Learn · reason · practise

Build the foundations for your interview.

Choose a role and build the foundations in a useful order. Fourteen original mini-lessons connect worked examples, independent checks, guided lessons and free interview questions. Start with what you need, then return to test your reasoning without hints.

Quantitative tradingQuantitative researchSoftware engineering

Quantitative research

Connect probability, linear models and honest empirical evaluation, supported by coding fundamentals.

13 ordered modules · start where your prerequisites are solid

Read the explanation, work through the example, attempt the check, then solve fresh questions. Revisit them later without hints. This sequence is a study recommendation; it does not certify interview readiness.

  1. 1

    Number sense and explicit models

    Guided arithmetic lessons plus a short modeling bridge.

    Mini lesson · 3 practice questions · 5 guided course lessons

    What you’ll learn

    • Keep units consistent.
    • Translate words into equations.
    • Check magnitude before exact arithmetic.

    The idea

    Choose a representation before calculating. A rate is a quantity per unit time; adding travel times is valid, averaging unlike rates usually needs weights. State assumptions, solve, and substitute the result into the original quantities.

    Worked example

    A car covers 60 km at 30 km/h and 60 km at 60 km/h. Find its average speed.

    1. The first leg takes 60/30=2 hours; the second takes 60/60=1 hour.
    2. Total distance is 120 km and total time is 3 hours.
    3. Average speed is 120/3=40 km/h; the slower leg gets more time weight.

    40 km/h

    Try it yourself

    Compute 48×25 using a convenient scale.

    Show a hint

    25 is 100/4.

    Check your reasoning

    1200

    48×100/4=4800/4=1200. This also lies just below 50×25=1250.

    Learn with guided practice

    Place value and magnitude →Split a product →Simplify and compare fractions →Weighted averages →Spoken calculation →

    Apply it to interview questions

    Further study beyond this mini lesson
    • No assessment certification; timing drills measure arithmetic rather than full interview readiness.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  2. 2

    Counting and complements

    Guided counting lessons with finite sample-space practice.

    Mini lesson · 3 practice questions · 4 guided course lessons

    Build on: Number sense and explicit models

    What you’ll learn

    • Separate ordered from unordered outcomes.
    • Use complements.
    • Prove an information lower bound.

    The idea

    Counting requires a precise object: a sequence differs from a subset. When counting favorable outcomes, ensure all denominator outcomes are equally likely. For guarantees, count how many distinct states an observation can distinguish.

    Worked example

    Select two different people uniformly from five. Two specific people are siblings. What is the probability the pair contains both siblings?

    1. There are 5×4 ordered draws, but each pair is counted twice. Hence 10 unordered pairs.
    2. Exactly one unordered pair contains both specified siblings.
    3. The probability is 1/10; using 20 in the denominator would require counting two favorable orders.

    1/10

    Try it yourself

    How many length 3 binary strings contain at least one 1?

    Show a hint

    Subtract the all-zero string from all strings.

    Check your reasoning

    7

    There are 2³=8 strings. Exactly 000 has no 1, leaving 7.

    Learn with guided practice

    Counting and the complement rule →Stars and bars →Grid paths and reflection →Derangements and subfactorials →

    Apply it to interview questions

    Further study beyond this mini lesson
    • Advanced generating functions and asymptotic combinatorics remain outside this mini-lesson.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  3. 3

    Conditioning and base rates

    Guided Bayes lessons plus a population-count mini-example.

    Mini lesson · 3 practice questions · 2 guided course lessons

    Build on: Counting and complements

    What you’ll learn

    • Define the conditioning event.
    • Use likelihood and prior separately.
    • Normalize over the observed population.

    The idea

    Conditioning restricts attention to outcomes consistent with what you observed. A positive result is not a diagnosis: its probability depends on how often the condition occurs and how often alternatives generate positives. Population counts make the denominator visible.

    Worked example

    A condition occurs in 1% of people. A test detects 90% of cases and falsely flags 5% of noncases. In a representative population of 10,000, what fraction of positive tests are true cases?

    1. There are 100 cases and 9900 noncases. Expected true positives are 90 and false positives 495.
    2. Positive results total 585; the relevant population is positives, not all people or all cases.
    3. The desired fraction is 90/585=2/13, approximately 15.38%.

    2/13

    Try it yourself

    A fair coin is flipped twice. Given at least one head, what is P(two heads)?

    Show a hint

    List the equally likely surviving ordered outcomes.

    Check your reasoning

    1/3

    HH, HT, TH survive; onlyHH has two heads. The observation protocol is exactly the event stated.

    Learn with guided practice

    Bayes' rule and base rates →Bayesian quote update after a trade →

    Apply it to interview questions

    Further study beyond this mini lesson
    • Continuous posterior densities and conjugate-prior derivations need further study.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  4. 4

    Expectation, variance and dependence

    Guided expectation/statistics lessons.

    Mini lesson · 3 practice questions · 4 guided course lessons

    Build on: Conditioning and base rates

    What you’ll learn

    • Use indicators and linearity.
    • Include covariance.
    • Separate average payoff from risk.

    The idea

    Expectation averages values using probabilities. Linearity does not require independence. Variance measures squared deviations and is not linear: Var(X+Y)=Var(X)+Var(Y)+2 Cov(X, Y). Check dependence before discarding the covariance term.

    Worked example

    Flip one fair coin. Let X=1 for heads and 0 for tails; let Y=X. Find E[X+Y] and Var(X+Y).

    1. X+Y equals 0 or 2 with probability 1/2 each, so its mean is 1.
    2. Squared deviations from 1 are both 1, hence variance 1.
    3. Adding individual variances gives only 1/2; the missing 1/2 is 2 Cov(X, Y), because X and Y are identical.

    mean=1; variance=1

    Try it yourself

    Let X be fair Bernoulli and Y=1-X. What is Var(X+Y)?

    Show a hint

    Inspect the sum before expanding a formula.

    Check your reasoning

    0

    X+Y is always 1. Negative covariance exactly cancels the two individual variances.

    Learn with guided practice

    Linearity of expectation →Expected value →Variance of linear combinations →Pairwise correlation bounds →

    Apply it to interview questions

    Further study beyond this mini lesson
    • Limit theorems, concentration bounds and statistical confidence intervals require a longer course.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  5. 5

    Vectors, projections and rank

    New mini-lesson plus authored linear-algebra questions; no mapped guided course.

    Mini lesson · 4 practice questions

    Build on: Number sense and explicit models

    What you’ll learn

    • Compute dot products and projections.
    • Identify dependent columns.
    • Explain rank before inversion.

    The idea

    A vector projection extracts the component parallel to a chosen direction. Least squares chooses coefficients so the residual is orthogonal to every design column. Dependent columns prevent unique coefficients even when fitted predictions are meaningful.

    Worked example

    Project y=(2,0) onto the line spanned by x=(1,1). Find the coefficient, fitted vector and residual.

    1. Minimize (2-b)²+(0-b)². Its derivative 4 b-4 vanishes at b=1.
    2. Equivalently b=(x·y)/(x·x)=2/2=1, so the fitted vector is(1,1).
    3. Residual is(1,-1), and its dot product with x is 0, checking the normal equation.

    coefficient=1; fitted=(1,1); residual=(1,-1)

    Try it yourself

    The matrix has columns(1,2) and(2,4). What is its rank?

    Show a hint

    Check whether one column is a scalar multiple of the other.

    Check your reasoning

    1

    The second is twice the first and the first is nonzero, so only one independent direction exists.

    Apply it to interview questions

    Further study beyond this mini lesson
    • No full guided linear-algebra course currently mapped; add elimination, eigenvectors, SVD and conditioning lessons.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  6. 6

    Covariance and portfolio aggregation

    Guided covariance and volatility practice.

    Mini lesson · 3 practice questions · 2 guided course lessons

    Build on: Vectors, projections and rank · Expectation, variance and dependence

    What you’ll learn

    • Compute quadratic portfolio variance.
    • Recognize covariance constraints.
    • Distinguish risk factors from return guarantees.

    The idea

    Portfolio variance is wᵀΣw. Off-diagonal entries count twice in a two-asset expansion because both orderings of a cross term occur. A covariance matrix must produce nonnegative variance for every weight vector; arbitrary correlation guesses can violate that requirement.

    Worked example

    Two returns have standard deviations 2 and 3, with correlation 0.5. A portfolio puts weight 0.5 on each. Find variance and standard deviation.

    1. Covariance is 0.5×2×3=3.
    2. Variance is 0.25×4+0.25×9+2×0.5×0.5×3=19/4.
    3. Standard deviation is sqrt(19)/2≈2.17945; taking the weighted average of volatilities would incorrectly ignore diversification.

    variance=19/4; sd=sqrt(19)/2

    Try it yourself

    Equal-weight assets both have standard deviation 2 and correlation-1. What is portfolio variance?

    Show a hint

    The perfectly opposing returns cancel.

    Check your reasoning

    0

    0.25×4+0.25×4+2×0.25×(-4)=0; this is a valid singular covariance boundary.

    Learn with guided practice

    Two-asset portfolio volatility →Regression beta and R^2 →

    Apply it to interview questions

    Further study beyond this mini lesson
    • Estimation error, changing correlations and factor-model diagnostics need longer empirical exercises.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  7. 7

    Calculus readiness: slopes and gradients

    A short derivative and partial-derivative check before learning optimization; this is a mini-lesson, not a full calculus course.

    Mini lesson · 4 practice questions

    Build on: Number sense and explicit models

    What you’ll learn

    • Interpret a derivative as a local rate of change.
    • Differentiate a polynomial and hold other coordinates fixed for a partial derivative.
    • Use a gradient direction with an explicit step size and verify the resulting loss.

    The idea

    A derivative describes the local change in a function per small change in its input. For a polynomial, differentiate each term: the derivative of x² is 2x, the derivative of ax is a, and constants contribute zero. With several coordinates, a partial derivative changes only one coordinate while the others stay fixed. The gradient collects these partial derivatives. Moving against it decreases a differentiable loss for a sufficiently small step; the step size still matters, so check the actual loss afterward.

    Worked example

    Let f(x)=x²−6x+10. Find f′(x). Starting at x=1, take one gradient-descent step x_new=x−ηf′(x) with η=1/4. Compute the new x and compare losses.

    1. Differentiate term by term: f′(x)=2x−6. At x=1 the derivative is −4, so a small increase in x lowers f locally.
    2. Apply the specified update: x_new=1−(1/4)(−4)=2. The derivative is a direction and rate, not the new parameter itself.
    3. The losses are f(1)=5 and f(2)=2. Completing the square gives f(x)=(x−3)²+1, so the update moves toward the minimum at x=3 without reaching it.

    derivative=2x−6; new x=2; loss 5→2

    Try it yourself

    For g(u,v)=(u−1)²+2(v+1)², find the gradient at (u,v)=(3,0). Take one simultaneous step of size 1/4 using that gradient. Give the new point and its loss.

    Show a hint

    Hold v fixed when differentiating u, and hold u fixed when differentiating v. Evaluate both derivatives at the original point before updating either coordinate.

    Check your reasoning

    gradient=(4,4); new point=(2,−1); loss=1

    The partial derivatives are 2(u−1) and 4(v+1), so the gradient is (4,4) at (3,0). Subtract (1/4)(4,4) simultaneously to get (2,−1). Its loss is 1, below the original loss 6. At the minimizer (1,−1), both partial derivatives are zero.

    Apply it to interview questions

    Further study beyond this mini lesson
    • No full guided calculus lesson is currently mapped. Continue with limits, chain rule, multivariable derivatives and curvature before treating this readiness check as full optimization preparation.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  8. 8

    Regression, loss and honest validation

    New research mini-lesson supported by question practice.

    Mini lesson · 5 practice questions

    Build on: Vectors, projections and rank · Expectation, variance and dependence · Calculus readiness: slopes and gradients

    What you’ll learn

    • Derive a simple loss update.
    • Keep preprocessing inside training data.
    • Respect chronology in evaluation.

    The idea

    A loss defines what is optimized; gradients tell how a small parameter change alters that loss. Evaluation asks a separate question: does the learned rule generalize to unseen information? Fit transformations on training data only, and use chronological splits when future observations would reveal unavailable information.

    Worked example

    A predictor is yhat=b*x with no intercept. Data are(1,2),(2,4). Use mean squared error L=((b-2)²+(2 b-4)²)/2. Starting b=0, take one gradient-descent step of size 0.1.

    1. Expand L=(5/2)(b-2)², so dL/db=5(b-2).
    2. At b=0 the gradient is-10. Update b_new=b-0.1*gradient=1.
    3. Loss falls from 10 at b=0 to 2.5 at b=1. This checks one step, not convergence for arbitrary learning rates.

    b=1; loss=2.5

    Try it yourself

    Training feature values are 1,3; a future test value is 100. What mean should center the training data?

    Show a hint

    The test value must not affect fitting.

    Check your reasoning

    2

    Use(1+3)/2=2. The all-data mean 104/3 leaks test information into preprocessing.

    Apply it to interview questions

    Further study beyond this mini lesson
    • No mapped full guided ML course. Multivariable calculus, likelihood, regularization and time-series validation need sustained study.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  9. 9

    Algorithms: contracts and invariants

    New engineering mini-lesson plus executable Python question solutions.

    Mini lesson · 4 practice questions

    Build on: Number sense and explicit models

    What you’ll learn

    • Specify input and duplicate semantics.
    • Prove a loop invariant.
    • Analyze time and memory separately.

    The idea

    An algorithm needs a contract before code: what is returned, how duplicates count, and what empty input means. A loop invariant states what the stored state means before each iteration. Complexity follows from the total number of operations, not just the number of nested loops visible in code.

    Worked example

    Does array[3,3] contain two distinct indices whose values sum to 6? Explain a one-pass set algorithm and the ordering of its check and insert.

    1. Before reading index i, the seen set contains values only from indices smaller than i.
    2. At each value x, test whether 6-x is already seen, then insert x. The first 3 is inserted and the second finds it.
    3. Thus the answer is true using distinct indices. Inserting before checking would wrongly let a single[3] match itself. Expected time is O(n), additional space O(n).

    true

    Try it yourself

    Under that distinct-index contract, does[3] contain a pair summing to 6?

    Show a hint

    There is no earlier element at its only index.

    Check your reasoning

    false

    Check before insertion: seen is empty, so no partner exists; the loop then ends.

    Apply it to interview questions

    Further study beyond this mini lesson
    • No full guided programming course mapped. Need implementation feedback, debugging sessions and adversarial performance practice.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  10. 10

    Graphs and shortest-path states

    New graph mini-lesson plus programming practice.

    Mini lesson · 4 practice questions

    Build on: Algorithms: contracts and invariants

    What you’ll learn

    • Choose BFS for unit-cost edges.
    • Maintain discovered-state invariants.
    • Distinguish directed reachability and weighted distance.

    The idea

    A graph state must describe what determines legal next moves. Breadth-first search processes unweighted states by increasing distance; discovering a vertex once is enough because all earlier queue layers are no farther away. Weighted edges require a different ordering such as Dijkstra when weights are nonnegative.

    Worked example

    A directed unweighted graph has edges A→B, A→C, B→D, C→D, D→E. Find the shortest distance from A to E and describe BFS layers.

    1. Layer 0 is{A}. Its newly discovered neighbors form layer 1={B, C}.
    2. Both lead to D, but D is marked discovered once; layer 2={D}.
    3. E is discovered from D at layer 3, so shortest distance is 3 edges. No earlier layer reaches E.

    3

    Try it yourself

    Add edge A→E. What is the new shortest distance?

    Show a hint

    BFS first discovers E while expanding A.

    Check your reasoning

    1

    A direct edge is a one-edge path; a nontrivial path cannot use fewer than one edge.

    Apply it to interview questions

    Further study beyond this mini lesson
    • No full guided graph course mapped; add traversal visualization, heap-based Dijkstra and union-find implementations.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  11. 11

    Sequential decisions and first-step analysis

    Guided stopping, coupon and Markov lessons.

    Mini lesson · 4 practice questions · 5 guided course lessons

    Build on: Expectation, variance and dependence

    What you’ll learn

    • Compare stopping value with continuation value.
    • Write a state-dependent recurrence.
    • Check terminal and impossible states.

    The idea

    A sequential problem needs a state containing all information relevant to future outcomes. Work backward when the horizon is finite; for waiting times, condition on the next event. Count the current step exactly once and make absorbing states explicit.

    Worked example

    Roll a fair six-sided die. You may accept it or pay 1 to reroll exactly once; the second value must be accepted. Find the optimal expected net payoff before the first roll.

    1. The last roll has expectation 3.5. Paying 1 makes continuation worth 2.5.
    2. At the first roll retain 3,4,5,6; reroll 1 or 2, since each is below 2.5.
    3. Average the six optimal values: (2.5+2.5+3+4+5+6)/6=23/6.

    23/6

    Try it yourself

    If the reroll costs 3, which first-roll outcomes should be rerolled?

    Show a hint

    Continuation is 3.5 minus the cost.

    Check your reasoning

    none

    Continuation is 0.5, below every possible first roll, so accept all outcomes.

    Learn with guided practice

    Two-stage stopping →Stopping with search cost →Coupon collector and waiting times →First-step analysis →Coin pattern waiting times →

    Apply it to interview questions

    Further study beyond this mini lesson
    • General measure-theoretic optimal stopping is outside the guided sequence.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  12. 12

    Quotes, adverse selection and sizing

    Guided market-making and sizing lessons.

    Mini lesson · 3 practice questions · 6 guided course lessons

    Build on: Sequential decisions and first-step analysis

    What you’ll learn

    • Distinguish bid/ask from value.
    • Update value after informative flow.
    • Account for risk and sizing assumptions.

    The idea

    A bid is a price at which you buy and an ask a price at which you sell. A spread only creates profit when the acquired inventory can be unwound favorably. If counterparties have information, a fill changes your estimate of value; quoting against the unconditional mean can lose money.

    Worked example

    A claim pays either 0 or 100, equally likely. A buyer purchases at your ask with probability 0.8 in the 100 state and 0.2 in the 0 state. You ask 60. Find expected P&L conditional on a purchase, ignoring fees.

    1. The purchase probability is 0.5×0.8+0.5×0.2=0.5.
    2. Given a purchase, probability of payoff 100 is 0.4/0.5=0.8, so conditional fair value is 80.
    3. You receive 60 for a claim worth 80 conditional on this fill: expected P&L is-20.

    -20

    Try it yourself

    With the same buyer behavior and ask 80, what is conditional expected P&L?

    Show a hint

    Use conditional value 80 rather than unconditional value 50.

    Check your reasoning

    0

    80 received minus conditional value 80 equals 0, before fees and other frictions.

    Learn with guided practice

    Markets 101: bids, asks and trades →Spreads and P&L →Two-sided quoting →Inventory and skew →Winner's curse and conditioning on the fill →Bet sizing and the Kelly criterion →

    Apply it to interview questions

    Further study beyond this mini lesson
    • Toy fills omit queue priority, fees, latency and impact; simulated profit is not a trading recommendation.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

  13. 13

    Proof, ambiguity and collaborative explanation

    New proof mini-lesson with puzzle practice.

    Mini lesson · 4 practice questions · 1 guided course lesson

    Build on: Counting and complements · Algorithms: contracts and invariants

    What you’ll learn

    • Separate a construction from an optimality proof.
    • State ambiguity rather than inventing certainty.
    • Explain a lower bound and a matching construction.

    The idea

    A construction proves feasibility; a lower bound proves necessity. Optimality requires both. Start by stating the model and what the answer means, then give the shortest argument establishing it. When observations do not determine a unique result, exhibit two consistent models rather than silently choosing one.

    Worked example

    Exactly one of 9 identical coins is known heavier. An ideal balance compares equal-size groups. Prove the minimum worst-case weighings.

    1. One weighing has only three outcomes, insufficient for nine possible heavy coins. Thus at least two weighings are needed.
    2. Weigh three coins against three. The heavier group is selected, or the unweighed group if balanced.
    3. Among the resulting three, compare one against one. The result identifies the heavy coin, meeting the lower bound.

    2

    Try it yourself

    Exactly one of 3 coins is known heavier. Minimum worst-case weighings?

    Show a hint

    Compare one against one and interpret all three outcomes.

    Check your reasoning

    1

    Left heavy, right heavy, or balance identify the first, second, or unweighed coin respectively.

    Learn with guided practice

    Spoken calculation →

    Apply it to interview questions

    Further study beyond this mini lesson
    • This mini-lesson does not replace live mock interviews or assess communication under pressure.
    Sources and scope

    Learnr’s wording, examples and role sequence are original. References support the topic and method; this is not an employer’s prescribed syllabus.

Curriculum research updated 2026-10-15. Role paths share foundations and branch by the work you want to do.