Problem 807

Loops of Ropes

Given a circle $C$ and an integer $n > 1$, we perform the following operations.

In step $0$, we choose two uniformly random points $R_0$ and $B_0$ on $C$.

In step $i$ ($1 \leq i < n$), we first choose a uniformly random point $R_i$ on $C$ and connect the points $R_{i - 1}$ and $R_i$ with a red rope; then choose a uniformly random point $B_i$ on $C$ and connect the points $B_{i - 1}$ and $B_i$ with a blue rope.

In step $n$, we first connect the points $R_{n - 1}$ and $R_0$ with a red rope; then connect the points $B_{n - 1}$ and $B_0$ with a blue rope.

Each rope is straight between its two end points, and lies above all previous ropes.

After step $n$, we get a loop of red ropes, and a loop of blue ropes.

Sometimes the two loops can be separated, as in the left figure below; sometimes they are “linked”, hence cannot be separated, as in the middle and right figures below.

Let $P(n)$ be the probability that the two loops can be separated.

For example, $P(3) = \frac{11}{20}$ and $P(5) \approx 0.4304177690$.

Find $P(80)$, rounded to $10$ digits after decimal point.

绳环

考虑圆$C$和整数$n>1$，我们进行如下操作。

在第$0$步，我们在圆$C$上均匀随机地选择两个点$R_0$和$B_0$。

在第$i$步($1 \leq i < n$），我们先在圆$C$上均匀随机地选择一个点$R_i$并将点$R_{i-1}$和点$R_i$用红绳连接，再在圆$C$上均匀随机地选择一个点$B_i$，并将点$B_{i-1}$和点$B_i$用蓝绳连接。

在第$n$步，我们先将点$R_{n-1}$和点$R_0$用红绳连接，再将点$B_{n-1}$和点$B_0$用蓝绳连接。

每条绳在其连接的端点之间都是直线，且都位于所有现有绳的上方。

在完成第$n$步之后，我们得到了一个红绳环和一个蓝绳环。

有时这两个绳环可以拆开，如下图左所示。有时这两个绳环相互缠绕，无法拆开，如下图中和下图右所示。

记$P(n)$为两个绳环可以拆开的概率。

例如，$P(3) = \frac{11}{20}$，$P(5) \approx 0.4304177690$。

求$P(80)$，并将你的答案保留小数点后$10$位。