What is $P(A|B,C)$ when $B$ and $C$ are independent? Let's say that I know the following:

*

*$P(A|B)$ is the probability that a storm is coming given it's cloudy.

*$P(A|C)$ is the probability that a storm is coming given that the dogs bark.

*$P(B)$ and $P(C)$ are independent.

How do I compute the following?:

*

*$P(A|B,C)$, the probability of a storm coming given that it's cloudy AND the dogs are barking.

In layman's terms, I know that there is some likelihood that a storm is coming if it's cloudy. And, I know that there is some likelihood that a storm is coming if the dogs are barking. Therefore, shouldn't I have more confidence that a storm is coming if it's cloudy AND the dogs are barking? How do compute this?
The reason that I ask this question is because I am trying to combine measurements from two different sensors that measure the same thing. If I combine the measurements, shouldn't I expect greater confidence in my measurement?
This post and this post are related to my question, but the answers fall short in that I do not know the general probabilities of $P(B)$ and $P(C)$ to compute $P(A|B,C)$.

given values of
P(A|B,C)
P(A|B,!C)
P(A|!B,C)
P(A|!B,!C)
given
P(B), P(C) and P(A|B) and P(A|C)
given B and C are independent
Then:
P(A|B) = P(A|B,C) * P(C) + P(A|B,!C) * P(!C)
P(A|C) = P(A|B,C) * P(B) + P(A|!B,C) * P(!B)
This only gives you two equations but three unknowns P(A|B,!C), P(A|!B,C) and P(A|B,C).
