In Example 5.12, a systematic sampling scheme was proposed for detecting hazardous wastes in landfills. How far apart should sampling points be placed? Suppose that if there is leakage, it will spread to a circular region with radius R. Let D be the distance between adjacent sampling points in the same row or column.

a. Calculate the probability with which a contaminant will be detected. Hint: Consider three cases, with R < D, D ≤ R ≤ √ 2D, and R > √ 2D.

b. Propose a sampling design that gives a higher probability that a contaminant will be detected than the square grid, but does not increase the number of sampling points.