What is the relationship between nonempty intersection and independence?

Conditions

• Events A and B have a non-empty intersection ($AB \neq \emptyset$).

Reasoning, step by step

1. Recognize that $AB \neq \emptyset$ implies the events are compatible and can happen at the same time.
2. Understand that compatibility is a necessary but not sufficient condition for independence.
3. Acknowledge that events with a non-empty intersection can still be heavily dependent (e.g., if one is a subset of the other).
4. Conclude that a non-empty intersection only allows for the possibility of independence, not certainty.

Example

If event A is 'rolling a 2 on a die' and event B is 'rolling an even number', their intersection is non-empty. However, A occurring guarantees B occurs, making them dependent, not independent.

Common misconceptions

• Assuming that if two events can happen together, they must be independent.
• Believing that a non-empty intersection precludes any form of statistical dependence.

Watch the explanation

Explore next

• Why, when AB ≠ ∅, can we only say the events 'might possibly' be independent—not 'definitely' independent?
• How does the video use a Venn diagram to refute the claim that non-empty intersections imply independence?