How to compute conditonal probability on multiple conditions on the right side?

Question

Is there a formula for a conditional probability of many "conditions" I.e: $P(X|Y,Z,\ldots)$ That is how to compute $P(X)$ given $X,Y,\ldots$ Or given separately $P(X|Y)$ and $P(X|Z)$ is there a formula to compute $P(X|Y,Z)$

Answer 1

In principle, for two conditions, by the usual formula, we have $$\Pr(X|Y,Z)=\frac{\Pr(X\cap Y\cap Z)}{\Pr(Y\cap Z)}.$$ From simply knowing the conditional probabilities $\Pr(X|Y)$ and $\Pr(X|Z)$, and say $\Pr(Y)$ and $\Pr(Z)$, and even $\Pr(Y\cap Z)$, we cannot determine $\Pr(X|Y,Z)$.