To determine the probability that the dart hits a shaded region on the target, we will follow these steps:

Step 1: Analyze the figure and its regions.

The target has four concentric circles with radii $1, 2, 3,$ and $4$. This creates three shaded regions (blue) and two unshaded regions (white), as shown:

1. Region 1: The inner circle with radius $1$ (unshaded).
2. Region 2: Between circles of radius $1$ and $2$ (shaded).
3. Region 3: Between circles of radius $2$ and $3$ (unshaded).
4. Region 4: Between circles of radius $3$ and $4$ (shaded).

Step 2: Calculate the areas of the regions.

The area of a circle is given by $A = \pi r^2$.

1. Area of the entire target (outermost circle of radius $4$):
   $A_{\text{total}} = \pi \cdot 4^2 = 16\pi.$

2. Areas of the individual circles:

   • Circle with radius $1$: $A_1 = \pi \cdot 1^2 = \pi$
   • Circle with radius $2$: $A_2 = \pi \cdot 2^2 = 4\pi$
   • Circle with radius $3$: $A_3 = \pi \cdot 3^2 = 9\pi$
   • Circle with radius $4$: $A_4 = \pi \cdot 4^2 = 16\pi$

3. Areas of the regions (difference between consecutive circle areas):

   • Region 1 (unshaded): $A_1 = \pi$
   • Region 2 (shaded): $A_2 - A_1 = 4\pi - \pi = 3\pi$
   • Region 3 (unshaded): $A_3 - A_2 = 9\pi - 4\pi = 5\pi$
   • Region 4 (shaded): $A_4 - A_3 = 16\pi - 9\pi = 7\pi$

Step 3: Total shaded area.

The shaded regions are Region 2 and Region 4. Add their areas:

$\text{Shaded Area} = 3\pi + 7\pi = 10\pi.$

Step 4: Probability of hitting a shaded region.

The probability is the ratio of the shaded area to the total area:

$P(\text{shaded}) = \frac{\text{Shaded Area}}{\text{Total Area}} = \frac{10\pi}{16\pi} = \frac{10}{16} = \frac{5}{8}.$

Final Answer:

The probability that the dart hits a shaded region is $\frac{5}{8}$.

Would you like me to explain any specific step in more detail or explore further? Here are some related questions you might consider:

1. What is the probability of hitting an unshaded region instead?
2. How would the answer change if more circles were added?
3. Can we generalize this for $n$ concentric circles?
4. How would the areas change if the radii increased linearly?
5. What is the probability of hitting each individual region?

Tip: Always check for symmetry when working with concentric figures. It simplifies calculations!