Let X and Y are two events such that P(X Y=) P X (Y . ) Statement 1: P(X Y^ = P X^ (Y=0 ). Statement 2: P(X) P Y ∈ 2) P X Y( )

Correct answer: B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1. Let X and Y be two events such that P(X Y)=P(X Y) We know aligned & P(X Y)=P(X)+P(Y)-P(X Y) & P(X Y)=P(X)+P(Y)-P(X Y) & (from (1) & P(X)+P(Y)=2 P(X Y) aligned Hence, Statement -2 is true. Now, P(X Y^ )=P(X)-P(X Y) and P(X^ Y)=P(Y)-P(X Y) This implies statement-1 is also true.

JEE Main 20127 MayOnline

Exam: JEE Main
Year: 2012
Date: 7 May
Mode: Online
Subject: Mathematics
Chapter: Sets and Relations

JEE Main 2012 · 7 May · Online · Mathematics · Sets and Relations

Let X and Y are two events such that P(X Y=) P X (Y . ) Statement...

Let

X

and

Y

are two events such that

P(X \cup Y=) P X \cap(Y . \quad)

Statement 1:

P\left(X \cap Y^{\prime}=\dot{P} X^{\prime} \cap(Y=0 \quad)\right.

Statement 2:

P(X) P Y \in 2) P X \cap Y(\quad)

Options

AStatement 1 is false, Statement 2 is true.
BStatement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
CStatement 1 is true, Statement 2 is false.
DStatement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

Correct answer

B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

Step-by-step solution

Let

X and Y

be two events such that P(X \cup Y)=P(X \cap Y) We know

\begin{aligned}& P(X \cup Y)=P(X)+P(Y)-P(X \cap Y) \\ & P(X \cap Y)=P(X)+P(Y)-P(X \cap Y) \\ & \text{ (from }(1) \\ & \Rightarrow P(X)+P(Y)=2 P(X \cap Y) \end{aligned}

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.

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