Renato is the personnel manager in a galvanized products factory and must prepare the production report for the month. According to reports, operator 1 has 40% of the production, operator 2 25% and operator 3 35%. Production defects were 5% for operator 1, 3% for 2, and 6% for 3. In a random review, what will be the probability that a report with a defect is observed, if it is from an operator? 2?

To calculate the probability of a report with a defect being observed, we need to take into account the defect rates for each operator. According to the reports, operator 1 has a defect rate of 5%, operator 2 has a defect rate of 3%, and operator 3 has a defect rate of 6%. We can use these defect rates to calculate the probability of a report with a defect being observed, given that it is from an operator. Let D1, D2, and D3 be the events that a report from operator 1, 2, and 3 respectively has a defect. We want to calculate P(D1 ∪ D2 ∪ D3), the probability that a report with a defect is observed, given that it is from an operator. We can use the law of total probability to calculate this probability as follows: P(D1 ∪ D2 ∪ D3) = P(D1)P(operator 1) + P(D2)P(operator 2) + P(D3)P(operator 3) where P(operator 1) = 0.4, P(operator 2) = 0.25, and P(operator 3) = 0.35 are the probabilities of a report being from operator 1, 2, and 3 respectively Substituting the defect rates, we get: $$P(D_{1} ∪ D_{2} ∪ D_{3}) = (0.05 x 0.4) + (0.03 x 0.25) + (0.06 x 0.35) = 0.0485$$ Therefore, the probability that a report with a defect is observed, given that it is from an operator, is 0.0485 or 4.85%.