Why are mutually exclusive events with nonzero probabilities generally not independent?

Conditions

• Events A and B are mutually exclusive ($AB = \emptyset$).
• The probability of event A is greater than 0 ($P(A) > 0$).
• The probability of event B is greater than 0 ($P(B) > 0$).

Reasoning, step by step

1. Assume events A and B are mutually exclusive, so their intersection is the empty set ($AB = \emptyset$).
2. If A occurs, B cannot occur, which means the conditional probability of B given A is zero ($P(B|A) = 0$).
3. Since $P(B) > 0$, we have $P(B|A) \neq P(B)$.
4. By the definition of independence, if the conditional probability differs from the unconditional probability, the events are not independent.
5. Conclude that mutually exclusive events with nonzero probabilities are necessarily dependent.

Example

Consider a single coin toss. Let A be 'heads' and B be 'tails'. They are mutually exclusive ($AB = \emptyset$). If tails occurs (B), the probability of heads (A) becomes 0, which is not equal to its original probability of 0.5. Thus, they are not independent.

Common misconceptions

• Believing that if two events cannot happen at the same time, they are unrelated or independent.
• Assuming that mutual exclusivity always implies independence.

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