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A point is selected at random inside an equilateral triangle with a side 1 mile (that means that, if X and Y are the coordinates of the chosen point, then the joint density of X and Y is constant over the triangle). What is the probability that the point will be within 1/4 mile of some corner of the triangle?

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One would think this would be the ratio of areas. The area within a distance $r$ from a corner is just: $$A= \frac{60^\circ (\pi r^2)}{360^\circ}=\frac{\pi r^2}{6}$$ So the area of three corners is $\pi r^2 / 3$, assuming $2r$ is smaller than the side length, so that the circles don't overlap. In your case this condition is satisfied, since $2r=1/2 < 1$. Now, if you know the area of a triangle, can you finish the calculation?

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