Quant Interview Expected Value Roadmap
A quant interview expected value roadmap covering linearity, indicators, stopping rules, betting games, and fair prices.
Candidates who need to turn probability reasoning into valuation and decision-making.
Start with direct expectation
Begin with direct expected value: list the outcomes, assign probabilities, multiply payoff by probability, and sum. A fair die that pays its face value in dollars has expectation (1+2+3+4+5+6)/6 = 3.5, so 3.50 is the fair price to play it once. This builds the habit of valuing uncertainty instead of guessing from the most likely outcome, and it makes the key point immediately: 3.5 is a value the die can never actually roll.
Use linearity early
Linearity of expectation is the highest-return tool in quant interviews, because it holds whether or not the variables are independent. The expected sum of three dice is 3 x 3.5 = 10.5, with no enumeration of 216 outcomes. The expected number of sixes in ten rolls is 10 x 1/6 = 5/3. And if n hats are returned to n people at random, the expected number of people who get their own hat back is n x (1/n) = 1, whatever n is - the events are heavily dependent and the answer is still exactly 1. Practice indicator variables until that reads as obvious rather than surprising.
Add stopping rules
Stopping questions need a state definition and a recurrence. If a game ends after a pattern, a threshold, or a choice, write the value of the current state and condition on the next event. The fair-coin waiting time is the cleanest template: E = 1 + (1/2) x 0 + (1/2) x E, so E = 2 flips to the first head, and 1/p for a coin with head probability p. Once you can write that equation without hesitating, most stopping problems become the same equation carrying more states.
Connect EV to trading decisions
Expected value prep is incomplete until it becomes a decision. After computing the value of a game, ask four follow-ups: what price makes it fair, how much variance comes with the edge, how size should change, and what would make you decline the trade entirely. A game worth plus five cents per play is an excellent trade a thousand times over and a terrible one once, if a single loss removes the bankroll. That distinction between expectation and survival is exactly what the follow-up questions are testing.
Concrete roadmap example
An eight-session EV block: two sessions on direct expectation over dice and card payoffs; two on indicator variables and linearity, including the hat-matching and coupon-collector patterns; two on one-step stopping games such as waiting times and the reroll die; and two on betting prompts where you must state a price, a size, and a reason you would decline. The reroll die makes a good milestone - given one optional reroll you should reroll on 1, 2, or 3, which is worth (1/2) x 3.5 + (4+5+6)/6 = 1.75 + 2.5 = 4.25.
Common mistakes
Candidates over-enumerate where linearity would answer in one line, confuse the expectation with the most likely outcome, or stop at "positive EV" without discussing risk at all. The stopping-problem version is failing to define when the game ends, which turns a two-line recurrence into an infinite sum. If the game can terminate early, condition on the next event before you write anything long.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.