Dice Sum to 10 Probability

This classic quant prep probability question is about understanding how a fixed total can be produced by several independent, discrete random variables. Here, the variables are identically distributed and symmetric, which makes it ideal for practicing clean combinatorial reasoning. It sits at the intersection of basic combinatorics and elementary probability, so it is approachable yet rich enough to expose gaps in core quant interview skills.

It trains careful counting in a constrained combinatorial space, comfort with discrete distributions, and translating a verbal probability question into a precise mathematical model. You practice identifying all valid patterns, handling repetitions, and mapping unordered structures to ordered outcomes in a systematic way, without missing or double-counting possibilities.

This matters for quant interviews because many trading and risk problems reduce to similar discrete probability setups. Interviewers use it to see whether you can reason rigorously under simple rules, generalize the structure to more complex scenarios, and communicate a clean, error-free argument under time pressure.

What it tests

Problems involving the probability of a specific sum from rolling multiple dice are governed by the principle of counting ordered outcomes that satisfy a constraint, then dividing by the total number of possible ordered outcomes. The key is to enumerate all integer solutions to the equation $x_1 + x_2 + \ldots + x_n = S$ where each $x_i$ is constrained by the die's minimum and maximum values. The structure of the problem is combinatorial: the sum constraint partitions the sample space into subsets of outcomes, and the symmetry of the dice means each ordered outcome is equally likely. The challenge is to count the number of ordered arrangements for each unordered combination, accounting for repeated values, which is a classic application of multinomial coefficients. This approach generalizes to any number of dice and any target sum, provided you respect the dice's value constraints.