I have read the solutions to the popular problem: with n points uniformly distributed on the circle, what is the probability that that they lie on the same semicircle? The general approach is to define $E_i$ the event that all points lie on the same semicircle and the semi-circle is started at point i. Then because these events are disjoint, you have that $$P(semicircle) = \sum P(E_i) = n/2^{n-1}$$
I am approaching this question from another angle but getting it wrong $$P(semicircle) = \sum P(semicircle | semicircle \ started \ at \ P_i) P(semicircle \ started \ at \ P_i)$$
This yields $$n * 1/2^{n-1} 1/n$$ which is clearly wrong. What do I need to do here to resolve this issue?
What do you mean by the first factor ?
$$P(semicircle | semicircle \ started \ at \ P_i)$$
I cannot make sense of it. If the conditioning event is "the points form a semicircle starting at $P_i$"... then the probability of the other event is $1$.