Conditional probability of a variable conditioned by more than one variable

Question

I'm learning statistics and I can find any example of how to calculate this $P(x|y,z)$.

I think is simply than that:

$$P(x|y,z) = \frac{P(x,y,z)}{P(y,z)}$$

Is that correct?

I will appreciate a suggestion for a good book of probability.

Answer 1

I assume you agree that $Pr(A \cap B) = Pr(A|B)Pr(B)$ for events $A$ and $B$.

First, we can write your $P(x,y,z)$ in a more standard way as $Pr(x \cap y \cap z)$.

Then we can re-bracket, making use of the associativity of set intersection, so that we have $Pr(x \cap (y \cap z))$. From this, using the above rule, it follows that $Pr(x \cap y \cap z) = Pr(x|y \cap z)Pr(y \cap z)$.

Rearrange this equation to obtain $$Pr(x|y \cap z) = \frac{Pr(x \cap y \cap z)}{Pr(y \cap z)}$$ as you had. So, yes—you are correct.

I hope this helps your intuition about extending this idea to as many events as you like.