(Solution) - Suppose that S is a subset of m units from -(2025 Original AI-Free Solution)

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Suppose that S is a subset of m units from U, and suppose S has mh units from stratum h, for h = 1, . . . , H. Let Zi = 1 if unit i is selected to be in the final sample and 0 otherwise; similarly, let Fi = 1 if unit i is selected to be in the stratified sample and 0 otherwise, and let Di = 1 if unit i is selected to be in the subsample chosen from the stratified sample and 0 otherwise. Then the probability that S is chosen to be the sample is