AbstractFor a d-dimensional random vector X, let $$p_{n, X}(\theta )$$ p n , X ( θ ) be the probability that the convex hull of n independent copies of X contains a given point $$\theta $$ θ . We provide several sharp inequalities regarding $$p_{n, X}(\theta )$$ p n , X ( θ ) and $$N_X(\theta )$$ N X ( θ ) denoting the smallest n for which $$p_{n, X}(\theta )\ge 1/2$$ p n , X ( θ ) ≥ 1 / 2 . As a main result, we derive the totally general inequality $$1/2 \le \alpha _X(\theta )N_X(\theta )\le 3d + 1$$ 1 / 2 ≤ α X ( θ ) N X ( θ ) ≤ 3 d + 1 , where $$\alpha _X(\theta )$$ α X ( θ ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point $$\theta $$ θ . We also show several applications of our general results: one is a moment-based bound on $$N_X(\mathbb {E}\!\left[ X\right] )$$ N X ( E X ) , which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of X, where our combinatorial approach allows us to generalize existing results in random matrix community significantly.