Suppose $A$ and $B$ are two events with $P (A) = 0.5$ and $P (A \cup B) = 0.8$ . Let $P (B) = p$ if $A$ and $B$ are mutually exclusive and $P (B) = q$ if $A$ and $B$ are independent events, then the value of $q / p$ is

2

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1

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3

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0.1

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Solution

The correct option is C $2$
$P (A) = 0.5$ and $P (A B) = 0.8$
When A and B are independent events,
$P (A \cap B) = P (A) \times P (B)$
$\Rightarrow P (A) + P (B) - P (A \cup B) = 0.5 \times q$
$\Rightarrow 0.5 + q - 0.8 = 0.5 q$
$\Rightarrow 0.5 q = 0.3$
$\Rightarrow q = 0.6$
When A and B are mutually exclusive events
$P (A \cap B) = 0$
$P (A) + P (B) = P (A \cup B)$
$0.5 + p = 0.8$
$\Rightarrow p = 0.3$
Hence the ratio of $q$ and $p$ is 2

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