We are tasked with finding the expected value of the sum of the points scored by three darts thrown at a target with three concentric circles, with radii 1, 3, and 5. The regions are assigned the following point values:

• The inner circle (radius 1) is worth 10 points.
• The middle circle (radius 3) is worth 5 points.
• The outer circle (radius 5) is worth 2 points.

Step 1: Understanding the Problem

The target consists of three circular regions:

1. The innermost region, with a radius of 1, has an area $A_1 = \pi \times 1^2 = \pi$.
2. The middle region, between radii 1 and 3, has an area $A_2 = \pi \times 3^2 - \pi \times 1^2 = 9\pi - \pi = 8\pi$.
3. The outer region, between radii 3 and 5, has an area $A_3 = \pi \times 5^2 - \pi \times 3^2 = 25\pi - 9\pi = 16\pi$.

The total area of the target is the sum of these areas: $A_{\text{total}} = A_1 + A_2 + A_3 = \pi + 8\pi + 16\pi = 25\pi.$

Step 2: Probability of Each Dart Landing in a Region

The probability of a dart landing in a particular region is the ratio of the area of that region to the total area of the target.

• The probability of landing in the inner circle (radius 1), worth 10 points, is: $P_1 = \frac{A_1}{A_{\text{total}}} = \frac{\pi}{25\pi} = \frac{1}{25}.$

• The probability of landing in the middle region (radius 3), worth 5 points, is: $P_2 = \frac{A_2}{A_{\text{total}}} = \frac{8\pi}{25\pi} = \frac{8}{25}.$

• The probability of landing in the outer region (radius 5), worth 2 points, is: $P_3 = \frac{A_3}{A_{\text{total}}} = \frac{16\pi}{25\pi} = \frac{16}{25}.$

Step 3: Expected Value for One Dart

The expected value of the points scored by a single dart is calculated by multiplying the point values by their respective probabilities and summing the results: $E(\text{single dart}) = 10 \times \frac{1}{25} + 5 \times \frac{8}{25} + 2 \times \frac{16}{25}.$ Simplifying this expression: $E(\text{single dart}) = \frac{10}{25} + \frac{40}{25} + \frac{32}{25} = \frac{10 + 40 + 32}{25} = \frac{82}{25}.$

Step 4: Expected Value for Three Darts

Since the darts are thrown independently, the expected value of the sum of the points scored by three darts is three times the expected value for a single dart: $E(\text{three darts}) = 3 \times \frac{82}{25} = \frac{246}{25}.$

Thus, the expected value of the sum of the points scored by the three darts is $\frac{246}{25}$, which simplifies to: $\boxed{9.84}.$

Step 5: Questions for Further Exploration

1. What would happen if the dartboard had more regions with different point values?
2. How would the expected value change if the radius of the innermost region was increased to 2?
3. How can the variance and standard deviation of the sum of points scored by three darts be calculated?
4. If the radius of the middle region were decreased, how would the probabilities change?
5. How would the expected value change if the area of the outermost region were halved?

Tip:

When calculating probabilities based on areas, always remember that the probability is proportional to the area of the region relative to the total area.