Drawers I and II contain black and red pencils as follows Dr

Drawers I and II contain black and red pencils as follows: Drawer I: b1 black pencils and r1 red pencils, Drawer II: b2 black pencils and r2 red pencils. A drawer is chosen at random and then a pencil is also chosen at random from that drawer. (i) What is the probability that the pencil is black? (ii) If it is announced that the pencil is black, what is the probability it was chosen from drawer I? (iii) Give numerical values in parts (i) and (ii) for b1 = 36, r1 = 12, b2 = 60, r2 = 24

Solution

i)

Let B = the pencil drawn is black
R = the pencil draw is red
D1 = drawer 1 is selected
D2 = drawer 2 is seletced

Thus, by Bayes\' Rule,

P(B) = P(D1) P(B|D1) + P(D2) P(B|D2)

P(B) = (1/2) (b1/(b1+r1)) + (1/2)(b2/(b2+r2))

P(B) = (1/2)[(b1/(b1+r1)) + (b2/(b2+r2))]

********************

ii)

Now,

P(D1|B) = P(D1 n B) / P(B)

As

P(D1 n B) = P(D1) P(B|D1) = (1/2)(b1/(b1+r1))

Then

P(D1|B) = [(b1/(b1+r1))]/[(b1/(b1+r1)) + (b2/(b2+r2))] [answer]

***************************

iii)

Plugging in these given,

For part i:

P(B) = 41/56 [answer]

For part ii:

P(D1|B) = 21/41 [answer]