An urn contains three white, six red and five black balls. Six balls are selected randomly. Let
X and Y, respectively, be the numbers of white and black balls selected.
a) Find p (X 1) X Y
b) Find E[X | Y = 2]
c) Are X and Y independent?
b) Y = 2, so the situation is the same as choosing 4 balls out of three white and six red. C(n, k) means n choose k.
E[X|Y=2] = P(X=1|Y=2)*1 + P(X=2|Y=2)*2 + P(X=3|Y=2)*3
= (1/C(9,4))*(C(3,1)*C(6,3)*1 + C(3,2)*C(6,2)*2 + C(3,3)*C(6,1)*3)
c) No. As a simple counterexample, P(X=2) > 0, but P(X=2|Y=5) = 0.