Question about independency of random variables let X, Y, Z be 3 random variables. if X and Y are independent and X and Z are independent, is X and (Y,Z) also independent? (if $$P(X \in A, Y \in B) =P(X \in A)P(Y\in B)$$ and $$P(X \in A, Z \in B) =P(X \in A)P(Z\in B)$$ is $$P(X \in A, (Y,Z) \in B) =P(X \in A)P((Y,Z) \in B)$$ )?

No. See, for example, this answer for a simple example (in which $Y$ and $Z$ also are independent), that is, $X, Y, Z$ are _pairwise_ independent but not mutually independent, and so $X$ is not independent of the bivariate random variable $(Y,Z)$.