A dartboard is made of three concentric circles of radii 3 cm, 6 cm and 9 cm. A dart lands at a random point of the largest circle. What is the probability that it lands in the outer ring, between the 6 cm circle and the 9 cm circle?

For a random point the probability is a ratio of areas. The whole board has area π$(9{)}^2=81$π and the circle of radius 6 has area 36π, so the outer ring has area 81π − 36π = 45π and the probability is 45/81 = 5/9. The other options are the neighbouring regions: 4/9 = 36/81 is the whole inner circle of radius 6, 1/3 = 27/81 is the middle ring and 1/9 = 9/81 the central circle. They check out, since 1/9 + 1/3 + 5/9 = 1. The outer ring is much bigger than its 3 cm width suggests, because area grows as the square of the radius.