JEE MAIN - Mathematics (2005 - No. 31)
Let $$A$$ and $$B$$ two events such that $$P\left( {\overline {A \cup B} } \right) = {1 \over 6},$$ $$P\left( {A \cap B} \right) = {1 \over 4}$$ and $$P\left( {\overline A } \right) = {1 \over 4},$$ where $${\overline A }$$ stands for complement of event $$A$$. Then events $$A$$ and $$B$$ are :
equally likely and mutually exclusive
equally likely but not independent
independent but not equally likely
mutually exclusive and independent
Explanation
Given that,
$$P(\overline {A \cup B} ) = {1 \over 6}$$, $$P(A \cap B) = {1 \over 4}$$, $$P(\overline A ) = {1 \over 4}$$
$$\because$$ $$P(\overline {A \cup B} ) = {1 \over 6}$$
$$ \Rightarrow 1 - P(A \cup B) = {1 \over 6}$$
$$ \Rightarrow 1 - P(A) - P(B) + P(A \cap B) = {1 \over 6}$$
$$ \Rightarrow P(\overline A ) - P(B) + {1 \over 4} = {1 \over 6}$$
$$ \Rightarrow P(B) = {1 \over 4} + {1 \over 4} - {1 \over 6}$$
$$ \Rightarrow P(B) = {1 \over 3}$$ and $$P(A) = {3 \over 4}$$
Clearly, $$P(A \cap B) = P(A)P(B)$$,
so, the events A and B are independent events but not equally likely.
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