Question

There are two boxes with red and blue balls in them. Box I has 1 red and 4 blue balls; Box II has 3 red and 2 blue balls. There is a fair coin with Box I written on one side and Box II written on the other. You toss the coin and then draw 2 balls without replacement out of the box that comes up on the face of the coin.

a. Let Y be the number of red balls that you have drawn. Write down the possible values and the distribution function of Y

b. Calculate the expected value and the variance of Y .

c. Given that you have drawn 1 red ball, find the probability that the coin said Box I.

d. Are the events A={coin said Box I}, B={Y=1} independent? - justify with formula.

Answer 1

Box I : 1 red and 4 blue balls ( total 6)  

Box II : 3 red and 2 blue balls (total 5)

a) Y can take values 0 , 1 ,2

each Box is selected with probability 1/2

If Box I is selected , then Y takes value 0 with probability

If Box 2 is selected , then Y takes value 0 with probability

If Box I is selected , then Y takes value 1 with probability

If Box 2 is selected , then Y takes value 1  with probability

If Box I is selected , then Y takes value 2 with probability 0

If Box 2 is selected , then Y takes value 2 with probability

Thus we get the following table

Y	Box I	Box II	Total
0	0.3	0.05	0.35
1	0.2	0.3	0.5
2	0	0.15	0.15

b) E(Y) = 0*0.35+1*0.5+2*0.15 = 0.8

V(Y) = 0*0*0.35+1*1*0.5+2*2*0.15- 0.8*0.8 =0.46

c) P( Box I I Y=1) = P( Y=1 and Box 1) / P( Y=1) = 0.2/ 0.5 = 0.4

d) A and B are independent if

P(A I B) = P( A)

P(A I B) = 0.4

P(A) = 0.3+0.2=0.5

Thus A and B are not independent