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. Find probability mass function for Y, i.e., for the number of red marbles that Alice gets, i.e., find pY(y) for y=0,1,2.

pY(0)=

pY(1)=

pY(2)=

b. Find E(Y).

Answer 1

(a)

Total Number of marbles = 8

Number of Red marbles = 2

Number of other marbles = 6

Number of marbles selected = 2

Number of ways of selecting 2 marbles from 8 marbles =

For Y = 0:

Number of ways of selecting 2 other marbles from 6 other marbles =

So,

P(Y=0) = 15/28 = 0.5357

For Y = 1:

Number of ways of selecting 1 Red marble from 2 Red marble =

Number of ways of selecting 1 other marble from 6 other marbles =

So,

P(Y =1) = 2 X 6/28 = 0.4286

For Y = 2:

Number of ways of selecting 2 Red marbles from 2 Red marbles =

So,

P(Y=2) = 1/28 = 0.0357

So,

the probability mass function of Y is given by:

Y                p(y)

0                   0.5357

1           0.4286         

2                    0.0357

(b)

y            p   yp

0          0.5357        0

1         0.4286         0.4286

2          0.0357        0.0714

-----------------------------------------

Total                  =   0.5

So,

E(Y) = 0.5