A small sub-triangle

HardProbability Random Variables~9m

A point $P$ is chosen uniformly at random inside triangle $\triangle ABC$ . It splits the triangle into three smaller triangles $\triangle ABP$ , $\triangle BCP$ , $\triangle CAP$ . What is the probability that at least one of these three has area at most $\tfrac14$ of the area of $\triangle ABC$ ?

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The three sub-triangle area-fractions are $P$ 's barycentric coordinates - uniform on the simplex, like three stick pieces.
Complement: all three exceed $1/4$ , probability $(1-3\cdot\tfrac14)^2=\tfrac1{16}$ .

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0.9375

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Asked at: Jane Street, Optiver

A small sub-triangle

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