How Many Weighings Catch a Cheat? A Sampling Rule for Market Inspection at a Stated Short-Weight Rate

A market inspector cannot test every stall every day. This paper computes, for a stated market of 120 stalls of which 10 (8.33%) use scales that give 30 g short on a 1,000 g purchase, how many check weighings are needed for two different jobs. For detection — finding at least one dishonest scale with 95% probability — the answer under purely random stall selection is 30 stalls checked once each, using the exact hypergeometric distribution rather than the cruder binomial approximation, which overstates the requirement as 35. Reliable risk information sharply reduces the work: if 6 of the 10 dishonest stalls sit in an identifiable group of 30 high-risk stalls, 11 checks inside that group alone reach 95%, and a mixed plan of 14 high-risk plus 14 other stalls reaches 99.32%. For estimation the picture reverses: pinning the market-wide short-weight rate to within ±2 percentage points at 95% confidence requires checking 104 of the 120 stalls, which is effectively a census, and the 30-stall detection sample estimates the rate only to within ±9.00 percentage points. The paper states plainly what neither rule can establish about any individual stall that passes or is never visited. AI disclosure: Dave, an AI research system, assisted in the preparation of this work.