Risky Dice Showdown

This interview question sets up a choice between two high-stakes dice games that are constructed to have the same expected monetary value but very different risk profiles. The candidate is asked, under explicit risk aversion, which gamble is preferable and why. The scenario highlights the contrast between a single, extremely volatile payoff and a large number of much smaller, independent payoffs that aggregate into a more predictable outcome. It is a typical style of question in quant and trading interviews, where understanding the difference between average outcome and distribution of outcomes is central to real-world decision-making.

To answer well, a candidate needs to use basic properties of expectation and variance for independent, identically distributed random variables, and to reason clearly about how dispersion scales with the number of trials. It leans on ideas like the law of large numbers, relative versus absolute risk, and the relationship between mean, variance, and standard deviation. Interviewers listen for comfort translating these statistical properties into intuitive risk–return trade-offs, and for an ability to connect mathematical calculations to the qualitative preferences of a risk-averse decision maker.

What it tests

When comparing random processes with the same expected value, the key to understanding risk is how variance scales with aggregation. For independent, identically distributed random variables, the variance of their sum grows linearly with the number of variables, while the mean grows linearly as well. However, the standard deviation of the average (not the sum) decreases as $1/\sqrt{n}$, which means that aggregating many small, independent risks reduces relative uncertainty, but the absolute variance of the sum still increases with more trials. For risk-averse decision-making, the distribution's spread (variance) is as important as its mean, because higher variance means more uncertainty about the outcome, even if the average payout is the same.