Box A contains four white marbles and five black marbles Box

Box A contains four white marbles and five black marbles. Box B contains seven white marbles and three black marbles. An experiment consists of first selecting a marble at random from Box A. The marble is transferred to Box Band then a second marble is drawn from Box B. What is the probability that the first marble was white given that the second marble was white? (Round your answer to three decimal places.)

Solution

Let

1W = first was white
2W = second was white

Thus,

P(1W|2W) = P(1W) P(2W|1W) / P(2W)

As

P(2W) = P(1W) P(2W|1W) + P(1W\') P(2W|1W\') = (4/9)(8/11) + (1-4/9)(7/11) = 67/99

Thus,

P(1W|2W) = P(1W) P(2W|1W) / P(2W) = (4/9)(8/11) / (67/99)

= 32/67 = 0.47761194 = 0.478 [ANSWER]