Consider a random set $S$ constructed as follows. For any $n>2$, $\mathbb p(n\in S) = 1/\log n$, all independently.
Show that $S$ almost surely satisfies the twin prime property: there are infinitely many pairs $n,n+1\in S$.
Show that $S$ almost surely for all but finitely many $n$ there is a solution to $n=x+y$ with $x,y\in S$.
For the first part, I have shown that for $A_{2n}=\{2n,2n+1 \in S\}$, $A_{2n}$ are independent. $\mathbb P(A_{2n})=\frac{1}{log(2n)log(2n+1)}$. By comparing with $\frac{1}{nlog(n)}$, $\sum\frac{1}{nlogn}$ diverges then $\sum\frac{1}{log(2n)log(2n+1)}=\infty$. Thus we have $\mathbb P(A_{2n},i.o.)=1$ by Borel_Cantelli. Since $A_{2n}$ is a subsequence of events of $A_n=\{n,n+1\in S\}$, $A_n$ also happens infinitely often.
For the second part. I feel it's to show the probability of liminf is 1, but I wonder what to do next. Can someone give me a hint?