring.

In probability theory, the likelihood of an event taking place is expressed as a value between $0$ and $1$. If an event is guaranteed to occur, its probability is $1$. Conversely, if an event is guaranteed not to occur, its probability is $0$. For any event denoted as $A$, the probability of $A$ occurring is represented as $P(A)$. To determine the probability of $A$ not occurring, one subtracts the probability of $A$ occurring from $1$. This relationship holds because the total probability of all potential outcomes must equal $1$.

For instance, if the probability of rain tomorrow is $0.3$ (i.e., $P(\text{rain}) = 0.3$), then the probability of it not raining is calculated as $1 - 0.3$, which equals $0.7$. This can be expressed as:

This relationship proves useful, as it allows for the straightforward calculation of the probability of the complementary event (the event not happening) once the probability of the event itself is known.

Grasping this concept is essential in probability, as it enables a more comprehensive analysis of various situations. For example, if the probability of drawing a red card from a standard deck of cards is $0.5$, you can immediately deduce that the probability of not drawing a red card is also $0.5$, since:

This principle is universally applicable to all types of probability problems, making it a fundamental tool in your mathematical toolkit.