Solving problem N4122 Bayesian probability theory

Solving problem N4122 Bayesian probability theory

4122. On four machines with identical and independent conditions, parts of the same name are made. On the first machine make 15%, on the second - 24%, on the third 38%, on the fourth - 23% of all parts. On the first machine, the probability of every detail to be defect-free is 0.68; for the remaining machines, these probabilities are respectively equal: 0.74; 0.89; 0.92. Find the probability that:
1) at random, the taken part will be defect-free;
2) and this part is made on the fourth machine.

Additional information:

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