Question

Three sides of a regular polygon with 8 sides are chosen at random. Find the probability that, when these sides are extended, they form a triangle containing the polygon.

Answer 1

`|\Omega|={8\choose 3}=(8!)/(3!5!)=(6\cdot7\cdot8)/(6)=7\cdot8=56`

There are two possibilities of choosing three sides of the polygon, so when they are extended, they form a triangle containing the polygon (pictures in the attachment). Multiplying it by the number of sides, gives us 16 possibilities in total.


`|A|=16\\\\ P(A)=(|A|)/(|\Omega|)\\ P(A)=(16)/(56)=(2)/(7)`