Let's say there are 1095 balls, 23 of them are black. Randomly assign each ball into 365 buckets, each has a maximum capacity of 3 balls. What is the probability of at least one bucket having at least 2 black balls?
Initially I thought the order does not matter, so we can imagine assigning the black balls to the buckets first. This will give probability of 0.5, just like the birthday problem. Then I think this is incorrect, as the bucket size is limited, the order of assignment actually matters. We cannot assign black balls first and the other balls later.
Could the probability simply be:
Probability of at least one unlimited bucket with at least 2 black balls - probability of at least one unlimited bucket with at least 4 black balls?
(By unlimited bucket I mean bucket with no capacity limitations.)