Solving problem N3885 Bayesian probability theory

Solving problem N3885 Bayesian probability theory

3885. The collector receives an average of 50% of parts of plant number 1, 35% of plant number 2, the remaining parts from plant number 3 The probability that a part of plant number 1 of excellent quality is 0.8 for parts of the second and third plants, these probabilities, respectively equal to 0.76 and 0.92. Find the probability that:
1) at random taken detail of excellent quality;
2) and this part is made by factory №2.

Additional information:

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