HMMT 2025 U16

The Cantor set is defined as the set of real numbers $x$ such that $0 \leq x < 1$ and the digit 1 does not appear in the base-3 expansion of $x$ . Two numbers are uniformly and independently selected at random from the Cantor set. Compute the expected value of their absolute difference.

(Formally, one can pick a number $x$ uniformly at random from the Cantor set by first picking a real number $y$ uniformly at random from the interval $[0,1)$ , writing it out in binary, reading its digits as if they were in base- 3, and setting $x$ to 2 times the result.)