1
$\begingroup$

If we observe that it is cloudy and raining. What is the probability that the grass is wet? The answer would be:

P(W=T|C=T,R =T) = P(W=T|R=T,S=T)*P(S=T|C=T)+P(W=T|R=T,S=F)*P(S=F|C=T)

But if we observe that the sprinkler is on and the grass is wet, then what would be the probability that it is raining? I'm not sure what would be the solution query to this problem?

enter image description here

$\endgroup$
2
  • $\begingroup$ If it is homework, please tag it as self-study $\endgroup$
    – jpmuc
    Jan 3, 2018 at 15:10
  • $\begingroup$ @jpmuc It's a group of self-study questions. $\endgroup$
    – Tak
    Jan 3, 2018 at 15:56

1 Answer 1

1
$\begingroup$

You first need to work out what the probability of the model is, $$ P(W, S, R, C) = P(W|S, R)P(S|C) P(R|C)P(C) $$ Then you ask for, $$ P(R=1|S=1, W=1) $$

Now, $$ P(R|S,W) = \frac{P(S, W|R)P(R)}{P(S,W)} $$ Let us first consider the numerator, $$ P(S, W|R)P(R) = \\ P(W|R, S)P(R)P(S|R) = \\ P(W|R, S)P(R,S) = \\ P(W|R, S)\sum_{C}P(R|C)P(S|C)P(C) $$ Similarly, $$ P(S,W) = \sum_{R}P(W|R,S) \sum_{C}P(R|C)P(S|C)P(C) = \\ \sum_{C} P(S|C)P(C) \sum_{R}P(W|R,S) P(R|C) $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.