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.
Pr=3/4Consider the position on the pond in polar coords. Project the position of the duck out to the circumference. The first two ducks define a sector, and the 3rd duck must be inside the sector or within 90 degrees of it on either side in order to be on one side of a diameter.
The average sector of the 1st two ducks is 90 degrees. (That's because after the first duck lands, the 2nd duck's angle is uniformly distributed between 0 and 180 deg on either side of the first.)
So, on average, the 3rd duck must land in a sector 90+90+90 degrees out of 360, to be on one side of a diagonal, and so has a 3/4 chance.
Edited on January 24, 2023, 9:53 am
3 duck soln
