I'm having some trouble figuring out a homework assignment which requires me to find the probabilities of two different variables that have a common parent. In order to better understand how to do this, I was wondering if someone could tell me how I could find P(S, R) since they both depend on C. This is a common Bayesian network example, which finds the probability of the grass being wet.
If C = T, then I know P(S|C) = .1 and P(R|C) = .8, so would P(S, R) = .08 since we know C = T? Alternatively, if C = F, then P(S|C) = .5 and P(R|C) = .2, so would P(S, R) = .1 since C = F? I thought that if this is the case, and since C = T or C = F, would P(S, R) regardless of C be .18?
This is as far as I got, and I would really appreciate some guidance!
You need to use the fact that Sprinkler and Rain are conditionally independent given Cloudy (this is implied by the structure of the network). In other words, $P(S,R|C)=P(S|C)P(R|C)$. Now you can work things out:
$$P(S,R)=P(S,R|C=T)P(C=T)+P(S,R|C=F)P(C=F)=...$$
can you finish it from here?