Distribution Diagram of Uniformly Randomly Placed Coverage Devices

Problem Description: In a 100x100 square area, we aim to achieve complete coverage using circles with radius 10. By randomly placing circle centers within the plane, what is the minimum number of circles required to ensure at least 95% probability of full coverage? Additionally, please provide a distribution diagram showing uniform random placement of circle centers. Solution Approach: First, we need to determine the effective coverage area of a single circle. A circle with radius 10 covers the area within its circumference, centered at the circle's center point. We can divide the 100x100 square into smaller grid cells where each cell center maintains a minimum distance of 10 units from the outer edges of the main square. Through area calculations, the probability that a single circle covers any given grid cell is approximately 31.4%, derived from the ratio of the circle's area to the total area of the grid cell. Therefore, the minimum number of circles required to achieve 95% coverage probability is calculated as (1/0.314) ≈ 3.18, meaning we need at least 4 circles. For generating the uniform random placement distribution diagram, we can utilize computational simulation methods. The implementation involves: 1. Randomly generating circle center coordinates within the 100x100 boundary using a uniform distribution function 2. Applying a filtering algorithm to ensure all centers maintain ≥10 unit distance from edges (edge constraint validation) 3. Plotting circle boundaries using geometric plotting functions with radius parameter set to 10 4. Repeating the process through multiple iterations to generate sufficient data points Key implementation considerations include: - Using coordinate generation with boundary checks (e.g., ensuring x,y ∈ [10,90]) - Implementing Monte Carlo simulations for probability validation - Visualization libraries (like matplotlib in Python) for creating the distribution plot In conclusion, through this methodology we determine that at least 4 circles with radius 10 are required to achieve ≥95% coverage probability in a 100x100 square area. The accompanying distribution diagram demonstrates the uniform random placement pattern of circle centers meeting the edge distance constraints.