Consider $3$ points randomly chosen on a circle, find the probability that all of them lie in some semi circle
I tried to solve this problem but I am getting an incorrect answer. I have seen correct solutions to this problem but I am unable to see what is wrong with my reasoning which is worrying me because I may make similar mistakes in future.
Number the points $1,2,3$ (the order in which I choose). After choosing first point, the event that the points $2$ and $3$ lie on the same semicircle is equivalent to the event that points $2,3$ lie either on the semicircle measured counter clockwise from point $1$ or clockwise from point $1$. Both of them have probabilities $1/4$. Since the events the disjoint, the total probability is $1/2$.
What is wrong with the above solution?