Probability of a bad partition

[isheff]

Stated generally, given N validators, of which f are faulty:

1. If we assign each validator (uniformly and independently at random) one of K partitions, what are the odds that:
   1. no partition has more than x faulty validators?
   2. every partition has at least y non-faulty validators?
   3. no partition has a ratio of faulty validators / non-faulty validators greater than z?
   4. no partition is empty?
2. If we choose K (possibly overlapping) subsets of validators (uniformly at random), each of size S, what are the odds that:
   1. no subset has more than x faulty validators?
   2. every subset has at least y non-faulty validators?
   3. no subset has a ratio of faulty validators / non-faulty validators greater than z?
   4. every validator is in a subset?
   5. Is there anything convenient about the special case where K×S=N?
3. If we (uniformly at random) partition the validators into K (non-overlapping) partitions, each of size S (so K×S=N), what are the odds that:
   1. no partition has more than x faulty validators?
   2. every partition has at least y non-faulty validators?
   3. no partition has a ratio of faulty validators / non-faulty validators greater than z?
   4. what if instead of all being size S, each partition has size S or S+1, and K×S<N< K×(S+1)?

These seem like classic Balls & Bins problems, but I don't know the answers.

These probabilities could be important in calculating the odds that a randomly-selected "shard" of validators is correct.