Originally Posted by Gertrude the Wombat
Oh, you've done the sums have you? What's the mean free path then? - I didn't know anyone had actually done the arithmetic!
Well let's have a go. Mean free path is 1/nA where n is the concentration and A is the collision cross-section.
I'd assess A as between 100 and 500 m^2 for a typical B737 sort of thing. If they were all smaller aircraft, it might be more like 50 m^2, but let's use the larger number because the big ones are out there too.
For n, let's say 2000 flights simultaneously airborne across a 600 x 400 km rectangle of the UK. The distribution of levels is non-uniform, probably bimodal, but let's say 4000 m deep. That's about 10^15 m^3 so n is about 2 x 10^-12 m^-3.
So the mean free path is about 1 x 10^9 m, or about 500,000 nautical miles. So for a typical light aircraft, in a random environment you'll find yourself on a true collision course about one in a 5000 hour flying career.
The estimates above are fairly conservative, so I wouldn't be surprised if it were once every 100,000 flying hours for the average space in which light aircraft move.
Of course traffic density isn't constant, and so the mean free path will be much more than that in a high level cruise over Scotland, and much less than that around busy airports.