The problem:
A shop sells 2600 pieces of a certain product, supplied by factories A, B, and C, in the following quantities: 3000 pieces from factory A, 2600 pieces from factory B, and 4200 pieces from factory C. Knowing that the factories supply products with the same quality, find the probability that exactly 820 pieces sold come from factory A, 500 from factory B, and 1280 from factory C.
Solving attempt:
Let S be the event that a piece is sold, and A,B,C, the events that the piece comes from the factory A, B and C, respectively.
The total quantity in the 'warehouse' is 3000+2600+4200=9800 pieces.
So P(S) = 2600/9800 = 26/98.
Also, P(A)= 26/98, P(B) =30/98, and P(C)=42/98.
As all the three factories supply products with the same quality, every single piece has the same chance of being sold, so P(S|A)=P(S|B)=P(S|C) = 1/3.
Using Bayes' formula: P(A|B)= (P(A) * P(B|A))/P(B), I tried to calculate the probability for each factory, adapting the formula as : P(A|S)=(P(A)*P(S|A))/P(S), getting the following results:
P(A|S)= 10/26, P(B|S)=12/26 and P(C|S)=14/26.
Which is obviously wrong, as their sum is 36/26, clearly higher than 1.