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State the equal-sized-group example
The frame contains eight branches, each with fifty distinct records. Select two branches uniformly without replacement. Within each selected branch, select ten records uniformly without replacement. This is a synthetic design with positive selection at both stages.
Follow a particular record
| Event | Probability |
|---|---|
| Its branch is selected | 2/8 = 0.25 |
| Record selected given its branch | 10/50 = 0.2 |
| Both selection stages | 0.25 × 0.2 = 0.05 |
Check the expected and actual final count
Each experiment contains two branches × ten records = twenty records. The expected total from summing 400 marginal probabilities is 400 × 0.05 = twenty. This agreement is a useful check on the inclusion calculation.
Do not infer independent selection from equal chances
Two records in the same branch share the first-stage event. Their inclusion indicators are not automatically independent. Unequal branch sizes or different within-group allocations would also change marginal probabilities, so carry the actual stages into any weights or variance calculation.