Let $b \ge 2$ be an integer. According to a conjecture of \'Emile Borel, the $b$-adic expansion of any irrational algebraic number behaves in some respect `like a random sequence'. We give a contribution to the following related problem: let $\alpha$ and $\alpha'$ be irrational algebraic numbers, prove that their $b$-adic expansions either have the same tail, or behave in some respect `like independent random sequences'.