Calculation of Probabilities of Events and Conditional Probabilities

To determine the correct options, we analyze the given information step-by-step using the basic laws of probability and conditional probability.

1. Given Data: $P(X) = \frac{1}{3}$ $P(X|Y) = \frac{1}{2}$ $P(Y|X) = \frac{2}{5}$

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2. Calculating $P(X \cap Y)$: By definition of conditional probability: $P(Y|X) = \frac{P(X \cap Y)}{P(X)}$

Substitute the given values into the formula: $\frac{2}{5} = \frac{P(X \cap Y)}{\frac{1}{3}}$ $\implies P(X \cap Y) = \frac{2}{5} \times \frac{1}{3} = \frac{2}{15}$

Thus, Option (C), which states $P(X \cap Y) = \frac{1}{5}$, is incorrect.

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3. Calculating $P(Y)$: By definition of $P(X|Y)$: $P(X|Y) = \frac{P(X \cap Y)}{P(Y)}$

Substitute the known values: $\frac{1}{2} = \frac{\frac{2}{15}}{P(Y)}$ $\implies P(Y) = 2 \times \frac{2}{15} = \frac{4}{15}$

Thus, Option (A) is correct.

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4. Calculating $P(X'|Y)$: Since $P(X'|Y) + P(X|Y) = 1$: $P(X'|Y) = 1 - P(X|Y) = 1 - \frac{1}{2} = \frac{1}{2}$

Thus, Option (B) is correct.

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5. Calculating $P(X \cup Y)$: Using the addition theorem of probability: $P(X \cup Y) = P(X) + P(Y) - P(X \cap Y)$

Substitute the calculated values: $P(X \cup Y) = \frac{1}{3} + \frac{4}{15} - \frac{2}{15}$ $P(X \cup Y) = \frac{5}{15} + \frac{4}{15} - \frac{2}{15} = \frac{7}{15}$

Since $\frac{7}{15} \neq \frac{2}{5}$, Option (D) is incorrect.

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Conclusion: The correct options are A and B.