[Math] Bucket Puzzle Probability Problem

Question

You have 2 buckets. One full of white marbles and the other full of black marbles
(equal amounts). How do you allocate the marbles into two buckets in a way that maximizes your probability of picking 2 white ones when you pick 1 marble from each bucket?

I would assume its 50/50? But how to justify it?

Answer 1

Hint: You could guarantee one bucket gave a white marble, while making the other bucket about (slightly less than) $50\%$ likely to give a white marble.