Solving problem N3886 Bayesian probability theory

Solving problem N3886 Bayesian probability theory

3886. The collector receives an average of 50% of the details of the plant number 1, 30% of the plant number 2, 20% of the plant number 3. The probability that a part of the first factory of excellent quality is 0.7; for the second and third plants, these probabilities are 0.8 and 0.75, respectively. Find the probability that:
1) at random the part taken by the assembler turned out to be of excellent quality;
2) the selected part of excellent quality made by the factory №3.

Additional information:

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