# Question #290f5

Sep 6, 2016

$P \left(A | B\right) = \frac{5}{8} = 0.625 , \mathmr{and} , P \left(A \cap B '\right) = 0.175$

#### Explanation:

We know that, $P \left(A \cup B\right) = P \left(A\right) + P \left(B\right) - P \left(A \cap B\right) \ldots \ldots \ldots . \left(1\right)$.

But, we are given that, $P \left(A \cup B\right) = 3 P \left(A \cap B\right)$.

$\therefore \text{ By } \left(1\right) , 3 P \left(A \cap B\right) = P \left(A\right) + P \left(B\right) - P \left(A \cap B\right) , i . e . , 4 P \left(A \cap B\right) = P \left(A\right) + P \left(B\right)$

Here, $P \left(A\right) = 0.3 , P \left(B\right) = 0.2 \text{, so that, } P \left(A \cap B\right) = \frac{0.5}{4} = \frac{1}{8.} . . \left(\star\right)$

Now, $P \left(A | B\right) = \frac{P \left(A \cap B\right)}{P \left(B\right)} = \frac{\frac{1}{8}}{\frac{1}{5}} = \frac{5}{8} = 0.625$

$P \left(A \cap B '\right) = P \left(A - A \cap B\right) = P \left(A\right) - P \left(A \cap B\right) = 0.3 - 0.125 = 0.175$.

Enjoy Maths.!