Solving problem N3915 Bayesian probability theory

Solving problem N3915 Bayesian probability theory

3915. On three machines with identical and independent conditions, parts of the same name are manufactured. On the first machine make 10%, on the second - 30%, on the third - 60% of all parts. The probability of each part to be faultless is 0.7, if it is made on the first machine, 0.8 - if on the second machine, and 0.9 - if on the third machine. The detail taken at random turned out to be faultless. Find the probability that it is made on 2 machines.

Additional information:

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