Conditional probability doubt Assume
$$P(B|A) = 1/5,\ P(A) = 3/4,\ P(A \cap B) = 3/20,\ \textrm{and}\ P(¬B|¬A) = 4/7.$$
Find $P(B)$.
What I tried: $$P(B)=\dfrac{ P(A \cap B)}{P(B|A)}=(3/20)/(1/5) = 3/4.$$ 
Answer is $P(B)=9/35.$
Where have I made the mistake? 

The probability of B can be split into the probability given A and given not A
$$P(B) = P(B|A) \cdot P(A) + P(B|\neg A) \cdot P(\neg A)$$
The negations can be replaced by one minus the actual and vice versa
$$P(B) = \frac{1}{5} \cdot \frac{3}{4} + (1-P(\neg B| \neg A)) \cdot (1-P(A))$$
$$P(B) = \frac{3}{20} + (1-\frac{4}{7})\cdot (1-\frac{3}{4})$$ 
$$P(B) = \frac{3}{20} + \frac{3}{7} \cdot \frac{1}{4}$$
$$p(B) = \frac{3}{20} + \frac{3}{28}$$
$$p(B) = \frac{21}{140} + \frac{15}{140}$$
$$P(B) = \frac{36}{140}$$
$$P(B) = \frac{9}{35}$$
