TITLE: Lottery with coupon collecting - what prices are fair?
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: To follow Ross Millikan's comment that "the common ones have zero value and all the value is on the rare one" if trading is allowed, then (Q1) the grand prize should be worth $1/p_{min}$ where $p_{min}$ is the lowest of the $ p_i $.
(Q2a) If there is a single type with that probability, then its value is arbitrarily close to the value of the grand prize, and every other type has value arbitrarily close to zero. 
(Q2b) If there a $k$ types with the same minimum probability then their values are arbitrarily close to the value of the grand prize divided by $k$.
(Q3) They should share the prize in proportion to the number of minimum probability coupons they hold.