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# Bayes' Theorem

Let A1, A2,…,An a complete system of events with P(Ai) ≠ 0, i = 1, 2,…,n.

Let B another event about which we know P(B/Ai), i = 1, 2,…,n.

Then:

Example 1: Three machines, M1, M2 y M3, produce 45%, 30% y 25%, respectively, of the total parts produced in a factory. The percentages of defective production of these machines are 3%, 4% y 5%, respectively.
a) If we choose a part randomly, calculate the probability that it is defective.
b) Suppose now that we choose a part randomly and it is defective. Calculate the probability that it was produced by M2.

Example 2 (PAEG- June 2014) In a company, there are three robots A, B and C which solder products. 15% of the products are soldered by robot A, 20% by robot B and 65% by the C one. It is known that the probability of finding a defective product soldered by robot A is 0.02, 0.03 if it is soldered by robot B and 0.01 if it is soldered by robot C .
a) If we choose a product randomly, find out the probability that it is defective.
b) If we choose a product randomly, find out the probability that, if it is defective, it has been made by robot A.

Exercise: (PAEG- June 2013) A company produces two types of parts: A and B. 20% of parts are type A and 80% are type B. The probability that a type A part is defective is 0.02 and it is 0.1 for a type B part. If we choose a piece randomly and it is not defective, what is the probability that it is type A?

Solution: 0.214