Solving problem N3729 Bayesian probability theory

Solving problem N3729 Bayesian probability theory

3729. Parts from three machines arrived at the warehouse. The first machine made 40% of the parts of their total number, the second - 35% and the third 25%, and the first machine produced 90% of the first-class parts, the second 80% and the third 70%. What is the probability that the detail taken at random will be of the first grade?

Additional information:

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