Solving problem N3879 Bayesian probability theory

Solving problem N3879 Bayesian probability theory

3879. Some product in a random order may be received for processing on one of three machines with probabilities equal to 0.2, 0.3, 0.5. When processing on the first machine, the probability of marriage is 0.02, on the second 0.03, and on the third 0.05. The product after processing turned out to be defective. What is the probability that the product was actually processed on the first machine?

Additional information:

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