3 factories produce and ship reclining chairs.
Factory % of chairs produced by this factory % of production that is faulty
A 20% 10%
B 30% 20%
C 50% 10%
You order a chair and when you receive it, you see that it is faulty. Determine:
P(chair came from factory A)
P(chair came from factory B) B C
P(chair came from factory C)
To determine the probabilities, we can use Bayes' Theorem. Bayes' Theorem states that:
P(A|B) = P(B|A) * P(A) / P(B)
In this case, we want to find the probability that a chair came from a specific factory given that it is faulty. So, let's calculate each probability step by step.
P(chair came from factory A | the chair is faulty):
P(chair is faulty | chair came from factory A) = 10% (Given)
P(chair came from factory A) = 20% (Given)
P(the chair is faulty) = P(chair is faulty | chair came from factory A) * P(chair came from factory A)
+ P(chair is faulty | chair came from factory B) * P(chair came from factory B)
+ P(chair is faulty | chair came from factory C) * P(chair came from factory C)
= (10% * 20%) + (20% * 30%) + (10% * 50%)
= 2% + 6% + 5%
= 13%
Using Bayes' Theorem:
P(chair came from factory A | the chair is faulty) = P(chair is faulty | chair came from factory A) * P(chair came from factory A) / P(the chair is faulty)
= (10% * 20%) / 13%
= 0.2 / 0.13
= 1.54 or 15.4%
Similarly, we can calculate the probabilities for factory B and factory C:
P(chair came from factory B | the chair is faulty) = P(chair is faulty | chair came from factory B) * P(chair came from factory B) / P(the chair is faulty)
= (20% * 30%) / 13%
= 0.6 / 0.13
= 4.62 or 46.2%
P(chair came from factory C | the chair is faulty) = P(chair is faulty | chair came from factory C) * P(chair came from factory C) / P(the chair is faulty)
= (10% * 50%) / 13%
= 0.5 / 0.13
= 3.85 or 38.5%
Therefore, the probabilities are:
P(chair came from factory A) = 15.4%
P(chair came from factory B) = 46.2%
P(chair came from factory C) = 38.5%