A Collective Decision
Three friends attempt to collectively choose one of $n$ options, labeled $1,\dots,n$, based upon their individual preferences. They choose option $i$ if for every alternative option $j$ at least two of the three friends prefer $i$ over $j$. If no such option $i$ exists they fail to reach an agreement.
Define $P(n)$ to be the probability the three friends successfully reach an agreement and choose one option, where each of the friends' individual order of preference is given by a (possibly different) random permutation of $1,\dots,n$.
You are given $P(3)=17/18$ and $P(10)\approx0.6760292265$.
Find $P(20\,000)$. Give your answer rounded to ten places after the decimal point.