An enterprise has workers divided in 3 groups. $p_i$ is a proportion of the $i$-th group to all, $p_1+p_2+p_3=1$.
Select independently and equiprobably with replacement $n$ workers out of all, $X_i$ workers belong to $i$-th group. $n=X_1+X_2+X_3$. Use $Y_i=X_i/n$ as unknown proportion $p_i$. After find the joint probability function $p(y_1,y_2,y_3)$. If I have large size of sample, does the random variable $Y_i$ converge to the true value $p_i$? If $y_1>p_1$, which is more probable, $Y_2>p_2$ or $p_2>Y_2$?and what is a conditional mean E($Y_i/y_1$),(i=2,3)?