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.
Imagine the words: "On average" proceeding each sentence below...
From my original post:
The first two ducks define a 90 deg sector.
3/4 of the time the 1st 3 ducks are within a semicircle.
Case I. Of that 3/4 time, 1/3 of the time the 3rd duck also lands in the original sector (that central 90 deg). In that case, the 4th duck lands within a semicircle with 3/4 probability. This is true because it is the same situation as the 3 duck case, which has been solved.
Case II. Of that 3/4 time, 2/3 of the time the 3rd duck has landed in one of the 90 degree side sectors, and thus an average of 45 deg away. The 3 duck sector is now 135 deg in extent. So, the 4th duck might land near this sector. The 4th duck must land within it or 45 degrees on either side and be within a semicircle, and does so with 225/360 odds, or 5/8.
So, the total prob is Case I + Case II
(3/4) (1/3) (3/4) + (3/4) (2/3) (5/8) = (3/16) + (5/16) = 1/2
Pr = 1/2
(No ducks or computers were harmed in the making of this answer. :-)
(To be totally honest, I did check my original 90 deg claim by computer, after doubt was cast)
Edited on January 26, 2023, 11:05 pm
4 duck soln
