How should N white and N black balls be distributed in N boxes (each box containing at least 1 ball) so that after randomly choosing an urn and then randomly choosing a ball from it, the probability of drawing a white (or black) ball is maximum?
I have an intuition and a not-so-justified reasoning that: if only a white ball is distributed in each of the N boxes, and then, all the black balls are put in one of the box, so that the probability of drawing a white ball from those boxes containing white-only is maximum (that is 1), and this [somehow] compensates for the last box, in which the probability of drawing a white ball is low (but not minimum). Can anyone please help with proper reasoning? Can it be proved that this method maximizes the required probability?
Related: N white and black balls and N boxes Probability (I would have commented on the answer there, but my reputation does not permit me to do so.)