# coin toss space, uncountable set

Let $X$ be the infinite coin toss space, ie $X=\{0,1\}^\mathbb{N}$ and $A_q:=\{\omega \in X: \lim_{n \to \infty} 1/n \sum_{k=1}^{n} \omega_k = q \}$. I read that $X-\bigcup_{q \in [0,1]} E_q$ is uncountable but i have absolutely no clue how to prove that. Does anyone have any ideas?