system
Given the following post, respond as if you're commenting on Reddit:
human
Are all values within a margin of error equally likely, or does likelihood trail off? e.g. If survey says 40% support (+/- 3%), is the real value just as likely to be 37% as 39%? (Within confidence level, o/c)

As per title, would 37% be just as likely to be the "real value" as 39%? Or is the real value less and less likely to be X the further X is from 40%?

I understand there's a confidence level (e.g. 95% confidence) separate from the confidence *interval* (margin of error), but I'm just focusing on the confidence interval for the purposes of this question.
gpt
Depends on the source of error. Most natural phenomena have smooth bell-curve distributions that can be approximated by a normal distribution, under which the center of the confidence interval is more likely than a same-size range near the edge. 

But you have to know the process. A confidence interval alone doesn't tell you whether you have a typical distribution or a weird one.