# Find conditional PMF of a multinomial distribution

I am trying to find the conditional PMF of a multinomial analytically and though I know my result is wrong I can't seem to pinpoint where my argument is wrong. Seeking help to find my mistake.

Given: $$\vec{X} = Mult(n,\vec{p})$$ where $$\vec{X} = [X_1,X_2,X_3,...X_k]$$ and $$\vec{p} = [p_1,p_2,p_3,...p_k]$$. Find the conditional PMF of $$\vec{X}$$ given $$X_1 = n_1$$

I have tried to solve it like this:

A bag contains $$n$$ different balls belonging to $$k$$ categories and you know the PMF of the number of balls belonging to each category. A third person goes through the bag and removes all balls belonging to type 1 ($$n_1$$ in count) and you have to find the PMF of balls that remain in the bag

$$E_1$$ = A third person goes through the bag and finds $$n_1$$ balls belonging to $$X_1$$ and removes them

$$E_2$$ = You choose a random ball from the bag

To find: $$P(E_2 \in type 2|E_1)$$

Solution: Using baye's theorem:

$$P(E_2 \in type 2|E_1) = \frac{P(E_1|E_2 \in type 2)*P(E_2 \in type 2)}{P(E_1)}$$ $$= \frac{{n-1 \choose n_1}(p_1)^{n_1}(1-p_1)^{n-1-n_1}*{p_2}}{{n \choose n_1}(p_1)^{n_1}(1-p_1)^{n-n_1}}$$ $$\frac{(n-n_1)*p_2}{n*(1-p_1)}$$

Which is clearly not right. It cannot depend on $$n_1$$. Can someone please point me to what i am going wrong?

The joint probability mass function of the vector $$\mathbf X$$ is $$p(x_1,\ldots,x_k)=\binom{n}{x_1\,\cdots\,x_k}p_1^{x_1}\cdots p_k^{x_k}\mathbb I_{x_1+\ldots+x_k=n}$$ Hence $$\mathbb P((X_1,\ldots,X_k)=(x_1,\ldots,x_k))=\binom{n}{x_1\,\cdots\,x_k}p_1^{x_1}\cdots p_k^{x_k}$$ if $$n_1+\ldots+x_k=n$$ and the probability is zero otherwise. This implies that \begin{align*}\mathbb P((X_1,\ldots,X_k)=(x_1,\ldots,x_k)\vert X_1=x_1) &\propto \mathbb P((X_1,\ldots,X_k)=(x_1,\ldots,x_k)\\ &\propto \dfrac{n!}{x_1!\cdots x_k!}p_1^{x_1}\cdots p_k^{x_k}\\ &\propto \dfrac{1}{x_2!\cdots x_k!}p_2^{x_2}\cdots p_k^{x_k}\\ &=\dfrac{(n-x_1)!}{x_2!\cdots x_k!}[p_2/\bar p_1]^{x_2}\cdots [p_k/1-p_1]^{x_k} \end{align*} the normalisation emerging from the Multinomial theorem.