Let $X$ and $Y$ be two events such that $P(X)=\frac{1}{3}$, $P(X|Y)=\frac{1}{2}$ and $P(Y|X)=\frac{2}{5}$. Then

$P(X) = \dfrac{1}{3},\quad P(X|Y) = \dfrac{1}{2},\quad P(Y|X) = \dfrac{2}{5}$ $P(X|Y) = \dfrac{P(X \cap Y)}{P(Y)}$ $\text{To find: which of } (1),\ (2),\ (3),\ (4) \text{ are TRUE}$ $1.\ P(X \cap Y) = P(Y|X)\,P(X)$ $2.\ P(Y) = \dfrac{P(X \cap Y)}{P(X|Y)}$ $3.\ P(X'|Y) = 1 - P(X|Y),\quad P(X \cup Y) = P(X)+P(Y)-P(X \cap Y)$ $\text{Venn diagram: universe } S,\ \text{events } X,\ Y$ $P(Y|X) = \dfrac{P(X \cap Y)}{P(X)}$ $P(X \cap Y) = P(Y|X)\cdot P(X) = \dfrac{2}{5}\cdot\dfrac{1}{3} = \dfrac{2}{15}$ $\dfrac{2}{15} \neq \dfrac{1}{5} \ \Rightarrow\ (3)\ \text{is FALSE}$ $P(X|Y) = \dfrac{P(X \cap Y)}{P(Y)} \Rightarrow P(Y) = \dfrac{P(X \cap Y)}{P(X|Y)}$ $P(Y) = \dfrac{2/15}{1/2} = \dfrac{4}{15} \Rightarrow (1)\ \text{is TRUE}$ $P(X'|Y) = 1 - P(X|Y) = 1 - \dfrac{1}{2} = \dfrac{1}{2} \Rightarrow (2)\ \text{is TRUE}$ $P(X \cup Y) = P(X) + P(Y) - P(X \cap Y)$ $= \dfrac{1}{3} + \dfrac{4}{15} - \dfrac{2}{15} = \dfrac{5 + 4 - 2}{15} = \dfrac{7}{15}$ $\dfrac{7}{15} \neq \dfrac{2}{5} \ \Rightarrow\ (4)\ \text{is FALSE}$ $(1)\ \checkmark \qquad (2)\ \checkmark \qquad (3)\ \times \qquad (4)\ \times$ $\boxed{(1)\ \text{and}\ (2)\ \text{are TRUE}}$