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G and H are mutually exclusive with P(G) = 0.5 and P(H) = 0.3. Why can P(H given G) not be 0.4?
ABecause P(H given G) must equal P(G), which is 0.5
BBecause G rules out H, so P(H given G) must be zero
CBecause P(H given G) must equal P(H), which is 0.3
DBecause the two events add to 0.8, not to 1.0 here
Answer & Solution
Correct answer: B. Because G rules out H, so P(H given G) must be zero
1. Mutually exclusive events cannot occur together, so the joint probability of G and H is zero.
2. The conditional is that joint probability divided by P(G), which is 0 divided by 0.5.
3. That gives exactly zero, so no positive value such as 0.4 is possible.
4. In words, once G has happened H is ruled out completely.
5. Setting the conditional equal to P(H) would assert independence, which mutually exclusive events with non-zero probabilities can never satisfy.
6. The two probabilities need not sum to one, since a third outcome outside both events may occur.
_Source: OpenStax Introductory Business Statistics (CC BY 4.0), Ch 3 "Probability Topics", section 3.2 Independent and Mutually Exclusive Events_
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