The addition theorem in probability, also known as the addition rule or sum rule, provides a way to calculate the probability of the union of two events. It is expressed as follows:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Where:
- \( P(A \cup B) \) is the probability of the union of events A and B.
- \( P(A) \) is the probability of event A.
- \( P(B) \) is the probability of event B.
- \( P(A \cap B) \) is the probability of the intersection of events A and B.
The formula accounts for the fact that when calculating the probability of the union of two events, the probability of their intersection is counted twice. By subtracting \( P(A \cap B) \), which represents the overlapping probability, we avoid double-counting.
The addition theorem is applicable whether the events A and B are mutually exclusive (no overlap) or not mutually exclusive (some overlap). In the case of mutually exclusive events (\( A \cap B = \emptyset \)), the formula simplifies to:
\[ P(A \cup B) = P(A) + P(B) \]
This represents the probability of either event A or event B occurring.
In summary, the addition theorem provides a general formula for calculating the probability of the union of two events, considering both overlapping and non-overlapping scenarios.
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