Probability Addition Rule

Statement

The probability of event \(A\) or event \(B\) happening is given by:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Why it’s true

• \(P(A)\) counts all outcomes in event A.
• \(P(B)\) counts all outcomes in event B.
• But if we just add them, the overlap (\(P(A \cap B)\)) is counted twice.
• So we subtract \(P(A \cap B)\) to avoid double-counting.

Recipe (how to use it)

1. Find \(P(A)\).
2. Find \(P(B)\).
3. Find \(P(A \cap B)\) if there is overlap.
4. Apply formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

Spotting it

Used whenever you are asked for the probability of "A or B" happening. The overlap must be subtracted if events are not mutually exclusive.

Common pairings

• Venn diagrams and set notation.
• Mutually exclusive events (\(P(A \cap B)=0\)).
• Conditional probability in more complex cases.

Mini examples

1. Given: \(P(A)=0.3\), \(P(B)=0.4\), \(P(A \cap B)=0.1\).
   Answer: \(0.3+0.4-0.1=0.6\).
2. Given: \(P(A)=0.5\), \(P(B)=0.2\), \(P(A \cap B)=0\).
   Answer: \(0.5+0.2=0.7\).

Pitfalls

• Forgetting to subtract the overlap.
• Assuming events are always mutually exclusive (when they may overlap).

Exam strategy

• Draw a Venn diagram to visualize overlaps.
• Check if events are independent or mutually exclusive.
• Probabilities must always stay between 0 and 1.

Summary

The addition rule ensures correct calculation of "A or B". It prevents double-counting overlap: \[P(A \cup B) = P(A)+P(B)-P(A \cap B)\].