Two Sigma 's Quant Researcher(QR) position has been called one of the "IQ ceiling" positions. Its screening process is extremely strict, and I have just finished a set of Two Sigma Quant Researcher OA with my students, so here is a detailed review of the questions I encountered and my thinking process. Overall, I feel that the difficulty of the questions is not low, examining probabilistic reasoning, expectation calculation, data structure design and geometric intuition, which is very comprehensive.
Part 1: Review of Online Assessment Questions
Problem 1: Probability Decision Problem
Question.
You start with $1. In each round, you can either stop and take your current amount, or continue.
If you continue, with probability p Your money doubles, and with probability (1 - p) You lose everything.
When should you continue?
Solution idea.
At its core, it compares the expected return of "continuing to play" with the certainty of "stopping".
Assume that the current funding is x:
- If stop → gain = x
- If continue → Expected return = 2x p + 0 (1 - p) = 2px
Therefore, when 2p > 1 (i.e. p > 0.5) When the expected gain from continuing the game is greater than stopping.
Conclusion:
- If p > 0.5, one can play all the way from the expectation point of view;
- In practice, however, if one plays infinitely, the ultimate probability of bankruptcy is still 100%, which begs the question of risk preferences and utility functions.
- If p ≤ 0.5, none of the first rounds should start.
This question is a classic, and Two Sigma just loves the way it examines both mathematical reasoning and risk awareness.
Problem 2: Expected Value of a Sequence
Question.
You have a biased coin with probability p You keep tossing until the pattern "HT" appears.
What's the expected number of tosses?
Solution idea.
This question is a classic question in the state expectation method. We define:
- E = the expected number of coin tosses from the initial state to the occurrence of an HT
- E_H = expected number of HTs after an H has been rolled
The state transfer is as follows:
from the initial state:
E = (1 - p)(1 + E) + p (1 + E_H)
The simplification is: E = 1/p + E_H
from the "Already H" state:
E_H = (1 - p) × 1 + p (1 + E_H)
The simplification is: E_H = 1/(1 - p)
Substitutions can be made:
E = 1/p + 1/(1 - p)
The result is beautiful, and in the "small talk" style that Two Sigma has always favored.
Problem 3: Median in a Dynamic Data Stream
Question.
Design a data structure that supports.
addNum(num)- add a numberfindMedian()- return the median of all elements
Requires the highest possible level of efficiency.
Solution idea.
This question is a standard algorithmic question, but its presence in QR OA means that it is not just the code implementation that is being examined, but also the logical thinking.
The optimal solution is to use Two piles.:
- max_heap Store the smaller half of the number (top of heap is the larger value)
- min_heap Store the larger half of the number (top of the pile is the smaller value)
Operational Logic:
- Inserts into the appropriate heap, then balances the two sides so that the difference between the number of elements in the two heaps does not exceed 1.
- When finding the median:
- If two piles are equal in size → return the average of the tops of the two piles
- If unequal → return the top of the heap with more elements
Complexity.
addNum: O(log n)findMedian: O(1)
This question also examines your sensitivity to "online processing of streaming data", which is a very common type of thinking question in QR interview.
Problem 4: Geometric Probability
Question.
A stick of length 1 is broken at two random points, forming three pieces.
What's the probability that these three pieces can form a triangle?
Solution idea.
Let the two breakpoint locations be x and y (assuming x < y), then the length of the three segments is:
a = x, b = y - x, c = 1 - y.
To form a triangle, it must be satisfied:
a + b > c, a + c > b, b + c > a.
By integration (or the geometric area method), the final result is 1/4.
This question is very much in the style of Two Sigma - combining mathematical intuition and probabilistic geometry to examine logical rigor.
Technical interview stage review (45min/round, 2-3 rounds)
After OA, went into technical interview, purely chatty: probability + modeling based, no whiteboard code. Interviewer (PhD mostly) will dig deeper:
- "How is a decision tree constructed for P dynamic updates (e.g., Bayesian posterior)?" (A: POMDP, value iteration.)
- "HT to TH or 'HH', state transfer change?" (Symmetry analysis, overlap penalty.)
- "Median plus delete operation?" (Tracking counts with HashMap, double heap adjustment.)
- "Non-uniformity of breakpoints in trigonometric probability?" (Dirichlet distribution, integral recalculus.)
Tips: don't just report answers, draw state diagrams/geometry as you go. Emphasize trade-offs (e.g., exact vs. approximate). Behavioral facets interspersed: talk about probability applications in projects (e.g., A/B testing).
Summarize
Two Sigma's QR OA can be described as a concentration of Math Intuition + Programming Logic + Rigor.
If you are preparing for a similar quantitative post test, more practice is highly recommended:
- probability and expectation type questions
- State Transfer + Equation Modeling
- Dynamic data structure maintenance issues
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