1. **Problem statement:** We have a dartboard with three concentric circles of radii 2 cm (red), 6 cm (yellow), and 8 cm (green). We want to find: (a) The probability that a dart hits the red area. (b) The probability that two darts, thrown one at a time, both hit the yellow area. 2. **Formula and rules:** The probability of hitting a certain colored area is the ratio of the area of that color to the total area of the dartboard. Area of a circle is given by $$A = \pi r^2$$. 3. **Calculate areas:** - Area of red circle: $$A_{red} = \pi \times 2^2 = 4\pi$$ - Area of yellow ring: area between radius 6 and radius 2: $$A_{yellow} = \pi \times 6^2 - \pi \times 2^2 = 36\pi - 4\pi = 32\pi$$ - Area of green ring: area between radius 8 and radius 6: $$A_{green} = \pi \times 8^2 - \pi \times 6^2 = 64\pi - 36\pi = 28\pi$$ - Total area of dartboard (largest circle): $$A_{total} = \pi \times 8^2 = 64\pi$$ 4. **(a) Probability of hitting red area:** $$P(red) = \frac{A_{red}}{A_{total}} = \frac{4\pi}{64\pi} = \frac{4}{64} = \frac{1}{16}$$ 5. **(b) Probability both darts hit yellow area:** Probability of hitting yellow area once: $$P(yellow) = \frac{A_{yellow}}{A_{total}} = \frac{32\pi}{64\pi} = \frac{32}{64} = \frac{1}{2}$$ Since darts are thrown one at a time independently, probability both hit yellow is: $$P(both\ yellow) = P(yellow) \times P(yellow) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$$ **Final answers:** (a) $$\frac{1}{16}$$ (b) $$\frac{1}{4}$$