1s vs 5s in n Dice Rolls
Probability of rolling 1 and 5 is a medium quant interview question on Covariance, reported to have been seen at Squarepoint Capital.
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This quant interview problem is about measuring the linear dependence between counts of different outcomes in repeated random experiments. It sits at the intersection of basic probability, discrete distributions, and the formal definition of correlation, all central themes in quant prep. It also checks whether candidates can translate a simple story into clean random variables with a clear joint structure.
It trains your understanding of covariance and correlation in multinomial settings, especially how different category counts relate to each other. You practice working with expectations of sums of indicator variables, interpreting negative dependence, and connecting marginal probabilities with joint behavior across many trials in a rigorous way.
It matters for quant interviews because similar logic appears in portfolio risk, factor models, and trade PnL attribution. Interviewers want to see you reason about dependence, not just compute it, and to move fluently between intuition and formal definitions in probabilistic models.
What it tests
When counting occurrences of mutually exclusive outcomes across independent trials, the joint probability of both outcomes occurring in the same trial is zero, while the probability of them occurring in separate trials is the product of their individual probabilities. This structure means that the covariance between counts of different outcomes is determined by the impossibility of their co-occurrence in a single trial, balanced against their independence across trials. The negative covariance arises because, for each trial, observing one outcome precludes the other, so their counts are slightly negatively related. This is a general property of multinomial or categorical sampling: the counts of distinct categories are negatively correlated because each trial can only contribute to one category. The sum of the counts over all categories is fixed (the number of trials), so an increase in one necessarily means a decrease in the others, creating this negative relationship.
Practise this question with written feedback, or hear it in a spoken mock interview.
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