Solving problem N3970 Bayesian probability theory

Solving problem N3970 Bayesian probability theory

3970. In the sale of the same type products from three plants. Marriage at each plant is 5, 5, 7%. The first plant supplies 25% of products, the second - 30%, the third - 45% of products. A single item was removed at random.
a) What is the probability of removing a defective product?
b) Which plant most likely produced the extracted good-quality product?

Additional information:

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