Player can extract card from deck (the size of deck is infinite) to obtain one of $k$ kinds of cards, and the possibility of obtaining each kind is given by $p_i$. (Obviously $\sum_{i=1}^{k} p_{i} = 1$). If you collect AT LEAST one card from all kinds, then you get 1 dollar. Actually, the amount of money you receive is the minimum number of collected cards of all kinds (In other words, the number of sets of cards of all kinds that you've completed)
My object is to calculate the expectation value of number of trials (namely $n$) to receive more than $m$ dollar, as a closed form of $p_{i}$s, $k$, $n$ and $m$.
How can I do this? Thanks.
Please search the web for the coupon collector's problem.
Let $ N$ be the number of cards to extract till we get all $ k$ kinds, and let $N_{i}$ be the number of extracts to collect the $i'th$ kind after $i−1$ kinds have been collected - note that $N=\sum_{i=1}^{k}N_{i}.$
We want to get more then m dollars so we'll calculate $n\geq E(m\cdot N)=m\cdot E(N)=m\sum_{i=1}^{k}E(N_{i}).$
We notice that:
$$N_{1} = 1$$
$$N_{2} \sim Geom\left(\frac{k-1}{k}\right)$$
$$\vdots $$
$$N_{k} \sim Geom\left(\frac{1}{k}\right)$$
Also we know that for $$X\sim Geom(p)\Rightarrow E(X)=\frac{1}{p}$$ therefore:$$m\sum_{i=1}^{k}E(N_{i})=m\cdot k\sum_{i=1}^{k}\frac{1}{i}=m\cdot k\cdot H_{k}$$