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Question

Let A and B be two events such that $$P(A \cup B) \geq 3/4 $$ and $$1/8 \leq P( A \cap B) \leq 3/8$$.
Statement 1: $$P(A) + P(B) \geq 7/8$$
statement 2: $$P(A) + P(B) \leq 11/8$$


A
Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1.
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B
Both the statements are TRUE and STATEMENT 2 is not the correct explanation of STATEMENT 1.
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C
STATEMENT 1 is TRUE and STATEMENT 2 is FALSE.
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D
STATEMENT 1 is FALSE and STATEMENT 2 is TRUE.
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Solution

The correct option is B Both the statements are TRUE and STATEMENT 2 is not the correct explanation of STATEMENT 1.
$$P\left( A\cup B \right) =P\left( A \right) +P\left( B \right) -P\left( A\cap B \right) \\ \because \dfrac { 3 }{ 4 } \le P\left( A\cup B \right) \le 1\\ \Rightarrow \dfrac { 3 }{ 4 } \le P\left( A \right) +P\left( B \right) -P\left( A\cap B \right) \le 1\\ \Rightarrow \dfrac { 3 }{ 4 } +P\left( A\cap B \right) \le P\left( A \right) +P\left( B \right) \le 1+P\left( A\cap B \right) \\ \Rightarrow \dfrac { 3 }{ 4 } +\dfrac { 1 }{ 8 } \le P\left( A \right) +P\left( B \right) \le 1+\dfrac { 3 }{ 8 } \\ \Rightarrow \dfrac { 7 }{ 8 } \le P\left( A \right) +P\left( B \right) \le \dfrac { 11 }{ 8 } $$
both the statements are correct but statement $$1$$ is not the correct explanation of statement $$2$$

Mathematics

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