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 Ducks in a pond (Posted on 2023-01-24)
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.

 No Solution Yet Submitted by Jer Rating: 5.0000 (2 votes)

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 Calculus Solution for 3 ducks | Comment 12 of 17 |
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 (2π-Θ)/2π probability of success.

So integrate over Θ from 0 to π, ((2π-Θ)/2π) dΘ/π
or  (1/(2*π^2) integral 0 to π of (2π-Θ) 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 2023-01-25 13:27:42

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