This comment was what made this article make sense to me.

Just to add to add to the mathematical intuition here (please correct me if I'm wrong): if you're thinking of it as a unit line/square/cube then total n-dim area is 1^n, and the portion in the 50% range is (1/2)^n, where n = number of dimensions. So that should simplify to 2^(-n).

Note this works out to 10th dimension as 0.09765% or 0.1 person per 1000.

ETA: As one of the comments below points out, you can also model it as a binomial distribution.

Probability of getting all heads, given p=0.5 is (n!)/((n-k)!k!) / 2^n. Since n=k since we're looking to get all heads at all times, this also simplifies to 2^(-n).