Solving problem N3680 Bayesian probability theory

Solving problem N3680 Bayesian probability theory

3680. Three machines make parts that go to a common conveyor. The performance of the first, second and third machines are referred to as 13:14:10. The probability that the part made by the first machine is of excellent quality is 0.9; for the second and third automata, these probabilities are equal to 0.8 and 0.7, respectively. Find the probability that: a) at random, the part taken from the conveyor is not of excellent quality; b) the part was made by the second machine, if at random the part taken from the conveyor is not of excellent quality.

Additional information:

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