Why, when AB ≠ ∅, can we only say the events 'might possibly' be independent—not 'definitely' independent?

Conditions

• Events A and B have a non-empty intersection ($AB \neq \emptyset$).
• Event A is a subset of event B ($A \subset B$).

Reasoning, step by step

1. Acknowledge that $AB \neq \emptyset$ means A and B can occur at the same time.
2. Construct a counterexample where A is entirely contained within B ($A \subset B$).
3. Observe that if A occurs, B must also occur, meaning the conditional probability of B given A is 1 ($P(B|A) = 1$).
4. Note that unless $P(B) = 1$, $P(B|A) \neq P(B)$, proving they are not independent.
5. Conclude that a non-empty intersection only allows for the possibility of independence, not certainty.

Example

Imagine rolling a standard six-sided die. Let A be 'rolling a 2' and B be 'rolling an even number'. A is a subset of B, so $AB \neq \emptyset$. However, if A occurs, B is guaranteed to occur, making them highly dependent, not independent.

Common misconceptions

• Assuming that if two events can happen together, they must be independent.
• Confusing the ability to co-occur with the absence of statistical influence.

Watch the explanation

Explore next

• How does the video use a Venn diagram to refute the claim that non-empty intersections imply independence?
• What is the relationship between nonempty intersection and independence?