# Probability to hit a real number

Given the function $$y=mx$$ defined in $\mathbb{R^2}$ with $m\in\mathbb{R}$ is it possible to give a proof that the probability for a dart to hit the line defined by the previus function is zero? The dart is supposed to hit randomly only one point in $\mathbb{R^2}$

Line is a set of zero area in $\mathbb R^2$. – user21436 Feb 27 '12 at 17:08
There is no uniform distribution on $\mathbb{R}^2$, thus it is nonsense to think of a fair dirt. Instead, suppose we throw a dirt in such a way that the dirt does not deviate too far. The corresponding probability measure $\nu$ would be absolutely continuous with respect to the Lebesgue measure $\mu$ on $\mathbb{R}^2$. Then $$\nu(y = mx) = \int\limits_{\{y = mx \}} \frac{d\nu}{d\mu} \; d\mu = 0.$$ – Sangchul Lee Feb 27 '12 at 17:29
In order to make the question precise, we have to put a probability measure on $\mathbb{R}^2$. But then the answer will depend on the measure. As an extreme example, let $m=0$ (this part doesn't really matter) so our line is the $x$-axis. Define a probability measure on $\mathbb{R}^2$ by setting the mass outside the $x$-axis to be $0$, and putting your favourite distribution on the $x$-axis. Then the probability your dart hits the $x$-axis is $1$. You can change that example to make the probability anything you like. – André Nicolas Feb 27 '12 at 17:33