Solving problem N3887 Bayesian probability theory

Solving problem N3887 Bayesian probability theory

3887. The collector receives 25% of the details of plant number 1, 35% of plant number 2 and 40% of plant number 3. The probability that a part of factory number 1 of excellent quality is 0.8, for factory number 2, 0.7, for factory number 3, 0.9. For good luck, the part taken by the assembler turned out to be of excellent quality, what is the probability that it was developed by plant No. 1?

Additional information:

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