Same Answer, Two Roads

Running the screening problem by formula reproduces the count to the digit — because the formula is the count with the thousands canceled.

The three quantities, read off the story. Likelihood: the test flags carriers at 0.9. Prior: 4 percent carry — 0.04. Evidence overall: both routes to a flag, multiply-along and add-across — 0.9 × 0.04 plus 0.1 × 0.96, which is 0.036 plus 0.096: 0.132. Divide: 0.036 over 0.132 — 3 in 11, the same 27 percent the room count gave. Line the roads up side by side: 36-in-a-thousand over 132-in-a-thousand, thousands canceled, is exactly 0.036 over 0.132. Two notations, one count — and whichever you run, the base rate sits inside the arithmetic where it cannot be forgotten.