This is a good example of Bayes Rule:
$$P(B_1|V)=\frac{P(V|B_1)P(B_1)}{P(V)}$$

Your calculation gives $P(V|B_1)P(B_1)=P(B_1\cap V)$, but the denominator needs to be calculated as well:

$$P(V)=P(V|B_1)P(B_1)+P(V|B_2)P(B_2)$$