We have two companies $A$ & $B$ that produces an item, let's say for the sake of argument, that they both produces light bulbs.
Light bulbs of company $A$ have a probability of $p(A)$ of being defective, while $B$'s light bulbs are defective with probability $p(B)$.
I always buy light bulbs from these two companies with the same proportions, $x(A)$ & $x(B)$ with, of course, $x(A) + x(B) = 1$.
Now, if I pick $n$ light bulbs to test them and I got that $m$ (with $m \leq n$) were defective, what is the conditional probability that the $n$ lights were produced by company $A$?
Should I use Bayes theorem to answer? And if so, how?
For $i=0,1,\ldots,n$ let $a_i$ represent the event that $i$ of the $n$ selected light bulbs come from company $A$. Let $d_m$ represent the event that $m$ of the $n$ bulbs are defective. You're asking for $P(a_n|d_m)$, if I understand correctly. By Bayes' theorem, that is given by:
$P(a_n|d_m)=\frac{P(d_m|a_n)P(a_n)}{\displaystyle{\sum_{i=0}^n}\left(P(d_m|a_i)P(a_i)\right)}$
In order to use this, you need to be able to calculate each $P(a_i)$ and each $P(d_m|a_i)$. Do you know how to calculate those?