I am trying to understand a problem solution but I can't get an answer for my confusion. Also, this question is asked before in another post . Below is the problem and solution description:
Each of 2 cabinets identical in appearance has 2 drawers. Cabinet A contains a silver coin in each drawer, and cabinet B contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the probability that there is a silver coin in the other drawer?
The reference solution:
A = the cabinet A is chosen
B = the cabinet B is chosen
S = a silver coin is chosen
$P(A|S) = \frac{P(S|A)P(A)}{(P(S|A)P(A)+P(S|B)P(B)}$
$P(A|S) = \frac{1\times0.5}{1\times0.5 + 0.5\times0.5}$
$P(A|S) = \frac{2}{3}$
My confusion:
As the question stated a cabinet is randomly selected, why the verified solution calculate the cabinet A is chosen $P(A|S)$ instead of the cabinet B is chosen $P(B|S)$? I am stucked at this point. Is that any reasoning behind?
The question asked: what is the probability that there is a silver coin in the other drawer?
The other drawer will have a silver coin if and only if cabinet A was selected.
That is why the solution computed $p(A|S).$