Solving problem N4000 Bayesian probability theory

Solving problem N4000 Bayesian probability theory

4000. In stock there are devices manufactured by three plants; among them, 50% are manufactured by plant number 1, 30% by plant number 2, 20% by plant number 3. It is known that a device manufactured by plant No. 1 may be defective with a probability of 0.05, and plant No. 2 with a probability of 0.10; plant number 3 - with a probability of 0.15.
a) Find the probability that the device taken at random is standard;
b) Find the probability that a standard device taken at random is manufactured by plant No. 2.

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