Assume there is a lottery where you can buy lots for 1\$ each. To win the grand price you have to collect $n$ different coupons $C_1, \ldots, C_n$ where $C_i$ occurs with probability $p_i$. You may assume that there are "infinitely" many lots, i.e. the $p_i$ do not change over time and successive drawings are independant. And of course $\sum p_i\le 1.
I specifically want to consider the case where the p_i are far from being equal.
Q1: What would the grand prize be worth if the lottery is fair?
Q2: What would be a fair price to sell a coupon of type C_i$ to other players? The obvious answer $1\over p_i$ seems to be wrong because in order to collect all other coupons one has to buy so many lots anyway that it is likely to find a $C_i$ while doing that (unless $p_i\ll p_j$ for $j\ne i)
Q3: Assume two players have collected subsets A$, $B$ of $\mathcal C=\{C_1, \ldots, C_n\}$ such that $A\cup B=\mathcal C$. If they cooperate, what would be a fair method to share the grand prize?