集合组合中不包含指定m个k长子组合的概率计算问询

Hey there, let's unpack this combinatorics probability problem step by step:

We start with a set of N distinct numbers, and we want to find the probability ( P(m,k,l,N) ). This probability tells us how likely it is that a randomly selected l-length combination (no repeated elements) from the original set does not include any of our pre-chosen m distinct k-length combinations (each of these m subsets is part of the original N-element set).

已明确的特殊场景计算结果 #

We already have concrete formulas for a few simple cases:

当仅排除1个k长子组合（m=1） #

The complement probability (meaning the random l-length subset does contain the single k-length subset we're checking for) is given by:
$$
1-P(m=1,k,l,N) = \binom{l}{k} \Big/ \binom{N}{k} = \binom{N-k}{l-k} \Big/ \binom{N}{l}
$$
Rearranging this gives us ( P(m=1,k,l,N) = 1 - \frac{\binom{l}{k}}{\binom{N}{k}} ).

当随机子集长度等于被排除子集长度（k=l） #

In this scenario, the complement probability (the random subset is exactly one of our m k-length subsets) is straightforward:
$$
1-P(m,k,l = k,N) = m \Big/ \binom{N}{k}
$$
So ( P(m,k,l=k,N) = 1 - \frac{m}{\binom{N}{k}} ).

当排除多个子集且k < l时（m>1且k < l） #

This is the more complex case: the value of ( P(m,k,l,N) ) depends entirely on the specific m k-length combinations we've chosen. For example, if some of these m subsets overlap a lot (share many elements), the calculation will differ from when all m subsets are completely disjoint. The original question cuts off here, but this key dependency is critical to keep in mind for further analysis.

备注：内容来源于stack exchange，提问作者siserman