Are impossible events independent of any event?

Conditions

• Event B is the impossible event ($B = \emptyset$).
• Event A is any arbitrary event.

Reasoning, step by step

1. Identify the impossible event as the empty set ($B = \emptyset$).
2. Calculate the intersection of A and B, which is $A\emptyset = \emptyset$.
3. Determine the probability of the intersection: $P(A\emptyset) = P(\emptyset) = 0$.
4. Calculate the product of their individual probabilities: $P(A)P(B) = P(A) \cdot 0 = 0$.
5. Compare the two results: $P(A\emptyset) = P(A)P(B)$, proving they are independent.

Example

Let A be 'heads appears when flipping a coin' and B be the impossible event (the empty set ∅). The intersection AB is ∅, satisfying mutual exclusivity. However, since $P(B) = 0$, P(AB) = $0 = P(A)P(B)$, meaning A and B are also independent.

Common misconceptions

• Assuming that because an impossible event is mutually exclusive with any event, it cannot be independent.
• Thinking that independence requires both events to have a nonzero probability of occurring.

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Explore next

• Why are mutually exclusive events with nonzero probabilities generally not independent?
• Can two events be both mutually exclusive and independent?