Solving problem N4254 Bayesian probability theory

Solving problem N4254 Bayesian probability theory

4254. In two identical boxes are the "Designer" pencils. It is known that a fifth of the pencils in the first box and a third in the second box have a hardness of TM. A box is selected at random, one pencil is drawn from it at random. It turns out to be hardness tm. What is the likelihood that it is removed from the first box?

Additional information:

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