HMMT-Nov 2009 guts, Problem 11

Lily and Sarah are playing a game. They each choose a real number at random between -1 and 1 . They then add the squares of their numbers together. If the result is greater than or equal to 1, Lily wins, and if the result is less than 1, Sarah wins. What is the probability that Sarah wins?

Solution: If we let $x$ denote Lily's choice of number and $y$ denote Sarah's, then all possible outcomes are represented by the square with vertices $(-1,-1),(-1,1),(1,-1)$, and $(1,1)$. Sarah wins if $x^{2}+y^{2} \leq 1$, which is the area inside the unit circle. Since this has an area of $\pi$ and the entire square has an area of 4 , the probability that Sarah wins is $\frac{\pi}{4}$. $\fbox{\frac{\pi}{4}}$.

Answer: \frac{\pi}{4}