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Let $X$ and $Y$ are two events such that $P(X \cup Y=P)\,(X \cap Y).$
Statement $1:$ $P(X \cap Y' = P)\,(X' \cap Y = 0).$
Statement $2:$ $P(X) + P(Y = 2)\,P\,(X \cap Y)$
Statement $1$ is false, Statement $2$ is true.
Statement $1$ is true, Statement $2$ is true,Statement $2$ is not a correct explanation of Statement $1$ .
Statement $1$ is true, Statement $2$ is false
Statement $1$ is true, Statement $2$ is true;Statement $2$ is a correct explanation of Statement $1.$
Solution
Let $X$ and $Y$ be two events such that
$P(X \cup Y)=P(X \cap Y)$ …..$(1)$
We know
$P(X \cup Y)=P(X)+P(Y)-P(X \cap Y)$
$P(X \cap Y)=P(X)+P(Y)-P(X \cap Y)$
(from $( 1)$)
$\Rightarrow P(X)+P(Y)=2 P(X \cap Y)$
Hence, Statement – $2$ is true
Now, $P\left(X \cap Y^{\prime}\right)=P(X)-P(X \cap Y)$
and $P\left(X^{\prime} \cap Y\right)=P(Y)-P(X \cap Y)$
This implies statement- $1$ is also true
