The location of a mole rat is Uniform inside a circular enclosure that has diameter $40$ feet. What is the probability that the mole rat is within $2$ feet from the edge of the enclosure?
So since it's uniform I figured that $f(x,y) = 1/{400\pi}$ for $x^2+y^2 \leq 400$
So wouldn't Probability (Rat is within two feet from the edge of the enclosure) be $= 1 -$ P(Rat is not within two feet from the edge of the enclosure) $= 1 - \cfrac{1}{400\pi}\pi19^2 = 39/400 $ where I used $\pi r^2 = \pi19^2$ since if the rat is not within two feet from the edge of the enclosure then the diameter of the related circle is $38$ and thus radius is $19$
But the answer is $.19$
Where did I go wrong?
The radius of the enclosure is $20$, so if the mole rat is within $2$ feet of the edge, that means it is outside of a circle of radius $20 - 2 = 18$ that is concentric with the enclosure. This means the desired probability is $$\frac{\pi (20)^2 - \pi (18)^2}{\pi (20)^2} = 1 - \left(\frac{18}{20}\right)^2 = 0.19.$$ Your error is a result of incorrectly interpreting the question and the subsequent radius of the circle from which the mole rat is excluded.