Solving problem N4087 Bayesian probability theory

Solving problem N4087 Bayesian probability theory

4087. The assembly received 7 boxes of the same type of parts: three boxes were made by factory No. 1, in which parts of the highest quality make up 78%, and four boxes are made by factory No. 2, in which parts of the highest quality are 92%. The collector took one of the boxes at random and took the part out of it. Find the probability that:
1) selected item of the highest quality;
2) the part is selected from the box received from plant No. 2.

Additional information:

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