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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Probability
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Let $A$ and $B$ be two events, such that: $ P(\overline{A \cup B} ) $$= \large\frac{1}{6}\:$,$\;\; P( A \cap B ) $$= \large\frac{1}{4} $,$\;$and $\;\;P(\overline A)$$=\large\frac{1}{4}$ where $\overline A$ stands for complement of event $A$. Then events $A$ and $B$ are

(A) Independent but not equally likely $\quad$(B) Mutually exclusive and independent$\quad$ (C) Equally likely and mutually exclusive $\quad$ (D) Equally likely but not independent $\quad$

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  • $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
$ P( \overline{A \cup B}) = \large\frac{1}{6}$
$ P( A \cap B) = \large\frac{1}{4}$
$ P( \overline A) = \large\frac{1}{4}$
$ \Rightarrow P(A) = 1-P(\overline A)$
$ = \large\frac{3}{4}$
$ P( \overline{A \cup B}) = 1-P(A \cup B)$
$\qquad\qquad = 1-P(A)-P(B)+P(A \cap B)$
$\Rightarrow\: \large\frac{1}{6} = \large\frac{1}{4}$$-P(B)+\large\frac{1}{4}$
$ \Rightarrow P(B)= \large\frac{1}{3}$
Since $P(A \cap B ) = P(A) \: P(B)\: and \: P(A) \neq P(B)$
$ \therefore $ A and B are independent but not equally likely.
Ans : (A)
answered Jan 3, 2014 by thanvigandhi_1
edited Mar 26, 2014 by rvidyagovindarajan_1

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