Solving problem N3682 Bayesian probability theory

Solving problem N3682 Bayesian probability theory

3682. Two machines produce the same parts, which are fed to a common conveyor. The performance of the first machine is twice the performance of the second. The first machine produces an average of 60% of parts of excellent quality, and the second - 84%. Randomly taken from the conveyor part was of excellent quality. Find the probability that this part is produced by the first machine gun.

Additional information:

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