Solving problem N3890 Bayesian probability theory

Solving problem N3890 Bayesian probability theory

3890. The collector receives an average of 40% of parts of plant number 1, 25% of plant number 2, 35% of plant number 3. The probability that part of plant number 1 of excellent quality is 0.9, for plants number 2 and number 3 these probabilities are equal, respectively, 0.7 and 0.95. a) Find the probability that the item retrieved at random will be of excellent quality. b) The item turned out to be of excellent quality. What is the probability that it came from the factory number 3?

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Detailed solution. Decorated in Microsoft Word 2003 (Quest decided to use the formula editor)