Unfortunately for the fart detector (but fortunately for us!), human farts average around 100ml [1], whereas an adult male breath averages about 500ml [2]. That could still be sorted out for your title, but we also have to consider that a fart is only about 7% methane [3]. This suggests the one mile sensitivity is going to be at around 18 simultaneous average farts, or 5 at the record-breaking end of the scale -- conditions which might indeed justify further surveying the region at altitude.

If we're willing to get closer than a mile (questionable in the above scenario), we can apply an inverse law for gas diffusion [4] (I was surprised it wasn't inverse square!) and find that the apparatus can can detect a single average fart at a distance of around 30 feet -- easily covering most elementary school classrooms and automating away the "whom smelt it dealt it" dilemma.

[1] https://www.chemistryworld.com/news/explainer-the-chemistry-... [2] https://en.wikipedia.org/wiki/Tidal_volume [3] https://en.wikipedia.org/wiki/Flatulence [4] https://physics.stackexchange.com/questions/290059/inverse-s...

Methane doesn't smell of flatulence; that smell consists of much larger molecules, which diffuse correspondingly much more slowly, and whose distribution is therefore more affected by advection.

Sounds like a real-life version of Futurama's Smelloscope?

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