You have 100 balls (50 black balls and 50 white balls) and 2 buckets. How do you divide the balls into the two buckets so as to maximize the probability of selecting a black ball if 1 ball is chosen from 1 of the buckets at random?

Just to be perfectly clear, you are assuming that one of the two buckets is chosen at random and then one of the balls from that bucket is chosen at random.  You want to put 1 black ball in 1 of the buckets and all of the other 99 balls in the other bucket.   This gives you just slightly less than a 75% change of having a black ball chosen.  The math works as follows:  There’s a 50% chance of selecting the bucket containing 1 ball with a 100% chance of selecting a black ball from that bucket.  And a 50% chance of selecting the bucket containing 99 balls with a ~49.5% (49/99) chance of selecting a black ball from that bucket.  Total probability of selecting a black ball is (50% % 100%) + (50% * 49.5%) = 74.7%.