PRACTICE GUIDE TWO SIGMA
Two Sigma Probability & Estimation Practice Test
Two Sigma does not publish the shape of its probability & estimation screen, and candidates describe it as 10 to 20 questions in 15 to 45 minutes, roughly one to three minutes per question, answered by picking one of the options.
Those are ranges across firms running this format rather than Two Sigma's own figures, so treat them as the shape to train against and check the instruction screen on the day for the marking rule.
Outcry is not affiliated with Two Sigma and has no access to their assessment content. This guide describes an assessment format that candidates report publicly; the questions here are generated by Outcry and are not Two Sigma’s own.
What it screens
A quantitative investment manager applying machine learning and distributed computing to markets.
- ✓Expected value computed fast and used to make a decision
- ✓Conditional probability and Bayes updates without formal notation
- ✓Ranking likelihoods rather than computing them exactly
- ✓Fermi estimation: decomposing an impossible question into three easy ones
Where it sits at Two Sigma
A quantitative investment manager applying machine learning and distributed computing to markets. What that means for the screen is that the questions tend to be drawn from the work rather than from a textbook, so the format below is the shape to train against rather than a syllabus.
Candidates also report 2 other screens at Two Sigma, covered separately on this site. Where a firm runs several, they usually sit in one round rather than spread across the process, so the pacing of the whole set matters more than any single section.
The format
Two Sigma does not publish this screen's shape, and it varies between firms, so these are the ranges candidates report rather than exact figures.
| Questions | 10 to 20 |
|---|---|
| Time | 15 to 45 minutes |
| Per question | one to three minutes |
| Negative marking | Varies by firm and year |
| Answer style | Multiple choice or a typed number |
| Where it sits | First round, or a second technical screen |
What it tests, with a worked example
Every example below is generated by Outcry, drawn from the same question generators the timed drills run. None of them is Two Sigma’s.
Conditional probability
Bayes' rule in disguise: a test result, a signal, a drawn card, and what it implies.
Example 1
How many distinct ways can 6 people line up single file?
Answer 720
6! = 720.
Example 2
A market is Calm or Volatile. Each day: Calm turns Volatile with probability 0.1, Volatile turns Calm with probability 0.3. Over the long run, what fraction of days are Calm?
- 1/4
- 3/4
- 1/2
- 9/10
Answer 3/4
Stationarity balances the flows: π(Calm)·0.1 = π(Vol)·0.3, so π(Calm) = 0.3/(0.1+0.3) = 3/4.
Example 3
Cards are dealt one at a time from a shuffled standard deck. Before each card you may call "red" - you win if that card is red, and the game ends. If you never call, the last card decides. What is your best possible probability of winning?
- It grows with patience
- 27/52
- 26/51
- 1/2
Answer 1/2
A martingale argument: whatever the strategy, the proportion of reds remaining is a martingale, so every stopping rule wins with exactly 1/2. No timing edge exists.
Distributions you should know cold
Binomial, geometric and uniform means and variances, applied rather than quoted.
Example
A fair 4-sided die is rolled until two 4s land back to back. How many rolls on average?
- 4
- 32
- 8
- 16
- 20
Answer 20
Let E be the answer from a fresh start. One roll hits with chance 1/4 and then needs one more hit. Solving gives 4² + 4 = 20.
Ranking outcomes
Ordering several events by likelihood, where only the exact order scores.
Example
An index trades at 44.16, the 6-month rate is 2.08% continuously compounded, and the index throws off a continuous dividend yield of 3.26%. What is the fair 6-month forward level?
Answer 43.9
Holding the index costs the funding rate and pays the dividend yield, so only the difference is carried: F = 44.16 x e^((0.0208 - 0.0326) x 0.5000) = 44.16 x 0.994117 = 43.900. The net carry is -1.18%, so the forward trades below spot. Leaving the dividend out gives 44.622. When the dividend yield exceeds the funding rate the whole forward curve inverts, and that is not a signal about direction, only about carry.
Waiting times and repeated trials
How long until the first success, and why the answer is rarely the average you expected.
Example
How many observations are needed for a 95% confidence interval of half-width 2.5, when the standard deviation is 25?
Answer 385
n = (1.96σ/w)². Call it 2: (2 × 25/2.5)² = 400, which overshoots by 4%.
Symmetry arguments
Questions that collapse to one line once you notice the two cases are mirror images.
Example
4 crates arrive at a receiving dock holding rivets, spacers, bushings and clips, one content each. Each crate carries a label naming one of those 4 contents, no two labels the same, and every label is on the wrong crate. Opening a crate shows what is in it. How many crates must be opened, in the worst case, to relabel all 4 correctly?
Answer 2
The label-to-contents map is a permutation with no fixed point, and there are 9 of those on 4 items. Opening a crate reveals its contents and splits the live set; picking the next crate after seeing the last result, a minimax search over every opening order gives 2 as the worst case. With 4 = 3 the two possible arrangements are the two rotations, and one look separates them. The general pattern: the last two crates settle themselves only when the two contents left both clash with their own labels.
Reading the question exactly
At least, exactly, and given that - the three phrases that change the answer entirely.
Example
Revenue by division, $m Division 2020 2021 2022 2023 Commodities 165 345 305 225 Macro 230 270 290 385 Rates 240 170 325 250 Credit 320 455 380 460
Which division added the most revenue in absolute terms between 2021 and 2023?
- Commodities
- Rates
- Macro
- Credit
Answer Macro
Changes: Commodities -120, Macro +115, Rates +80, Credit +5. Macro is the largest at +115.
What a good score looks like
Probability rounds are usually marked as part of a wider battery, so a standalone threshold is rarely reported. Where candidates do describe one it clusters around getting two thirds of the items right, with the caveat that these sections are short enough that two mistakes move you a long way.
How to train for it
- 01Practise setting up before calculating. Most wrong answers on these are a correct calculation of the wrong quantity.
- 02Sanity-check the magnitude before committing. A probability above one or an expectation outside the range of outcomes is a caught error, and these papers plant both as options.
- 03Work the complement first whenever the question says at least or at most. It is usually the shorter route and it is almost always the intended one.
TRAIN IT HERE
The drills that match each section
Likelihood Ranking
Order events from most to least likely - the exact screen-question format.
Fermi Estimation
Piano tuners in Chicago, atoms in a body - scored on order of magnitude.
Quitters Never Lose
A casino stopping-strategy game that is secretly a martingale lesson.
Tail Risk
Rare events and fat tails, and what they do to a plain expected-value estimate.
SIT THE FULL BATTERY
All the sections back to back on one clock, marked the way the real screen marks them, with a by-skill breakdown at the end. Included with any pass.
Also reported at Two Sigma
Common questions
- Is the Two Sigma probability & estimation test multiple choice?
- Reported as multiple choice or a typed number. Formats move, so treat this as the shape rather than a guarantee.
- How long is the Two Sigma probability & estimation test?
- Candidates report 10 to 20 questions in 15 to 45 minutes, roughly one to three minutes per question.
- Is there negative marking on the Two Sigma probability & estimation test?
- It varies by firm and by year. Where a paper does mark negatively, skipping beats guessing; where it does not, answer everything. Check the instruction screen before you start - it is always stated there.
- How do I practise for it free?
- Every drill linked on this page is free to play, with no account, inside a daily run cap. Questions are generated fresh each run, so there is nothing to memorise between attempts.