Expected Value in quant interviews
The single most common technique in quant interviews. Every problem reduces to it eventually.
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In the classical secretary problem, as n → ∞, what fraction should you reject outright before accepting the next best-so-far candidate?
WHAT IT IS
Expected value is the weighted average of possible outcomes, weighted by their probabilities. Formally, for a discrete random variable X with outcomes x₁, x₂, … and probabilities p₁, p₂, …, E[X] = Σ pᵢ xᵢ. In quant interviews, expected value is the lingua franca: almost every probability question ultimately asks for the expected value of some quantity (winnings, stopping time, number of events). Mastery is not just about the formula — it's about the reflexes you build around it. Good candidates reach for expected value with three tools pre-loaded: (1) linearity of expectation, E[X + Y] = E[X] + E[Y] whether or not X and Y are independent; (2) conditioning, E[X] = E[E[X | Y]]; and (3) recursion, setting up E[X] in terms of itself via the first step. Interviewers probe whether you understand when each tool applies and can swap between them when one framing stalls.
WHEN IT APPEARS IN INTERVIEWS
Universally. Expected value shows up in every round at every quant firm — it's the common substrate under brainteasers, Markov-chain problems, trading-style estimation, and even coding interviews that ask for algorithmic complexity in expectation. If you're strong on nothing else, be strong on expected value.
FIRMS THAT TEST THIS
SAMPLE PROBLEMS
You flip a fair coin until you see two heads in a row. What is the expected number of flips?
Set up a recurrence via first-step analysis: let E be the expected flips from reset, E_H the expected flips after one head. Then E = ½(1 + E_H) + ½(1 + E) and E_H = ½(1) + ½(1 + E). Solve: E = 6.
You roll a fair six-sided die once. You may reroll it exactly once (and are then stuck with the second value). What's the expected payoff if you play optimally?
Optimal strategy: keep the first roll if it exceeds E[uniform die] = 3.5, else reroll. Expected payoff: ½(average of 4, 5, 6) + ½(3.5) = ½(5) + ½(3.5) = 4.25.
You draw uniformly at random from n distinct coupon types (with replacement) until you've seen all n. What's the expected number of draws?
Break into n stages: stage k is the wait for a new coupon given k-1 have been collected. Each stage is geometric with success probability (n - k + 1)/n, so E[stage k] = n/(n - k + 1). Sum: n·(1 + ½ + ⅓ + … + 1/n) = n·H_n ≈ n ln n.
You roll a fair six-sided die repeatedly, summing the results until the running sum reaches at least 7. What is the expected final sum?
A classic recursive EV problem. Define f(k) = expected final sum given current running sum is k. Then f(k) = 1/6 · Σ_{d=1..6} f(k + d) for k < 7, and f(k) = k for k ≥ 7. Work backwards from k = 6 down to k = 0. Answer is approximately 8.03.
SOLVING STRATEGIES
- ·Before computing anything, ask: can I break this into a sum of indicator variables? Linearity of expectation is the most over-powered tool in probability.
- ·If the problem has a self-similar structure (after one step, you're in a smaller version of the same problem), set up a recursion on E.
- ·Condition on the first event — often the entire problem collapses.
- ·Check symmetries. If positions or roles are interchangeable, E is often equal across them; use this to avoid messy algebra.
- ·Sanity-check with bounds. If the true E is bounded by known values, use those to flag arithmetic errors.
COMMON VARIATIONS
- ·Conditional expectation: 'given that event A happened, what is E[X | A]?'
- ·Variance and second moments — E[X²], Var(X), and their decompositions.
- ·Expected value of the maximum or minimum of independent random variables (order statistics).
- ·Expected value under a stopping rule (e.g., optimal stopping problems — see the secretary problem).
- ·Expected number of something under a random permutation — derangements, cycle lengths, fixed points.
FAQ
Probability is how often an event occurs; expected value is the average numeric outcome, weighted by probabilities. You can compute probability without ever using EV, but almost every probability interview question eventually asks you to compute an expected value.
Yes, at least conceptually. Variance questions (especially for portfolio or regression contexts) appear at Citadel, Jane Street, and most quant research interviews. You don't need to derive it from scratch; you need to use it fluently.
Whenever you can decompose the target into a sum of simpler indicator variables. Classic example: expected number of fixed points in a random permutation. Without independence, you can't multiply expectations, but you can always add them.
Drill problems across the five main subtypes: simple discrete (dice, coin), recursive (first-step analysis), linearity (indicator variables), Markov chain, and conditional. QuantPrep covers all five with adaptive selection — it prioritises whichever subtype you're weakest on.
RELATED TECHNIQUES
If the problem has states and transitions, it's a Markov chain — and quant interviewers love them.
"Given X happened, what's the probability of Y?" — the second-most-common framing in quant interviews.
When to stop and take the offer — the secretary problem and its cousins.
CLASSIC EXPECTED VALUE PROBLEMS
Deep walkthroughs of named problems that test expected value.
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