Doubt on conditional probability

Question

It had been found from past experience that of the articles produced by a factory, 20% comes from Machine 1; 30% comes from Machine 2 and 50% from Machine 3. The percentages of satisfactory articles among those produced are 95 for machine 1, 85 for machine 2 and 90 for machine 3.

i) An article is chosen randomly from the lot. What is the probability that is satisfactory?

ii) Assume the article is satisfactory. What is the probability that it was produced by machine 1?

Comments

A) 0.895

B) 0.212....?

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For the first, we can do it this way: it is the sum of all probabilities where the item is satisfactory. So it can be written as:

0.2*0.95 + 0.3*0.85 + 0.5*0.85

For the second, just apply Baye's rule. The numerator would be the probability of satisfactory article produced by M1 times probability of M1 being chosen. The denominator is the same as we have calculated above - the probability of an item being satisfactory.