Let $X \in \text{Random}(\mathbb{R})$ be a D6096: Continuous random real number.

Then
\begin{equation}
\mathbb{P}(X \in \mathbb{Q})
= 0
\end{equation}

Result R5362
on D6096: Continuous random real number

*Subresult of R5360: Probability that a continuous random real number takes value in a countable set is zero*

Probability that a continuous random real number takes value in the rational numbers is zero

Formulation 0

Let $X \in \text{Random}(\mathbb{R})$ be a D6096: Continuous random real number.

Then
\begin{equation}
\mathbb{P}(X \in \mathbb{Q})
= 0
\end{equation}

Proofs

Let $X \in \text{Random}(\mathbb{R})$ be a D6096: Continuous random real number.

Due to R104: Set of rational numbers is countable, this result is a particular case of R5360: Probability that a continuous random real number takes value in a countable set is zero. $\square$