An 8-sided die is rolled and depending on the result a ball is then chosen from

Question

An 8-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 a ball is drawn from box A, if the roll results in a 2, 3, or 4 a ball is drawn from box B, and if the roll results in a 5, 6, 7, or 8 a ball is drawn from box C. If box A contains 5 red, 8 white, and 2 black balls, box B contains 8 red and 7 black balls, and box C contains 5 white and 10 black balls:    (Need help with the following. Please explain).

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

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

c) What is the probability that you drew a ball from box A given that the ball is white?

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

Explanation / Answer

Since die is 8 sided so probability of selecting box A is

P(A) = 1/8

probability of selecting box B is

P(B) = 3/8

probability of selecting box C is

P(C) = 4/8

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

Let R shows the event that red ball is selected, W shows the event that white ball is selected and E shows the event that black ball is selected.

Total number of balls in box A : 5+8+2 =15

Total number of balls in box B : 8+7 =15

Total number of balls in box C : 5+10=15

So we have

P(R|A) = 5/15, P(W|A) = 8/15, P(E|A) = 2/15

P(R|B) = 8/15, P(E|B) = 7/15

P(W|C) = 5/15, P(E|C) = 10/15, P(R|C) = 0

(a)

By the law of total probability the probability that you draw a red ball is

P(R) = P(R|A)P(A) + P(R|B)P(B) + P(R|C)P(C) = (5/15)*(1/8) + (8/15)*(3/8) + 0 = 29 / 120

(b)

By the law of total probability the probability that you draw a black ball is

P(E) = P(E|A)P(A) + P(E|B)P(B) + P(E|C)P(C) = (2/15)*(1/8) + (7/15)*(3/8) + (10/15)*(4/8) = 63 / 120

So the the probability that you draw a black ball that come from box B is

P(B|E) = [P(E|B)P(B)] / P(E) = 21/63 = 1/3

By the compement rule,  the probability that you draw a black ball that does not come from box B is

1 - (1/3) = 2/3

(c)

By the law of total probability the probability that you draw a white ball is

P(W) = P(W|A)P(A) + P(W|B)P(B) + P(W|C)P(C) = (8/15)*(1/8) + (0)*(3/8) + (5/15)*(4/8) = 28 / 120

So  the probability that you drew a ball from box A given that the ball is white is

P(A|W) = [P(W|A)P(A)] / P(W) = 8/28 = 2/7

(d)

The probability that you draw a black ball given that you drew the ball from box C

P(E|C) = 10/15

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