A circular garden bed of radius \(1 \mathrm{~m}\) is to be planted so that \(N\) seeds are uniformly distributed oser the circular area. Then we can talk about the number \(n\) of seeds in some particular area \(A\), or we can call \(n / N\) the probability for any one particular seed to be in the area \(A\). Find the probability \(F(r)\) that a seed (that is, some particular seed) is within \(r\) of the center.

A circular garden bed is essentially a garden patch shaped like a circle. In this problem, the garden bed has a radius of 1 meter. This implies that any point on the boundary of this garden is exactly 1 meter away from its center.

Understanding the structure of the garden bed is crucial. All calculations and probabilities will be based on this circular layout. A circle's geometry simplifies many assumptions and calculations due to its symmetrical properties.

The fundamental attributes of the circle, like its radius and area, will form the basis of our discussion.

When we say that seeds are uniformly distributed in the circular garden bed, it means every part of the garden has an equal chance of having seeds.

Imagine spreading seeds evenly across the garden. No matter where you look within the circle, the density of seeds remains consistent.

This uniform spread ensures that the probability calculations remain straightforward. Each section of the garden, regardless of its size, will have a proportionate share of seeds relative to its area.

The area is an essential property when dealing with circles. The total area of our circular garden bed can be calculated using the formula for the area of a circle: \[\text{Area} = \pi r^2\].

Given that the radius \((r)\) of the garden is 1 meter:

\[A_{\text{total}} = \pi (1)^2 = \pi \text{ square meters}\]

This area forms the foundation for understanding how seeds are spread across the garden.

Similarly, for a smaller circle within the garden with a radius \((r)\), the area can again be computed using the same formula:

\[A_r = \pi r^2\].

Probability helps us determine the chance that a particular event will occur. In this case, we are interested in finding the probability \(F(r)\) that a seed lies within a specific radius \((r)\) from the center of the garden.

The method involves comparing the area of the smaller circle (of radius \((r)\)) to the total area of the garden:

\[F(r) = \frac{A_r}{A_{\text{total}}} = \frac{\pi r^2}{\pi} = r^2\]

This result indicates a simple relationship: the probability \(F(r)\) that a seed falls within a radius \((r)\) from the center is directly proportional to \(r^2\). For example, if the radius \(r = 0.5\) meters, then:

\[F(0.5) = (0.5)^2 = 0.25 \(or 25%\)\]

This straightforward formula makes it easy to calculate the probability for any given radius within the circular garden.