Probability of selecting $k$ points such that the resulting polynomial $H$ satisfies $H(0)=0$

Question

I read that $k$ points on some finite field $\mathbb{F}$ completely determine a $k-1$ polynomial $H$ in $\mathbb{F}[x]$, and the polynomial can be computed using Lagrange interpolation.

Suppose that I uniformly sample these $k$ points and use Lagrange interpolation to get $H$. What is the probability that $H(0)=0$ ? ... My guess is that this probability is very small....