Surprisingly large. This isn't accurate even to within an order-of-magnitude: it's just to get some insight. The surface of the earth is (to the nearest power of 10) ~10^9 km^2. LEO satellites go up ~10^3 km, so that's ~10^12 km^3 of volume for them. A satellite traverses ~10^8 km/year (that's a lot!), so if it has ~10 m^2 cross section (10^-5 km^2), its path encompasses a volume of ~10^3 km^3/year. Under some simplifying assumptions, the probability of two arbitrary satellites colliding is something like ~(10^3 km^3/year / 10^12 km^3) = 10^-9/year. There's ~10^4 satellites, and the rate of collision goes as the number of pairs of satellites ~N^2, so the rate at which any two satellites collide is (10^4)^2 * 10^-9/year = 10^-1/year or once per decade.

I think the particularly "unintuitive" parts of this are, one, the incredible distances travelled when you sustain Mach 20 for years (10^8 km/year); and two, the quadratic scaling (10^4*10^4).