Addition Rule & Mutually Exclusive Events
The Formula: For any two events and , the probability of the union is given by: .
The Logic of Subtraction: When we add and , the outcomes in the intersection () are included in both totals. We subtract the intersection once to correct this double-counting and ensure the final probability is accurate.
Applicability: This general rule works for all pairs of events, whether they overlap or are entirely separate. It is the most robust tool for calculating 'OR' probabilities.
Simplified Addition Rule: If events and are mutually exclusive, their intersection is empty, meaning . Consequently, the formula simplifies to: .
Identifying Disjointness: To determine if events are mutually exclusive, ask if they can happen at the same time. For example, being in London and being in New York at the exact same moment are mutually exclusive events.
Extension to Multiple Events: For three or more mutually exclusive events, the probability of any one of them occurring is simply the sum of their individual probabilities: .
| Feature | General Events | Mutually Exclusive Events |
|---|---|---|
| Intersection | ||
| Formula | ||
| Visual | Overlapping circles | Separate circles |
| Relationship | Can occur together | Cannot occur together |
Check for Overlap: Always ask 'Is it possible for both of these to happen at once?' before choosing a formula. If the answer is yes, you must find and subtract the intersection probability.
The 'At Least One' Keyword: In exam questions, the phrase 'find the probability that at least one event occurs' is a direct signal to use the Addition Rule for the union of those events.
Probability Bounds: Remember that any calculated probability for must be between 0 and 1. If your sum exceeds 1, you likely forgot to subtract the intersection.
Using Complements: If calculating is complex, it may be easier to calculate , which is equivalent to the union.
Double Counting: The most frequent error is simply adding for events that are not mutually exclusive. This leads to an inflated probability that ignores the shared outcomes.
Misinterpreting 'OR': In probability, 'OR' is usually inclusive (A, B, or both). Students sometimes mistakenly use the exclusive 'OR' (A or B, but not both), which would require subtracting the intersection twice.
Confusing Symbols: Ensure you do not swap the union symbol () with the intersection symbol (). Think of as a 'cup' that holds everything (union) and as an 'arch' where only things underneath both sides meet (intersection).