A point near the cone's base

MediumProbability Calculus~5m

A point is chosen uniformly at random (by volume) inside a solid cone of base radius $1$ and height $2$ (base at the bottom, apex at the top). What is the probability the point lies within height $1$ of the base?

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Cross-section area at height $y$ is $\propto (1-y/2)^2$ ; integrate for volume below height $1$ over total.
$\mathbb{P}=1-(1-\tfrac12)^3 / 1 = 1 - \tfrac18$ .

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0.875 (± 0.001)

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A point near the cone's base

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