Addition Rule (Probability)
Statistics & ProbabilityThe 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.
Example
In a class, $60\%$ play soccer and $40\%$ play basketball, and $20\%$ play both. $P(\text{soccer or basketball}) = 60\% + 40\% - 20\% = 80\%$. We subtract $20\%$ because those students were counted in both groups.
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.
Definition
The general addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. For mutually exclusive events, $P(A \text{ and } B) = 0$, so $P(A \text{ or } B) = P(A) + P(B)$. The "or" in probability always means "at least one of," which is inclusive or.
Example
Drawing one card: $P(\text{heart or face card}) = P(\text{heart}) + P(\text{face card}) - P(\text{heart face card}) = 13/52+12/52-3/52 = 22/52 = 11/26$. The $3$ heart face cards (Jack, Queen, King of hearts) are subtracted because they were counted in both groups.
Key Insight
The addition rule uses the principle of inclusion-exclusion. For three events: $P(A \text{ or } B \text{ or } C) = P(A)+P(B)+P(C) - P(A \text{ and } B) - P(A \text{ and } C) - P(B \text{ and } C) + P(A \text{ and } B \text{ and } C)$.
Definition
The inclusion-exclusion principle generalizes the addition rule: $P\left(\bigcup_{i=1}^{n} A_i\right) = \sum P(A_i) - \sum P(A_i \cap A_j) + \sum P(A_i \cap A_j \cap A_k) - \ldots + (-1)^{n+1} P(A_1 \cap \ldots \cap A_n)$. This formula alternates between adding and subtracting intersection terms of increasing order.
Example
The probability of at least one success in $n$ Bernoulli trials each with probability $p$, when trials are independent: $P(\text{at least one success}) = 1 - (1-p)^n$. This is the complement rule applied to the intersection of failure events, which is simpler than direct inclusion-exclusion for large $n$.
Key Insight
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.