I have to solve the following problem:
Suppose we have a bayesian net in which we have the following variables: R, PA and PR
P(R) = 0.1, P(PA) = 0.5, P(PR|R, PA) = 0.6, P(PR|¬R, PA) = 0.4, P(PR|¬R, ¬PA) = 0.1 and P(PR|R, ¬PA) = 0.2
What is the probability of P(¬R, PR, ¬PA)?
I started with P(¬R) and P(¬PA), because I can compute them as follows:
P(¬R) = 1 - P(R) = 0.9 P(¬PA) = 1 - P(PA) = 0.5
Then I think I can compute P(PR|¬R) and use bayes rule, however:
P(PR|¬R, PA) = 0.4 $\Rightarrow$ P(PR|¬R) * P(PA) = 0.4 \Rightarrow P(PR|¬R) = 0.8
I also have P(PR|¬R, ¬PA) = 0.1 \Rightarrow P(PR|¬R) = 0.2
The same for P(PR|R)... I get different results, so I can't apply bayes rule. This means I am obviously doing something wrong, where is my mistake? How can I solve it?