How to use the coin-tossing example to prove that mutually exclusive events cannot be independent.

Conditions

• A single toss of a fair coin.
• Event A is 'heads up' and Event B is 'tails up'.

Reasoning, step by step

1. Define event A as getting heads and event B as getting tails in a single coin toss.
2. Establish that A and B are mutually exclusive because a single toss cannot yield both outcomes ($AB = \emptyset$).
3. Calculate the unconditional probability of A, which is 0.5.
4. Evaluate the conditional probability of A given B occurred, which is 0.
5. Compare the probabilities: since $P(A|B) = 0 \neq P(A) = 0.5$, the occurrence of B affects the probability of A.
6. Conclude that mutually exclusive events with nonzero probabilities are not independent.

Example

In a coin toss, let A = {heads} and B = {tails}. They are mutually exclusive ($AB = \emptyset$). If we know tails occurred (B), the probability of heads (A) drops to 0, which is different from its original probability of 0.5. This demonstrates strong dependence.

Common misconceptions

• Thinking that because heads and tails are separate physical outcomes, they are statistically independent.
• Assuming that mutual exclusivity means the events do not influence each other's probabilities.

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