A company has three departments. We will look at the wrong production of a certa

Question

A company has three departments. We will look at the wrong production of a certain type of item that now is manufactured in all of the departments. The three departments are A1, A2 and A3, and they account for respectively 10, 20 and 70% of the production of this item that we look at. Furthermore, it is stated that the error ratio of A1 is 1% error ratio in A2 is 0.5% and the error ratio of A3 is 0.1%.

a) Express the data by probabilities to find the probability that a random item of this kind is flawless.

b) Find the probability that an error produced item comes from A3.

Explanation / Answer

Here we have following information:

P(A1) = 0.10, P(A2) = 0.20, P(A3) = 0.70

Let E shows the error ratio. So

P(E|A1) = 0.01, P(E|A2) = 0.005, P(E|A3) = 0.001

(a)

By the total law of probability the probability that a random item of this kind has error is

P(E) = P(E|A1) P(A1) +  P(E|A2) P(A2) + P(E|A3) P(A3) = 0.01 * 0.10 + 0.005 * 0.20 + 0.001 * 0.70 = 0.0027

By the complement rule the probability that a random item of this kind is flawles is

P(flawless) = 1 - P(E) = 1 - 0.0027 = 0.9973

(b)

The probability that an error produced item comes from A3, by the conditional probability is

P(A3|E) = [P(E|A3) P(A3) ] / P(E) = [0.001 * 0.70] / 0.0027 = 0.2593

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