Player extracts card from the deck (which has infinite number of size) to obtain one of $k$ colors of cards. The possibility that the player pick a card with $i$th color is given by $p_i>0$. Of course, $\sum_{i=1}^{k}p_{i} = 1$. If you collect at least one card from all colors, then we call you completed *one set of cards*. After certain number of trials (as the player want to end), dealer gives player $m$ dollars, where $m$ is the number of set of cards the player have completed. My object is to calculate the expectation value of number of trials to get $m$ dollar, where $m$ is given.

To rewrite it more mathematically:

Let $X_1,X_2,\cdots,X_k$ be multinomial distribution with probability $p_i>0$ respectively. Let $g(k,m)$ be the least number of trials which making $\min(X_i)=m$. $g(k,m)$ forms a distribution of the number of trials, namely $n$. What is the explicit formula of $\mathbb{E}(g(k,m))$?

We know that if $m=1$, it becomes the Coupon Collector's problem. The exact formula is given by

$$\mathbb{E}(g(k,1)) = \int_{0}^{\infty}(1-\prod_{i=1}^{k}(1-\exp(-p_{i}t)))dt$$

At first, I tried the simplest case: when $k=2$, the probability that after exactly $n$ trials we get $m$ dollars is $$\mathbf{P}(X_{1}=m-1,X_{2}=n-m)\times p + \mathbf{P}(X_{1}=n-m,X_{2}=m-1)\times (1-p)$$ $$={n\choose m-1} p^{m}(1-p)^{m}(p^{n-2m}+(1-p)^{n-2m})$$ Therefore: $$\mathbb{E}(g(2,m))=\sum_{n=2m}^{\infty}{n\choose m-1} np^{m}(1-p)^{m}(p^{n-2m}+(1-p)^{n-2m})$$ Especially, $\mathbb{E}(g(2,1))=\frac{p^{2}-p+1}{p(1-p)}$, which attains minimum at $p=\frac{1}{2}$, and this result fits well, since the difference between probabilities from extracting cards of different colors increase, making one set with given number of trials will be significantly harder. I have two conjectures:

$\mathbb{E}(g(k,m))$ attains minimum value when $p_{1}=p_{2}=\cdots=p_{k}=\frac{1}{k}$.

$\mathbb{E}(g(k,m))$ will behave like $\frac{m}{\min(p_i)}$ asymptotically as $m \to \infty$.

But I failed to figure out for the general case, since it requires too many number of calculations of infinite sums over binomial coefficients.

Can I get your help? Thanks.