Given values for $P(B\mid A\cup B)$ and $P(A\mid B)$. What is $P(B)$?

Question

Let $A$ and $B$ be two independent events. How do you solve for $P(B)$?

Since they are independent, we can say that $P(A\mid B) = P(A)$, but I don't know what much else we can extract from this information.

Answer 1

Note that $$P(B|A\cup B)P(A\cup B)=P(B\cap(A\cup B))=P(B) \text{ (Eq. 1)}$$ where (given that $A$ and $B$ are independent) $$P(A\cup B) = P(A)+P(B)-P(A\cap B)=P(A)+P(B)-P(A)P(B) \text{ (Eq. 2)}$$ As you have correctly pointed out $P(A|B)=P(A)$ as $A$ and $B$ are independent, so we can express (Eq. $2$) as $$P(A\cup B) =P(A|B)+P(B)-P(A|B)P(B)$$ which can be substituted into (Eq. $1$) as follows:- $$P(B|A\cup B)(P(A|B)+P(B)-P(A|B)P(B))=P(B) \\\Rightarrow (1+P(B|A\cup B)P(A|B)-P(B|A\cup B))P(B)=P(B|A\cup B)P(A|B) \\\Rightarrow P(B)=\frac{P(B|A\cup B)P(A|B)}{1+P(B|A\cup B)P(A|B)-P(B|A\cup B)}$$