In a bolt factory, machines A, B and C manufacture respectively 20 %, 30 % and 50 % of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine C is :

Solution

Maths sir · 3 steps

Given that the bolt is defective, count only the defective bolts and ask what share of them came from C.

1. Picture 1000 bolts. Take 1000 bolts from the factory. Split them by each machine's share of production, then apply that machine's defect rate to count its defective bolts.

   Machine	Bolts made	Defect rate	Defective bolts
   A	200	3%	6
   B	300	4%	12
   C	500	2%	10
   Total	1000	not needed	28

2. Apply Bayes. The bolt is known to be defective, so it is one of the 28 defective bolts. Of these, 10 came from C.

   P(C | D) = (0.5 × 0.02)/(0.2 × 0.03 + 0.3 × 0.04 + 0.5 × 0.02) = 10/28

3. So. P(C | D) = 10/28 = 5/14, which is option (c).

The idea to keep. In a Bayes question, multiply each source's share by its chance of the outcome, add these up, and divide the one you want by the total. Counting per 1000 items keeps it easy.

Answer from the NTA final answer key. Solution by PrepWiser.