Question

I have two bags. Bag 1 contains 3 green balls, while bag 2 contains 2 green balls.
I pick one of the bags at random, and throw 5 red balls in it. Then I shake the bag and choose
4 balls (without replacement) at random from the bag.
(a) If bag 1 is picked, what is the probability that there are exactly 2 red balls among
the 4 chosen balls?
(b) If bag 2 is picked, what is the probability that there are exactly 2 red balls among
the 4 chosen balls?
(c) If there are exactly 2 red balls among the 4 chosen balls, what is the probability
that I have chosen bag 1?
(d) Given that there is at least 1 green ball in the 4 chosen balls, what is the
probability that there are exactly 2 red balls

Answer 1

a) No. of red = 5 ; Green = 3

P( 2 red) = 5C2 *3C2/ 8C4 = 10*3 *24/ (7*8*6*5) = 0.42857

b) No. of red = 5 ; Green = 2

P( 2 red) = 5C2 *2C2/ 7C4 = 10*1 *24/ (7*4*6*5) = 0.2857

c) P ( Bag 1) = 0.42857 / [ 0.42857 + 0.2857] = 0.60

d) If bag 1 is chosen . P ( 2 red / at least 1 green) = P( 2 red and 2 green ) =

0.42857

If bag 1 is chosen . P ( 2 red / at least 1 green) = P( 2 red and 2 green ) =

0.2857

Since, probability of choosing both bags is half. Probability = 1/2( 0.2857+0.42857) = 0.357135