Discussion
If E and F are events with \(P(E) \lt P(F)\) and \(P(E \cap F) \gt 0\), then :
Answer: D
None of the implications holds.
The conditions \(P(E) \lt P(F)\) and \(P(E \cap F) \gt 0\) say nothing about one event containing the other. Roll a die with E = {1, 2} and F = {2, 3, 4}. Then \(P(E) = \frac{1}{3} \lt \frac{1}{2} = P(F)\) and \(E \cap F = \{2\}\) has positive probability.
If 1 is rolled, E occurs but F does not, so E does not imply F. If 3 is rolled, F occurs but E does not, so F does not imply E. If 3 is rolled, E fails but F occurs, so non-occurrence of E does not imply non-occurrence of F.
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