Solving problem N4009 Bayesian probability theory

Solving problem N4009 Bayesian probability theory

4009. The shop receives the same type of products from three plants, with the i-th (i = 1,2,3) plant supplying mi percent of the products (m1 = 40%, m2 = 30%, m3 = 30%). Among the products of the i-th plant, ni percent are first-rate (80%, 80%, 90%). Bought one product. It turned out to be first-rate. Determine the probability that the purchased product is released by the 3rd plant.

Additional information:

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