Is there a way to think about Probability rules with multiple variables?

Question

I am working through Intro to AI, and several times so far Sebastian makes a jump from a rule defined in 2 variables to using the rule with three variables. There seems to be some understanding he is applying here, that I am not getting.

I'll give you another example. I was taking a look at an answer regarding one of the questions, and the answer'er says just use conditional probability. But it is not defined for three variables, only two, so how are they thinking about this?

My theory is posted below.

Answer 1

If you take the Conditional Probability question, every step is conditioned on S, so you can just ignore the S for the purpose of following the rule, and keep everything dependent on S.

To illustrate graphically. Since the original probability is dependent on S, keep all of your probabilities dependent on S, and apply the rule of conditional probability to the R & H variables.

You could have also ignored H, and applied conditional probability to R & S if that would have been more convenient for the solution you were looking for.