>>12214123Let's turn it into another question.
Consider a finite group , i.e. a set of size and a particular, fixed, group structure .
Here's a list of the first such groups.
https://en.wikipedia.org/wiki/List_of_small_groupsThe number of groups per finite cardinality is
1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, ...
The fractions of those values over are
1.0, 0.5, 0.333, 0.5, 0.2, 0.333, 0.143, 0.625, 0.222, 0.2, 0.091, 0.417, 0.077, 0.143, 0.067, 0.875, 0.059, 0.278, 0.053, 0.25, 0.095, 0.091, 0.043, 0.625, 0.08, 0.077, 0.185, 0.143, 0.034, 0.133, 0.032, 1.594, 0.03, 0.059, 0.029, 0.389, 0.027, 0.053, 0.051, 0.35, 0.024, 0.143, 0.023, 0.091, 0.044, 0.043, 0.021, 1.083, 0.041, 0.1, 0.02, 0.096, 0.019, 0.278, 0.036, 0.232, 0.035, 0.034, 0.017, 0.217, 0.016, 0.032, 0.063, 4.172, 0.015, 0.061, 0.015, 0.074, 0.014, 0.057, 0.014, 0.694, 0.014, 0.027, 0.04, 0.053, 0.013, 0.077, 0.013, 0.65, 0.185, 0.024, 0.012, 0.179, 0.012, 0.023, 0.011, 0.136, 0.011, ...[/math]
But since we fixed the group operation on in advance, it seems a little more complicated to compute the chance of a random subset equip it with restricted to forming a group. Correct me if I'm wrong and it's simpler.
In the original subgroup-subset variant, we can ask the possibly complicated question given B, I think, since some groups will be more likely to be subgroups than others
For n<100, say, we have all the data to answer all questions (even brute forcing it), so all these have concrete answers.
