Say there are $n$ dice in the game. I have $p$ dice. Among them, I have $m_1$ aces and $m_2$ three.
Let $X$ be the random variable: X="Number of dice which are either aces or three" and $B$ the event: B="I have $m_1$ aces and $m_2$ three".
What is the conditional expectation of $X$?
I propose:
$$ E(X|B) = \sum_{k=0}^n k\,P_B(X=k)=\sum_{k=1}^n k\,P(X=k-m_1-m_2)$$ $$ =\sum_{k=1}^n k\, C^{k-m_1-m_2}_{n-p}(\frac{1}{3})^{k-m_1-m_2}(\frac{2}{3})^{n-p-k+m_1+m_2} $$
I can check that the statement: $P_B(X=k)=P(X=k-m_1-m_2)$ can be found using: $P_B(X=k)=P(X=k|"m_1 \text{ aces } + m_2 \text{ three}")/P(B).$
Well, given all rolls are independent the expected number of aces or threes given you rolled $m_1+m_2$ is simply $m_1+m_2 + \frac{n-p}{3}$ since each of the remaining $n-p$ dies has a probability of $\frac{1}{3}$ of landing ace or three.