A 12sided die is rolled and depending on the result a ball i

A 12-sided die is rolled and depending on the result a ball is then chosen from one of three boxes. If the roll results in a 1, 2, 3, 4, 5, 6, or 7 a ball is drawn from box A, if the roll results in an 8 or 9 a ball is drawn from box B, and if the roll results in an 10, 11, or 12 a ball is drawn from box C. If box A contains 4 red, 5 white, and 6 black balls, box B contains 11 red and 4 black balls, and box C contains 8 white and 7 black balls:

a) What is the probability that you draw a black ball?

b) What is the probability that you draw a white ball that does not come from box C?

c) What is the probability that you draw a red ball that comes from box B?

d) What is the probability that you draw a black ball given that you drew the ball from box C?

e) What is the probability that you drew a ball from box A given that the ball is red?

Solution

P(A) = Probability of rolling a 4 = 1/8.

P(B) = Probability picking a black ball = 3/8 * 2/11(Probability of rolling box A and then pickign black from it) + 1/8 * 6/11 (Probability of rolling box B and then picking black from it) + 4/8* 3/11(Probability of rolling box C and then pickign black from it) = 3/11.

p(Rolling 4 and picking black) = P(A and B) = 1/8 * 6/11 = 3/44.

P(Rolling box B given picking black) = P(A|B) = P(A and B) / P(B) = (3/44)/(3/11) = 1/4