[BetterGeiger]

If you want to actually measure radon concentration quantitatively what you are describing is not the way. Typically devices simply measure alpha count rates coming from air, radon diffuses very effectively and no filtering or extra flow is necessary. Your technique would show "hey look there is a little bit of something radioactive here" but would be very difficult to turn into a concentration without a lot of extra complicated steps.

[DoctorOetker]

The relevance of the fan with filter is that it is analogous to your lungs which can also trap radioactive particles, like radon, causing say lung cancer.

[BetterGeiger]

Practically every radon measuring device on the market does it the way I described because of the reasons I described.

[java-man]

Radon is a gas.