The title pretty much says my question. More specifically, I was asked the following question: Consider 256 randomly-chosen vertices of a regular 2019-gon. What is the probability that the 256-gon does not contain the center of the original 2019-gon in its interior? I would guess it might be easier to find the probability that the center is contained, perhaps using induction? But I am unsure of how to prove such a formula
Any suggestions or help would be greatly appreciated!
The centre of the regular $2019$-gon is not contained in the $256$-gon exactly if all $256$ vertices are in one semicircle. We can split this event into $256$ events, namely that all $256$ vertices are in the semicircle clockwise from a given vertex. These $256$ events are mutually disjoint and together exhaust the event that the vertices are in any semicircle. The probability that all $256$ vertices are in the semicircle clockwise of a given vertex is
$$ \frac{\binom{1009}{255}}{\binom{2018}{255}}\;. $$
Thus the probability for the centre not to be contained in the $256$-gon is
\begin{eqnarray} &&256\cdot\frac{\binom{1009}{255}}{\binom{2018}{255}} \\ &=&\frac{15865309465706232832981575559545291327395059198698618829139882039912932479498809443780224}{361558702088650776590352424439455000541031270078646733695481454328575304744288927295247413821425064475196440245870105015601861838259842130006107502554951092437472018066755} \\[10pt] &\approx& 4.4\cdot10^{-83}\;. \end{eqnarray}