To solve this problem, we need to calculate the probability that a randomly selected point falls in one of the shaded regions of the concentric circles. Here's the step-by-step process:

Step 1: Areas of Each Circle

The area of a circle is given by the formula: $A = \pi r^2$ where $r$ is the radius of the circle.

For the six concentric circles, we calculate their areas using their respective radii:

• Area of circle with radius $1$: $A_1 = \pi(1^2) = \pi$
• Area of circle with radius $2$: $A_2 = \pi(2^2) = 4\pi$
• Area of circle with radius $3$: $A_3 = \pi(3^2) = 9\pi$
• Area of circle with radius $4$: $A_4 = \pi(4^2) = 16\pi$
• Area of circle with radius $5$: $A_5 = \pi(5^2) = 25\pi$
• Area of circle with radius $6$: $A_6 = \pi(6^2) = 36\pi$

Step 2: Total Area of the Largest Circle

The largest circle has a radius of $6$, so its total area is: $\text{Total Area} = A_6 = 36\pi$

Step 3: Identify Shaded Regions

From the description, every other circle is shaded, starting with the second one. This means the shaded regions correspond to the areas between:

• The circles with radii $2$ and $1$,
• The circles with radii $4$ and $3$,
• The circles with radii $6$ and $5$.

These regions are annular (ring-like), and their areas are calculated by subtracting the area of the smaller circle from the area of the larger circle in each pair.

• Shaded area between radii $2$ and $1$: $\text{Area} = A_2 - A_1 = 4\pi - \pi = 3\pi$
• Shaded area between radii $4$ and $3$: $\text{Area} = A_4 - A_3 = 16\pi - 9\pi = 7\pi$
• Shaded area between radii $6$ and $5$: $\text{Area} = A_6 - A_5 = 36\pi - 25\pi = 11\pi$

Step 4: Total Shaded Area

The total shaded area is the sum of the three shaded regions: $\text{Total Shaded Area} = 3\pi + 7\pi + 11\pi = 21\pi$

Step 5: Probability

The probability that a randomly selected point falls in one of the shaded regions is the ratio of the total shaded area to the total area of the largest circle. This is: $\text{Probability} = \frac{\text{Total Shaded Area}}{\text{Total Area}} = \frac{21\pi}{36\pi} = \frac{21}{36} = \frac{7}{12}$

Thus, the probability is $\boxed{\frac{7}{12}}$.

Would you like more details or have any further questions?

Here are some related questions:

1. What is the probability that a point falls in a non-shaded region?
2. How would the probability change if an additional circle with radius 7 cm is added, and it's shaded?
3. Can you generalize the formula for the probability of a shaded region for any number of concentric circles?
4. What is the area of the non-shaded regions?
5. How would the problem change if every other circle starting from the first is shaded?

Tip: To calculate areas between circles, subtract the area of the smaller circle from the area of the larger circle, treating them as annular rings.