Addition Rule (Probability)
The addition rule gives the probability that at least one of two events occurs, accounting for any overlap between them.
Formula
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
Definition
The addition rule tells you the probability that at least one of two events happens: if the events can both happen at the same time, you must subtract the overlap so it is not counted twice. The general addition rule is $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$; for mutually exclusive events, $P(A \text{ and } B) = 0$, so $P(A \text{ or } B) = P(A) + P(B)$, and "or" in probability always means "at least one of," which is inclusive or. Formally, the inclusion-exclusion principle generalizes this: $P\left(\bigcup_{i=1}^{n} A_i\right) = \sum P(A_i) - \sum P(A_i \cap A_j) + \sum P(A_i \cap A_j \cap A_k) - \ldots + (-1)^{n+1} P(A_1 \cap \ldots \cap A_n)$, alternating between adding and subtracting intersection terms of increasing order.
Example
In a class where $60\%$ play soccer, $40\%$ play basketball, and $20\%$ play both: $P(\text{soccer or basketball}) = 60\% + 40\% - 20\% = 80\%$, subtracting $20\%$ so those students are not counted twice. Drawing one card: $P(\text{heart or face card}) = 13/52+12/52-3/52 = 22/52 = 11/26$, subtracting the $3$ heart face cards counted in both groups. The probability of at least one success in $n$ independent Bernoulli trials each with probability $p$ is $P(\text{at least one success}) = 1 - (1-p)^n$, the complement rule applied to the intersection of failure events, simpler than direct inclusion-exclusion for large $n$.
Key Insight
The subtraction prevents double-counting: think of two overlapping circles, adding both circles counts the middle twice, so you subtract the middle once; for three events, $P(A \text{ or } B \text{ or } C) = P(A)+P(B)+P(C) - P(A \text{ and } B) - P(A \text{ and } C) - P(B \text{ and } C) + P(A \text{ and } B \text{ and } C)$. The Bonferroni inequalities provide bounds using partial sums of the inclusion-exclusion formula: the sum of the first $2k-1$ terms is an upper bound and the sum of the first $2k$ terms is a lower bound, enabling approximations when the full formula is computationally intractable.