Question

Suppose that a drawer contains 8 marbles: 2 are red, 2 are blue, 2 are green, and 2 are yellow. The marbles are rolling around in a drawer, so that all possibilities are equally likely when they are drawn. Alice chooses 2 marbles without replacement, and then Bob also chooses 2 marbles without replacement. Let Y denote the number of red marbles that Alice gets, and let X denote the number of red marbles that Bob gets.

a. For i=1,2, let Yi=1 if the ith ball that Alice selects is red, and Yi=0 otherwise. So we have Y=Y1+Y2. Find E(Yi) for i=1,2, and then find E(Y).

E(Y)=

b. Temporarily view one of the red balls as having a #1 painted on it, and the other red ball as having a #2 painted on it. For i=1,2, let Zi=1 if the ith red ball is picked by Alice (at any time, i.e., on either of her roles), and Zi=0 otherwise. So we have Y=Z1+Z2. Find E(Zi) for i=1,2, and then find E(Y).

E(Y)=

Answer 1

Given that the drawer contains 8 marbles: 2 are red, 2 are blue, 2 are green, and 2 are yellow.

(a.) Now, we have for i=1,2,

Yi=1 if the ith ball that Alice selects is red,

and; Yi=0 otherwise

Also, the variable Y is defined as Y=Y1+Y2

Thus, For we have:

and ;

So,

Similarly; For we have:

and ;

So,

So,

(b.) Now, we have for i=1,2,

Zi=1 if the ith red ball is picked by Alice (at any time, i.e., on either of her roles),

and; Zi=0 otherwise

Thus, For we have:

and ;

So,

Similarly; For we have:

and ;

So,

So,