# Bayesian networks and weird probabilities

I have to solve the following problem:

Suppose we have a bayesian net in which we have the following variables: R, PA and PR

Let:

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 ⇒⇒ 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?

$$P(R, PR, PA) = P(R)P(PA)P(PR|R,PA)$$
so you can calculate the joint probability directly from the conditional probabilities. Like you said you know $P(\neg r)$ and $P(\neg pa)$ and you also know $P(pr| \neg r, \neg pa)$.