Discussion
Suppose an unbiased coin is tossed.
Event A : Getting head.
Event B : Getting tail.
Which of the following statements is true ?
Answer: B
The events are mutually exclusive.
On a single toss the coin cannot show both a head and a tail, so A and B cannot occur together. That is the definition of mutually exclusive events: \(P(A \cap B) = 0\).
They are not independent. For independent events the product rule must hold:
\[ \begin{aligned} P(A)P(B) &= \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4} \\ P(A \cap B) &= 0 \ne \tfrac{1}{4} \end{aligned} \]
Indeed, knowing that A happened tells us for certain that B did not. Two events with non-zero probabilities can never be both mutually exclusive and independent, which rules out the third option, and the last option is wrong because they are mutually exclusive.
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