Consider the following probability space: \((\Omega,\mathcal{F},\mathbb{P})\), where \(\Omega=\mathbb{R}^{\mathbb{N}}=\{\omega=(\omega_1,\omega_2 ,\ldots)\}\), let \(x\in \mathbb{R}\), define \(A_x=\{\omega:x\in \omega\}\subset \Omega\). Let \(F\subset \mathbb{R}\) be finite or countable, let \(A_F=\cap_{x\in F}A_x\). Define \[\mathcal{F}=\{B\subset \Omega| \exists F,\ A_F\subset B \text{ or }\exists F,\ B\subset A_F^c\}.\] One can check that \(\mathcal{F}\) is a sigma-algebra, moreover, an element of \(\mathcal{F}\) is either with \(A_F\subset B\) for some \(F\) (called the first type) or \(B\subset A_F^c\) for some \(F\). Define the probability be \(\mathbb{P}(B)=1\) is \(B\) is of the first type, otherwise \(\mathbb{P}(B)=0\). Can check that \(\mathbb{P}\) is a probability measure.

Now sampel a sequence \(\omega\) with \(\mathbb{P}\), guess the number one by one in the sampled sequence, for any fix real \(x\), \(\mathbb{P}(\omega\in A_x)=1\), that is, almost surely, with finitely many guess we guess correctly the target.