Please help with all parts of the problem A 12sided die is r

A 12-sided die is rolled and depending on the result a hall is then (potentially) chosen from one of three boxes. If the roll results in a 1. 2. 3. or 4 a hall is drawn from box A. if the roll results in a 5.6. 7. 8. or 9 a hall is drawn from box B. if the roll results in a 10 or M a bail is drawn from box C. and if the roll results in a 12 ne hall is drawn from any of the boxes. If box A contains 4 red. 6 white. and 5 black halls, box B contains 3 red and 12 black halls, and box C contains 7 white and 8 black halls: What is the probability that you draw a black bail? What is the probability that you draw a red bail that comes from box C? What is the probability that you drew a bail from box B given that the bail is red? What is the probability that you draw a white hall that does not come from box A? c) What is the probability that you draw a bail of any color given that you did not draw the bail from box B?

Solution

a)

P(B) = P(A) P(B|A) + P(B) P(B|B) + P(C) P(B|C) + P(D) P(B|D)

= (4/12)*(5/15) + (5/12)*(12/15) + (2/12)*(8/15)

= 0.533333333 [ANSWER]

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b)

P(R n C) = P(C) P(R|C) = (2/12)*(0) = 0 [ANSWER, there are no red balls in C!]

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c)

P(B|R) = P(B) P(R|B) / P(R)

As

P(R) = P(A) P(R|B) + P(B) P(R|B) + P(C) P(R|B)

P(R) = (4/12)*(4/15) + (5/12)*(3/15) + (2/12)*(0)

P(R) = 0.172222222 [ANSWER]

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d)

P(W n A\') = P(B) P(W|B) + P(C) P(W|B)

= (5/12)*(0) + (2/12)*(7/15)

= 0.077777778 [ANSWER]

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