3 ducks have landed in a circular pond. What is the probability there is a diameter such that the semicircle on one side contains all of them?
Repeat for 4 ducks.
Note: consider the ducks to be randomly chosen points in a circle.
wlog, assume the first duck to be at 0 degrees.
wlog, assume the second duck to be between 0 and 180 degrees.
The second duck is in a tiny slice between Θ and Θ+dΘ.
The likelihood the duck has landed there is dΘ/π
The third duck can land anywhere from 0 to 2π. All 3 ducks being on the same side of a diameter means that duck 3 cannot land between π and π+Θ. There is a (2πΘ)/2π probability of success.
So integrate over Θ from 0 to π, ((2πΘ)/2π) dΘ/π
or (1/(2*π^2) integral 0 to π of (2πΘ) dΘ
1/(2*π^2) (2π*Θ  (Θ^2)/2) eval at π minus eval at zero
1/(2*π^2) * (2*π^2  π^2/2)
(1/(2*π^2))*(3*π^2/2) = 3/4

Posted by Larry
on 20230125 13:27:42 