math math math?

Question

Urn A contains six white and seven black balls. Urn B contains three white and eight black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. What is the probability that the transferred ball was black given that the second ball drawn was white?
If necessary, round your answer to two decimal places.

Answer 1

Let

P(ww) – probability that white ball is drawn from urn A
and then white ball drawn from urn B,

P(wb) – probability that white ball is drawn from urn A
and then black ball drawn from B,

P(bw) – probability that black ball is drawn from urn A
and then white ball drawn from urn B,

P(bb) – probability that black ball is drawn from urn A
and then black ball drawn from urn B.

P(ww) = (6/13)*(1/3) = 2/13 = 15.38 %
P(wb) = (6/13)*(2/3) = 4/13 = 30.77 %
P(bw) = (7/13)*(1/4) = 7/52 = 13.46 %
P(bb) = (7/13)*(3/4) = 21/52 = 40.38 %

The probability that the transferred ball was black
given that the second ball drawn was white is

P(bw) / [P(ww) + P(bw)] = (7/52) / (2/13 + 7/52)
= 7/15 = 46.67 %

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