A factory production line is manufacturing bolts using three...

'A factory production line is manufacturing bolts using three machines, A, B and C_ Of the total output, machine A is responsible for 25%, machine B for 35% and machine C for the rest. It is known from previous experience with the machines that 5% of the output from machine A is defective, 4% from machine B and 2% from machine € A bolt is chosen at random from the production line and found to be defective. What is the probability that it came from (i) machine A (ii) machine B (iii) machine C?'

Transcript

00:01 In this problem, it is said that a factory production line is manufacturing bolts using three machines a, b, and c.

00:08 We are given information about how much of the total output is produced by which machine.

00:12 And we are also given information about which machine produces how many defective bolts.

00:18 So a bolt is chosen at random.

00:20 It is found to be defective.

00:22 We need to determine the probabilities that it came from machine a, machine b, and machine c.

00:26 So let us consider a to be the event that the bolt came from machine a, let b be the event that the bolt came from machine b, let c be the event that the bolt came from machine c, and let d be the event that the bolt is defective.

00:43 Now, let us determine the probability of d.

00:48 Now, we will determine the probability of d using the law of total probability.

00:54 Now, to use the law of total probability, we need a set of mutually exclusive and exhaustive events, and we will use this set, a, b, and c.

01:01 They are mutually exclusive because a bolt cannot be produced by two machines at once, so only one event can happen at one time.

01:10 Two events cannot happen simultaneously.

01:13 And they are exhaustive because there are only these three machines, so it's not possible for any other machine to produce the bolt, so either one of these events must happen.

01:24 So they're mutually exclusive and exhaustive.

01:27 And so we can use the law of total probability to say that p of d is equal to p of a times p of d given a plus p of b times p of d given b plus p of c times p of d given c.

01:48 So p of a is the probability that the bolt came from machine a.

01:51 Now, machine a is responsible for 25 % of the work, so this probability is 25%, which is 0 .25.

02:00 Pfd given a is the probability that the bolt is defective, given that it is coming from machine a.

02:06 5 % of the output from machine a is defective, so this probability is 5 % or 0 .05.

02:12 Now, machine b produces 35 % of the bolt, so pfb is 0 .35, and a total of 4 % of the output from machine b is defective.

02:21 So p of b given b is 0 .04.

02:26 And the machine c produces all of the other bolts.

02:31 So 25 % is done by machine a, 35 is done by machine b.

02:36 So 25 % plus 35%...

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