The binary expansion with coefficients equal to Bernoulli random variables 
Let $X_1, X_2, . . .$ be i.i.d. Bernoulli(p) random variables with $p \ne 1/2$. Let $Y =\sum_{i=1}^{\infty}X_i/2^i$. Show that there is a set A of Lebesgue measure $0$ so that $\mathbb P(Y\in A)=1$.

I am thinking about using the second Borel-Cantelli Lemma to approach it, by finding a sequence $A_n$ with $\mathbb P(A_n)$ with the sum $\sum_{i=1}^{\infty} \mathbb P(A_n)=\infty$ and $Y^{-1}(A)=\limsup(A_n)$. Unfortunately I couldn't use the condition $p\ne 1/2$.
I also saw a hint somewhere: Try to find a set like that using the Strong Law of Large Numbers. It might be helpful to consider the sequence of the coefficients of the binary expansion. But I can't figure out the connection between the LLN (about the convergence of the average of i.i.d of r.v.'s) and this problem. 
Thanks in advance!
 A: We have $\mathbb{P}(X_1 = 1) = p$ and $\mathbb{P}(X_1 = 0) = 1-p$, and will use the notation $Y(\omega) = (X_1(\omega), X_2(\omega), ...)_2 \in [0,1]$ (with subscript $2$) for binary expansion of a point in $[0,1]$.
Consider the set
$$
\Omega_p : = \left\{ \omega \in \Omega: \ \frac{X_1(\omega) + ...+X_n(\omega)}{n} \to p  \right\}.
$$
In view of the SLLN (strong law of large numbers) we have $\mathbb{P}(\Omega_p) = 1$.
Observe that the set $\mathbb{X}: = \{(X_1(\omega),X_2(\omega),...): \ \omega\in \Omega\}$,
as a "raw" set (no probability involved) is precisely the set of all
binary sequences, i.e. $\{0,1\}^\mathbb{N}$. This is because each coordinate in $\mathbb{X}$ takes both values $0$ and $1$ independently of the others (just think about why $Y(\Omega) = [0,1]$). Now if we impose a restriction on coordinates of $\mathbb{X}$, such as the one used in $\Omega_p$, and apply the definition of $Y$ which simply identifies a binary sequence with a point of $[0,1]$, we get precisely those points of $[0,1]$ in the image of $Y$ whose binary expansion satisfy the restriction imposed on $\mathbb{X}$.
With this in mind we get
$$
\tag{1} Y(\Omega_p) = \left\{x = (b_1,b_2,...)_2 \in [0,1]: \ \frac{b_1+...+b_n}{n} \to p \right\}.
$$
Each individual bit $b_i$ in $(1)$ can be identified with a Bernoulli $1/2$ random variable defined on $[0,1]$ with usual Borel $\sigma$-algebra, and Lebesgue measure as the probability measure. With such identification of the bits $b_i$ of binary expansion of $x\in [0,1]$, $\{b_i\}$ forms an independent sequence of random variables. Hence, SLLN implies that the Lebesgue measure of $Y(\Omega_p)$ is $0$ as $p\neq \frac 12$. Thus $Y(\Omega_p)$ serves as the desired set $A$ of the problem.
