What are the odds of all 32 NFL teams finishing 8-8?

(quick uneducated guess)

1 in 17 ^32

To make the problem anywhere near tractable, you’ll need to begin with a simplifying assumption that every game is a 50-50 proposition. By the basic binomial theorem, the probability of any one team going 8-8 is then 19.64%. A lower bound for the probability of all 32 teams going 8-8 would be 0.1964^32, or 2.4 * 10^-23.

The actual probability would be somewhat higher than that, because the fact of one team going 8-8 increases the probability that others will. (In the extreme case, if 31 teams go 8-8, the 32nd team must.) The exact interdependencies depend on how the schedule is constructed. I’m afraid the math is beyond me.

Another approach is to say that there are 256 games in a season, and thus 2^256 possible sets of outcomes, and to ask how many of those sets result in the all 8-8 result. But I’m afraid that math is beyond me too.