Probability of Event ‘not A’

The Complement of an Event: What Does “Not A” Mean?

When we talk about the probability of an event, we often also need the probability that the event does not happen. This is the probability of the complementary event, written as “not A” or $A'$.

Consider a simple experiment: draw one card from a deck of ten cards numbered 1 through 10. The sample space is $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Let event $A$ be “the card shows an even number from the set $\{2, 4, 6, 8\}$.” If every outcome is equally likely, each outcome has probability $\frac{1}{10}$.

The probability of $A$ is the sum of the probabilities of its outcomes:

Now, the event “not A” is the complement $A' = \{1, 3, 5, 7, 9, 10\}$. Its probability is:

Notice that $\frac{3}{5} = 1 - \frac{2}{5} = 1 - P(A)$. This is not a coincidence — it is a fundamental relationship.