Complement Rule vs Addition Rule

Written in symbols, $P(A^c) = 1 - P(A)$, where $A^c$ is read A complement and means A does not occur. It works because an event and its complement cover the whole sample space without overlapping, so their probabilities must add to 1. If the probability of rain tomorrow is 0.35, the probability of no rain is $1 - 0.35 = 0.65$. The rule is often used backwards as $P(A) = 1 - P(A^c)$ when the complement is the easier side to count.

The general addition rule is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, where $\cup$ is the union symbol (A or B) and $\cap$ is the intersection symbol (A and B). You subtract the overlap because every outcome sitting in both events was already counted once in $P(A)$ and a second time in $P(B)$. That subtraction disappears only when the two events are mutually exclusive, because then they share no outcomes and $P(A \cap B) = 0$. Drawing one card from a standard 52-card deck, $P(\text{heart or king}) = 13/52 + 4/52 - 1/52 = 16/52 \approx 0.3077$.