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  4. If A and B are independent events such that P(B) = (2/7),P(A∪ barB) = 0.8, then P(A)

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Q. If $A$ and $B$ are independent events such that $P(B) = \frac{2}{7},P\left(A\cup\bar{B}\right) = 0.8, then\, P(A) $

AMUAMU 2016Probability - Part 2

Solution:

We know that,
$P(A \cup \bar{B}) = P(A) + P(\bar{B}) - P(A \cap \bar{B})$
$ = P(A) + [1 -P(B)] - P(A) \cdot P(\bar{B})$
$ = P(A) + [1 - P(B)] - P(A) [1 - P(B)]$
$\Rightarrow 0.8 = P(A) + (1 - \frac{2}{7} ) - P(A) ( 1 - \frac{2}{7}) $ [Given]
$\Rightarrow \frac{4}{5} = P(A) [ 1 - 1 + \frac{2}{7}] + \frac{5}{7}$
$\Rightarrow \frac{4}{5} - \frac{5}{7} = P(A) \cdot \frac{2}{7}$
$\Rightarrow \frac{28-25}{35} = P(A) \cdot \frac{2}{7}$
$\Rightarrow P(A) = \frac{3}{35} \cdot \frac{7}{2}$
$ = \frac{3}{5\times2}$
$ = 0.3$