$d$ ducks in a pool, the probability of "all $d$ ducks are within a half circle".

Analysis

Label the ducks as $x_{1},x_{2},…,x_{d}$. The probability of "all $d$ ducks are within a sector of angle $2πr$" is the integral of the probability density of "$d$ ducks forms a sector of exactly $2πr$", i.e.:

$P=∫_{0}f(r)dr$ The probability of "$d$ ducks forms a sector of exactly $2πr$" can be calculated as follows: let $g_{k}(r∣i,j)$ denote the probability density function $f(r)$ when there are $k$ ducks and the duck $i$ and $j$ are on the two borders of the sector. $g_{2}(r∣i,j)g_{k}(r∣i,j) ={2,0, r⩽0.5otherwise =r⋅g_{k−1}(r∣i,j) $ Therefore, $g_{k}(r∣i,j)=2⋅r_{k−2},r⩽0.5$ Since the probability of two ducks on the same radius is 0, every pair of $i$ and $j$ is disjoint from others. $f(r) =i,j⋃ g_{d}(r∣i,j)=i=j∑ g_{d}(r∣i,j)=2d⋅(d−1) ⋅g_{d}(r∣0,1)=d⋅(d−1)⋅r_{d−2} $$P=∫_{0}f(r)dr=∫_{0}d⋅(d−1)⋅r_{d−2}dr=d⋅r_{d−1}$