2023 WSMO Team Round Problems/Problem 5
Problem
A monkey is throwing darts at the dart board pictured below. The dart is equally likely to land anywhere on the board. Point values for the three regions are labeled and the radii the three circles are 
respectively. If the expected value of points the monkey gets from 5 dart throws is 
for relatively prime positive integers 
and 
find 
![[asy] size(6cm); fill(circle((0,0), 6), red); fill(circle((0,0), 4), green); fill(circle((0,0), 2), yellow); label("3",(0,5)); label("5",(0,3)); label("7",(0,0)); [/asy]](http://latex.artofproblemsolving.com/5/d/0/5d0f4813246775d7c34ea18d51b83596d9fd8557.png)
