[Someone]

> p(12) sits at 50%, of course it will be the mean!

Technically, “the mean is 12” does not follow at all from “p(12) sits at 50%”.

“p(12) sits at 50%” implies the median (https://en.wikipedia.org/wiki/Median) is 12, and that can differ from the mean (https://en.wikipedia.org/wiki/Arithmetic_mean) of the distribution, and the difference can be quite large.

For example, if that list were to continue

  p(13) = 0.44277496045533604
  p(14) = 0.44277496045533604
  p(15) = 0.44277496045533604
  p(16) = 0.44277496045533604
  p(17) = 0.44277496045533604
  p(18) = 0.44277496045533604
  …
  p(1000) = 0.44277496045533604
  P(1001]) = 0

the about 44% that makes it to 13 also will make it to 1000, and the mean value will be a bit more than 442.77496045533604 (the contribution of that 44.2%)

If, on the other hand, it were to continue

  p(13) = 0.44277496045533604
  p(14) = 0.0

No result would be higher than 13, and the mean would be lower than 12.