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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 $\mathbb{R^2}$

Thanks in advance.

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Line is a set of zero area in $\mathbb R^2$. –  user21436 Feb 27 '12 at 17:08
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Is it correct also from a formal point of view? No, since there is no way to choose a point randomly uniformly in the whole plane. –  Did Feb 27 '12 at 17:21
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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 $$\nu(y = mx) = \int\limits_{\{y = mx \}} \frac{d\nu}{d\mu} \; d\mu = 0.$$ –  sos440 Feb 27 '12 at 17:29
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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
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As Andre points out, there are measures for which this is not true. If you wanted to formalize it, you'd probably want to use translation-invariant measures or some other restricted class. Then, even though you can't do a uniform probability distribution over the reals, you can say something like the intersection of the line and the reals has measure 0 for any measure in the class of measures you picked. –  James Kingsbery Feb 27 '12 at 18:29

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