On randomly chosen arrangements of q + 1 lines with different slopes in Fq2

In this paper, we prove that the expected number of points in Fq2 of multiplicity m, for 0≤m≤q+1, with respect to a randomly chosen arrangement of q+1 lines with different slopes, is (1/(m!e))q2+O(q), as q→∞. We further state that the distance between the number of such points in a randomly chosen arrangement and (1/(m!e))q2 is lower than qln⁡q with probability close to 1 for large q.