Joint Probability Statistics Problem?

Question

This is my last problem and I don't know how to approach it. Any ideas?

Contracts for two athletes are randomly assigned to one or more of three firms A, B, and C. Let Y_1 denote the number of contracts assigned to team A and Y_2 denote the number of contracts assigned to team B.
a) Find the joint probability function for Y_1 and Y_2
b) Find F(1,0)

thanks!

Answer 1

The first athlete's contract has a 1/3 probability of being assigned to each of teams A, B and C. So does the second athlete's contract. So there are 9 possibilities, each with probability 1/9:
AA, AB, AC, BA, BB, BC, CA, CB, CC
where the first letter represents the first athlete's contract and the second letter represents the second athlete's contract.
Corresponding values for Y1 are
2, 1, 1, 1, 0, 0, 1, 0, 0
and for Y2 are
0, 1, 0, 1, 2, 1, 0, 1, 0
So we get the following table for the joint PDF:
Y1\Y2 . 0 . . 1 . . 2
. 0 . . 1/9 . 2/9 . 1/9
. 1 . . 2/9 . 2/9 . 0
. 2 . . 1/9 . . 0 . . 0

and F(1, 0) = 2/9.