Suppose X_{1}, X_{2,}X_{3}~ Multinomial(n, $\displaystyle \theta$_{1}, $\displaystyle \theta$_{2}, $\displaystyle \theta$_{3}).

I know that X_{1}, X_{2,}X_{3}are individually Binomial(n, $\displaystyle \theta$_{1}), Binomial(n, $\displaystyle \theta$_{2}), Binomial(n, $\displaystyle \theta$_{3}) respectively.

I'd like to know if:

1) X_{1}, X_{2,}X_{3}are independent

2) If they are, whether that means that the conditional distribution of X_{2}given X_{1}would be simply Binomial(n, $\displaystyle \theta$_{2})?

Thanks.