This is my first question on stackexchange - so apologies in advance if I haven't been able to follow the best practices while asking this question.
I am trying to understand the solution to question "Probability of n points lying within a semi-circle". The question has been asked and answered here - Probability that n points on a circle are in one semicircle.
I am struggling with a statement in this answer https://math.stackexchange.com/a/325168/927405. I understand that the n points are disjoint in a way that only one of the n points can have all the points within a semi-cicrle in particular angular direction. My question is - why are we not multiplying (1/n) for the (conditional) probability of selecting the right point?
I am drawing a parallel that while calculating probability of different outcomes in 2 coin tosses - we say that if I get heads on first toss, prob. of getting tails in 2nd toss is 1/2. Likewise we can get a TH. HT and TH are both disjoint but while calculating final answer - we don't say that if I choose heads first, then the prob of tails in second toss is 1/2 and we can also have a TH and since both are disjoint - let's add them up 1/2 + 1/2 and prob. of different outcomes would be 1 (which we know is incorrect answer because we need to multiply the probability of getting first heads / tails too!)
Similarly, why are we not multiplying 1/n for the probability of selecting the right starting point in this "N point in a semicircle question"? (which will lead to the answer $\frac1{2^{n-1}}$)
Thank you so much!