Four points A1, A2, A3 and A4 are chosen at random on a circle. Find the probability that chords A1A2 and A3A4 intersect.

Out of 4 ways, there is only one way to set up the chords so that A_1 A_2 and A_3 A_4 intersect, so the probability is 1/4.

"Anchor"  A1  at  any point on the  circle

Moving clock-wise.....we have the  following   possible configurations

A1   A2  A3   A4

A1   A2  A4   A3

A1   A3  A2   A4

A1   A3  A4  A2

A1   A4  A2  A3

A1  A4  A3   A2

Note  that onlt  2  of the 6 possibilites  have  chord A1 A2    intersecting chord  A3  A4

So....the probability is     2 / 6    =   1 / 3