Average Runs in Marble Draw Sequence

This quant interview question is about understanding the structure of randomness in a finite sequence generated by sampling without replacement. Instead of focusing on individual outcomes, it asks you to reason about the global behavior of an entire path, summarized by how often it switches type. It connects combinatorial randomness with a clean expectation over an entire sequence, which is central in many probability-heavy quant prep problems.

It trains your ability to translate a verbal description of a random sequence into indicator variables and expectations, and to see how local behavior aggregates into a global statistic. You practice working with dependence in draws, recognizing when linearity of expectation still applies, and reasoning quantitatively about changes rather than levels.

This matters in quant interviews because modeling order flow, trade signs, or regime switches often reduces to understanding transitions in categorical sequences. Being comfortable with this type of expected-value reasoning is crucial for designing, stress-testing, and simplifying models that involve discrete states, such as buy/sell sequences, volatility regimes, or signal flips.

What it tests

Problems involving the expected number of transitions or 'runs' in a sequence generated by random draws (with or without replacement) are governed by the principle that the expected number of such events can be decomposed into a sum of local, pairwise transition probabilities. Specifically, the expectation is additive over positions: each position after the first contributes the probability that its value differs from the previous one. This works because the expectation of a sum is the sum of expectations, regardless of dependencies between draws. The key is that the indicator for a new run at each position depends only on the immediate predecessor, so the global expectation is the sum of these local transition probabilities. This principle holds for any sequence where the rule for a 'run' is based on local changes, not just for marbles or colors, but for any categorical sequence.