2.6 Independent Events and Unions of Events

Testing for independence

Two events A and B are independent if the occurrence of one does not change the probability of the other: P(A|B) = P(A). Equivalently, events are independent if P(A ∩ B) = P(A) × P(B). If knowing B changes P(A), the events are dependent. Always check this multiplication rule to confirm independence rather than assuming it.

The addition rule (union of events)

To find the probability that A or B occurs, use the general addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtracting the overlap avoids double-counting outcomes in both events. If A and B are mutually exclusive (disjoint), P(A ∩ B) = 0, simplifying to P(A ∪ B) = P(A) + P(B).

Mutually exclusive vs. independent

These are different ideas! Mutually exclusive events cannot happen together (P(A ∩ B) = 0). Independent events don't influence each other's probability. Two events with nonzero probabilities that are mutually exclusive are actually dependent, since knowing one occurred makes the other impossible — changing its probability to 0.

Key terms

Independent events

Events where P(A|B) = P(A), or equivalently P(A ∩ B) = P(A) × P(B); one event's outcome doesn't affect the other's probability.

General addition rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); subtracts overlap to avoid double-counting.

Mutually exclusive (disjoint) events

Events that cannot happen at the same time; P(A ∩ B) = 0.

Mutually exclusive vs. independent

Mutually exclusive events with nonzero probability are dependent, not independent — they are distinct concepts.

Simplified addition rule for disjoint events

When A and B are mutually exclusive: P(A ∪ B) = P(A) + P(B).

Try an AP-style question

A and B are independent, P(A) = 0.5, and P(B) = 0.4. What is P(A and B)?

1. 0.2
2. 0.45
3. 0.9
4. 0.1