Box i contains 4 red and 8 blue marbles while box ii contains 5 red and 3 blue marbles. an unfair coin is tossed –whose probability of turning up heads is 40%. if the coin comes up heads box i is chosen and a random marble is chosen, otherwise if it is tails the marble is chosen from box ii.(a)find the probability a red marble is chosen.(b)if a red marble is chosen, what is the probability it came from box i? (c)if a blue marble is chosen, what is the probability it came from box i?

Denote by the event that a marble is drawn from box 1 or 2, respectively, and by the event that a red marble is drawn. If a blue marble is drawn, we'll denote that by the complement of , or .

a. If a red marble is drawn, either it's drawn from box 1 or box 2. The events and are mutually exclusive, so we have



Now invoke the definition of conditional probability:



We're explicitly given the probability of drawing from either box. We also know the probabilities of drawing a red marble from box 1 or 2 are, respectively,



and



So



b. Given that we draw a red marble, we're looking for the probability that it was drawn from box 1, i.e. . We use Bayes' theorem:



which is really just a matter of using the definition of conditional probability to rewrite the numerator on the the right side, as



We know from part (a), so we compute



c. Now we look for , which can be computed similarly.