Is independence preserved by conditioning? $X_1$ and $X_2$ are independent. $Y_1|X_1\sim\mathrm{Ber}\left(X_1\right)$, $Y_2|X_2\sim\mathrm{Ber}\left(X_2\right)$. Are $Y_1$ and $Y_2$ necessarily independent? (Assume $\mathrm{P}\left(0<X_1<1\right)=1$, $\mathrm{P}\left(0<X_2<1\right)=1$)

No. Let $X_1,X_2$ be independent uniform (0,1) random variables, and then define $$Y_1=1_{(X_1\geq X_2)}\,\mbox{  and }\, Y_2=1_{(X_2> X_1)}.$$
