Question

Suppose that we have two identical boxes: box 1 and box 2. box 1 contains 5 red balls and 3 blue balls. box 2 contains 2 red balls and 4 blue balls. a box is selected at random and exactly one ball is drawn from the box. (a) what is the probability that the ball is blue? (b) given that a selected ball is blue, what is the probability that it came from the rst box?

Answer 1

(a) The probability of drawing a blue marble at random from a given box is the number of blue marbles divided by the total number of marbles. We assume that the probability of selecting one of two boxes at random is 1/2 for each box.

... P(blue) = P(blue | box1)·P(box1) + P(blue | box2)·P(box2) = (3/8)·(1/2) + (4/6)·(1/2)

... P(blue) = 25/48 . . . . probability the ball is blue

(b) P(box1 | blue) = P(blue & box1)/P(blue) = (P(blue | box1)·P(box1)/P(blue)

... = ((3/8)·(1/2))/(25/48)

... P(box1 | blue) = 9/25 . . . . probability a blue ball is from box 1