Most curves in $\mathbb{R}^2$ miss $\mathbb{Q}^2$

Question

I would like a formal statement for the following fact.

Let $\gamma$ be an analytic curve from the open interval $(-1, 1)$ to $\mathbb{R}^2$. With probability $1$ (in some sense), the image of the curve does not intersect $\mathbb{Q}^2$.

How can I do it?