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Omni-MATH / Let n,k≥3 be integers, and let S be a circle. Let n blue points and k red points be chosen uniformly and independently at random on the circle S. Denote by F the intersection of th…

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problem

Let n,k≥3n, k \geq 3 be integers, and let SS be a circle. Let nn blue points and kk red points be chosen uniformly and independently at random on the circle SS. Denote by FF the intersection of the convex hull of the red points and the convex hull of the blue points. Let mm be the number of vertices of the convex polygon FF (in particular, m=0m=0 when FF is empty). Find the expected value of mm.
Plain-text mathematical notation (without MathML)
Let n,k≥3 be integers, and let S be a circle. Let n blue points and k red points be chosen uniformly and independently at random on the circle S. Denote by F the intersection of the convex hull of the red points and the convex hull of the blue points. Let m be the number of vertices of the convex polygon F (in particular, m=0 when F is empty). Find the expected value of m.
Original LaTeX notation
Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex polygon $F$ (in particular, $m=0$ when $F$ is empty). Find the expected value of $m$.

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Source and history

task_id
60603b20-96c9-5969-9480-8a7c648997d8
task_key
test--60603b20-96c9-5969-9480-8a7c648997d8
task_revision_id
1
config / split
/ test

Official source

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