Having the simple model, for example $P(W\mid R)= 0.8$ Is probability of wet grass given rain and $P(W\mid S)= 0.6$ Is probability of wet grass given sprinkler, It is correct, that the $P(W \mid S,R)$, if $S$ and $R$ would be statistically independent, is calculated as following: $$P(-W \mid R)=0.2$$ $$P(-W \mid S)=0.4$$ $$P(-W \mid S,R) = 0.2*0.4 =0.08$$ $$P(W \mid S,R) = 1-0.08 = 0.92$$
The questions are:
- Is this inference correct, again, given independence of $S$ and $R$?
- why do we calculate it using multiplication of negative probabilities and do not use positive probabilities in some way?
- If given $P(W \mid S,R)$ would be unequal to $0.92$, would that be that $S$ and $R$ have conditional dependencies towards each other?
- Why if $W$ is given, $S$ and $R$ became dependent anyway?