We define a bow-tie quadrilateral as a quadrilateral where two sides cross each other. An example of a bow-tie quadrilateral is shown below.

Seven distinct points are chosen on a circle. We draw all C(7, 2) = 21 chords that connect two of these points. Four of these 21 chords are selected at random. What is the probability that these four chosen chords form a bow-tie quadrilateral?

What would the conditions of the selected points be? For example, one point must be adjacent to this point, so find the probability of that and multiply it with the other conditions. But what would the conditions be? I do not know how to solve this.

Thanks in advance!

CentsLord Jun 21, 2020