Solving problem N3622 Bayesian probability theory

Solving problem N3622 Bayesian probability theory

3622. Three machines, the performance of which is referred to as 1: 2: 3, produce the same parts, while the first machine gives 80% of the highest grade parts, the second - 65%, the third - 50%.
a) Find the probability that a piece taken at random will be of the highest grade.
b) The detail taken at random turned out to be of the highest grade. Find the probability that the part is made on the second machine

Additional information:

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