Suppose you have $n$ fair coins and mark one of them for identification. Now you flip the coins with your eyes closed, while your friend looks on. After all $n$ flips have been made, your friend tells you the total number of heads across the $n$ coins is $k$.
In this situation, the probability the marked coin is heads equals $k/n$. Moreover, if the coins are unfair, but are unfair in the same way, the probability the marked coin is heads remains $k/n$.
This makes me wonder. Can something general be said about the probability the marked coin is heads, when coins are unfair in possibly different degrees? It is, of course, simple to see that various unfairness has no effect when probability(heads) = 0 for the marked coin. But in other instances, it surely does. But how?
Your question moves into a realm where philosophical attitudes about probabilities (the most prominent being the frequentist interpretation and the Bayesian interpretation) become relevant.
As long as the coins are known to be identical (even if unfairly so), the probability $k/n$ follows by mere symmetry, and it doesn't matter which school of thought you follow.
If the coins might be different, then it becomes a matter of interpretation whether you can say anything about the probabilities. The Bayesian interpretation views probabilities as subjective assignments of levels of beliefs to statements, so under this interpretation you could still specify probabilities – say, you have no reason to believe that the marked coin is more or less unfair than any other coin, so you assign probabilities to the possible degrees of unfairness such that the probability for the marked coin to be heads still comes out as $k/n$. By contrast, under the frequentist interpretation it no longer makes sense to specify the probability of the marked coin being heads, since this probability is viewed as an objective feature of the situation (namely the limiting value of the frequency of this result if you do the same experiment many times), not as an expression of your beliefs about the situation, and as long has you have no objective knowledge about the different degrees of unfairness of the coins, this probability is simply unknown.