Solving problem N3884 Bayesian probability theory

Solving problem N3884 Bayesian probability theory

3884. An indicator is used to signal that the operation mode of an automatic line deviates from the normal one. It belongs with probabilities of 0.2, 0.3, and 0.5 to one of three types, for which the probabilities of actuation when the normal operation is disturbed are 0.9, 0.7, and 0.6, respectively. Find the probability that an indicator taken at random will work if the normal line is broken.
From the indicator received a signal. Find the probability that the indicator belongs to the first type.

Additional information:

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