Citadel·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Quant interview at Citadel with a probability/combinatorics problem that looks deceptively simple until you actually sit down and grind through the cases. One question, clean setup, but the arithmetic at the end is where things get slippery.

Questions Asked (1)

Five people each independently pick one of two locations (A or B) with equal probability. Let X be the number of people at whichever location ends up less crowded. What is E[X]?

Algorithms & Data Structures

Author's notes

My first instinct was to overcomplicate it.

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AI HintsAI Generated

Suggested Approach

Model the number of people choosing location A as a Binomial(5, 0.5) random variable. Then express X as the minimum of K and 5-K, and compute its expected value by summing over all possible values of K.

Pro tip: After computing the answer, mention that you can verify it by symmetry and by checking that the expected number at the less crowded location is always less than or equal to 2.5, which adds credibility to your solution.

1. Define the random variable

Let K be the number of people who choose location A. Then K follows a Binomial distribution with n=5 and p=0.5.

2. Express X in terms of K

The number of people at the less crowded location is X = min(K, 5-K). This is because if K > 2.5, then location B has 5-K people and is less crowded, and vice versa.

3. List possible values and probabilities

Enumerate all possible values of K (0 to 5) and compute their probabilities using the binomial formula. Then determine the corresponding X values.

4. Compute expected value

Calculate E[X] by summing X * P(K=k) over all k from 0 to 5. Use symmetry to simplify the calculation if needed.

5. Verify and present

Check that the result is reasonable (e.g., between 0 and 2.5) and present the final answer clearly.

Key Points to Mention

Binomial distribution with n=5 and p=0.5
Definition of X as min(K, 5-K)
Symmetry of the binomial distribution around 2.5
Calculation of probabilities for each possible value of K
Expected value formula: E[X] = sum of x * P(X=x)
Final numerical answer: 45/32 or approximately 1.40625

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.