Three Uniform Dice Form a Triangle

This quant interview question is about understanding when three independent continuous random variables can represent the sides of a triangle, and how to translate that geometric requirement into a probabilistic event. It lives at the intersection of geometry and probability, which is very common in quant prep for continuous distributions and joint densities.

It trains your grasp of continuous random variables, joint distributions, conditioning on order statistics, and evaluating probabilities as measures of regions in higher-dimensional spaces. You need to be comfortable turning inequality constraints into regions in the unit cube and reasoning about their relative size. It also tests symmetry insights and abstraction beyond brute-force computation.

This matters for quant interviews because it mimics real quant work: formalizing an intuitive condition as a mathematical event, working in multiple dimensions, and quickly assessing probabilities for structured constraints in models and simulations.

What it tests

When determining whether randomly chosen values can serve as the sides of a triangle, the core structure is governed by the triangle inequality: for any three positive real numbers, a triangle is possible if and only if no single value exceeds the sum of the other two. In the context of independent, identically distributed random variables, the problem reduces to finding the measure (probability or volume) of the region in the sample space where all triangle inequalities are satisfied. The symmetry of the variables allows us to focus on the event where the largest value is greater than the sum of the other two, and then multiply by the number of variables, since each variable can play the role of the largest. The mutually exclusive nature of these failure events simplifies the calculation, as their probabilities can be summed directly. This approach generalizes to any problem where a set of random variables must satisfy a system of symmetric inequalities, and the solution is often found by partitioning the sample space into regions defined by these inequalities.