Uniform Sum Cuts Below One

This question considers the sum of many independent uniform variables on a bounded interval and asks for the probability that this sum stays below a small fixed threshold. Conceptually, the candidate must interpret a probabilistic event as a region in a high-dimensional space: each coordinate is one draw of the uniform variable, and the sum constraint carves out a region inside the unit hypercube. The focus is on recognizing that this type of event has a clean geometric meaning and that the probability depends only on the shape and volume of that region.

Solving it leans on multivariable integration, geometric probability, and comfort with high-dimensional simplexes. Strong candidates typically recast the problem as computing a volume in n dimensions and either derive it recursively or recall the general volume formula. Interviewers watch for an ability to move fluidly between algebraic integration and geometric intuition, to justify why the region is a simplex, and to generalize the reasoning beyond the specific threshold or distribution.

What it tests

When working with the sum of independent uniform random variables constrained to be less than a fixed value, the probability corresponds to the volume of a simplex within the unit hypercube. This is because the set of points $(x_1, ..., x_n)$ with $0 \leq x_i \leq 1$ and $x_1 + \cdots + x_n < c$ forms a simplex, whose volume is $c^n/n!$ when $c \leq 1$. The factorial in the denominator arises from the way the simplex's volume scales with dimension: each new variable adds another integration over a shrinking interval, recursively multiplying by $1/(n+1)$. This geometric structure is universal for problems where the sum of bounded independent variables is compared to a threshold, not just for uniform distributions but for any scenario where the constraint defines a simplex in the feasible region. The principle holds because the uniform measure distributes probability evenly, so the probability is proportional to the region's volume.