Consider a p-dimensional hypercube, What is the probability of the event...

Question

Considering a p-dimensional hypercube H_1^p = [-1,1]^p.

Assume we draw two p-dimensional points, a = (a_1, a_2, ..., a_p) and b = (b_1, b_2, ..., b_p) uniformly at random from this hypercube.

Denote H_r^p the p-dimensional hypercube [-r, r]^p for r<1.

What is the probability of the event a not in H_r^p?

r^p?

1 - r^p?

r^{p-1}?

What is the r value for which P(a not in H_r^p) - 90% as a function of p?

Answer 1