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.
For each duck in the pond we can consider its doppelganger located on the circumference of the pond instead.
When expressed in polar coordinates the doppelgangers will have the same angle, but different radii than the original ducks. But for the problem only the angle matters for determining if the ducks are all in the same sector - the radii are immaterial.
So looking at these doppelgangers will not change the probability we want to find.
If the doppelganger ducks are all on one common semicircle of the circumference (the condition we want to evaluate the probability for) then there is exactly one duck who is furthest counterclockwise.
Then for each doppelganger duck we ask "For this duck are all the other doppelganger ducks within 180 degrees clockwise?" For any one duck out of a set of D ducks that probability is 1/2^(D-1). We can multiply this by D to get a final probability of P(D)=D/2^(D-1).
Lets check this.
For D=2 then P(2)=1. This makes sense because if we have just two ducks they must have an angle between them of less than 180deg. All we need to do is rotate the 180deg semicircle sector until the two ducks are inside it.
For D=3 then P(3)=3/4 and D=4 then P(4)=1/2. Which agrees with earlier analyses by Steve and Charlie.
Solution
