Ducks | ylxdzsw's blog

Problem

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

Analysis

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

P = \int_{0}^{\frac{1}{2}} f (r) d r

The probability of "

d

ducks forms a sector of exactly

2 π r

" 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 ⩽ 0.5 otherwise = r \cdot g_{k - 1} (r ∣ i, j)

Therefore,

g_{k} (r ∣ i, j) = 2 \cdot 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 \neq = j \sum g_{d} (r ∣ i, j) = \frac{d \cdot ( d - 1 )}{2} \cdot g_{d} (r ∣ 0, 1) = d \cdot (d - 1) \cdot r^{d - 2}

P = \int_{0}^{\frac{1}{2}} f (r) d r = \int_{0}^{\frac{1}{2}} d \cdot (d - 1) \cdot r^{d - 2} d r = d \cdot r^{d - 1}

Calculator

Number of ducks (

\geq 2

)Portion of circle (

\in [0, 0.5]

)