Getting samples or subsets from a distribution

Question

I probably missed some basic basic math class, but I'm struggling a bit with finding a solution for this marble problem.

Suppose you want to divide $Y$ marbles into $Z$ categories. Every category has an unequal but known number of marbles, so if you would separate all marbles one by one, you'd get that exact distribution.

In reality, you take a random fist full of $X$ size from the population with size $Y$, and then do a simple majority vote (mode, if equal = random pick between winners) and throw it in the winners bin.

I need to find a way to make sure that there are more marbles in their correct place, but I can't change the sample size or the randomness of the marbles in the fist. I can only play with the majority vote... How can I tune this so that it performs optimally?

And what if the mistake is worse in category A than it is in category B, how would you add weights to the problem?

Answer 1

I am an idiot.

Of course the maximum nr of correct marbles will always be achieved by choosing the bin that corresponds with the maximum number of correct marbles per hand, if you cannot preselect the marbles. So if these constraints stay the same, the maximum has already been reached. Oh well... at least I found out on my own :P.