I'm trying to work my way through the following problem:
Suppose you have ten buyers, we'll call them and ten sellers, .
For each , let that buyer, have $i available to him/her. Similarly, each seller has the item the buyers want for sale at $i (they all want the same item).
We put all twenty participants in a room- if a buyer meets a seller and the buyer has enough money to purchase the item, both the buyer and seller make the sale and leave the room.
What is the expected number of sales?
My immediate problem is even modeling the problem, yet alone computing the probability distribution. I've started by trying to build the model with just two buyers and sellers, since we're in the simple case of just two permutations.
So, if N = 2, then if X represents the number of sales, we have
P(0) = 0,
P(1) = 0.5,
P(2) = 0.5
so E(X) = 1.5
However, the problem quickly expands combinatorially with increased buyers/sellers, and the problem isn't _really_ recursive, that is, the specifics of which buyers and sellers stay in the room is relevant.
Does anyone have any ideas on how they might approach this problem?