Complement vs Event

The complement of an event $A$, written $A^c$ (read A complement), collects all outcomes in the sample space where $A$ does not happen. Its probability is $P(A^c) = 1 - P(A)$, since an event and its complement together cover everything. For example, if the probability of rain is 0.3, the probability of no rain is $1 - 0.3 = 0.7$. The complement rule is handy when at least one is easier to compute as 1 minus none.

An event is any set of outcomes you might care about, and its probability is the chance that the outcome lands inside it. For example, when rolling a die, the event of rolling an even number is the subset $\{2, 4, 6\}$. For equally likely outcomes, $P(\text{event}) = \frac{\text{outcomes in the event}}{\text{total outcomes}}$, so this event has probability $3/6 = 0.5$. Events can be combined with and, or, and not to build new events.