# Probability of circle inside square

We pick a uniformly chosen random point on a unit square (a square with unit side length) and draw a circle of radius 2/22 around the point. Find the probability that the circle lies entirely inside the square.

For this question I said the answer was (20/22)^2 because you can pick an points inside of a (20/22) by (20/22) area, but this was incorrect. any suggestions on where I went wrong?

## 2 Answers

A unit square is 1 x 1. Any point within 2/22 of the edge of the square will overlap the square. So the side length of the square boundary within the 1 x 1 square would be $1 - 2/22 - 2/22$, which is $(18/22)^2$, because there is a boundary on all sides of the square. So the probability is $\frac{(18/22)^2}{1} = (18/22)^2$

Is someone randomly downvoting reasonable probability questions? This seems fine to me.

These fractions are weird, but that doesn't matter. I think what you need to do is draw the situation out, and you'll see your mistake: Your basic approach is right, but you must have missed that there is a forbidden zone of $1/11$ all the way around the square. In particular, that means that the width of the permitted square is not $10/11$, but $9/11$.

Square that and you get your answer.

ETA: I'm upvoting your question to compensate.