# Conditional Probability

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• Q1

In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?

Marks:3

Let events B1, B2, B3

B1 = the bolt is manufactured by machine A

B2 = the bolt is manufactured by machine B

B3 = the bolt is manufactured by machine C

And E = ‘the bolt is defective’.

P(B1) = 0.25 P(E|B1) = 0.05

P(B2) = 0.35 P(E|B2) = 0.04

P(B3) = 0.40 P(E|B3) = 0.02

By Baye's theorem,

P(B1|E) = 0.05

• Q2

Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.

Marks:3

Let E1 = choosing the bag I, E2 = choosing the bag II and A = drawing a red ball.

• Q3

If 2P(A) = P(B) = 5/13 and P(A|B) = 2/5, evaluate P(A B).

Marks:1

P(AB) = P(A|B).P(B) = 2/5 x 5/13 = 2/13.

Now, P(AB) = P(A) + P(B) - P(AB)

P(AB) = 5/26 + 5/13 - 2/13 = 11/26.

• Q4

If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find P(AB).

Marks:1