Every concept, endless questions
Each concept is a small financial model. Read how it works, then draw a real interview question from it: forwards, backwards or what-if, with fresh numbers and a worked solution every time.
Forward runs a concept the usual way. Inverse runs it backwards, which is what separates understanding from memorizing.
Probability
6 conceptsConditional probability and Bayes
A signal that is right 95% of the time can still be wrong most of the times it fires. What matters is not how accurate the signal is but how rare the thing it looks for is: if the event happens on one day in a hundred, then even a small false-alarm rate on the other ninety-nine days produces more false alarms than true ones. Bayes’ rule is the bookkeeping that weighs the true alarms against the false ones. Every question of the form "the test came back positive, how worried should I be" is this calculation, and the intuitive answer — the sensitivity — is almost always far too high.
Expected value of a game
The price of a bet is the average of what it pays, weighted by how often it pays it. That number — the expected value — is what you would earn per play if you played forever, and it is the only thing that decides whether a game is worth entering at a given fee. What it does not tell you is how bumpy the road is: two games with the same expected value can have wildly different standard deviations, and a small positive edge on a very volatile bet can bankrupt a player long before the average shows up. Repetition shrinks the noise relative to the edge, but only with the square root of the number of plays, which is why a casino wants millions of small bets and a gambler with one big bet is not "investing".
Dice sums and distributions
Two dice do not produce a flat distribution. Each face of each die is equally likely, but the totals are not, because a total in the middle can be made in many ways and a total at the edge in only one. Counting those ways — the ordered pairs — is the whole technique, and it generalises: every "probability of a sum" question, from dice to card totals to the number of defaults in a portfolio, is a question about how many equally likely outcomes land on each value. The counting also exposes the two habits interviewers test for: listing ordered pairs rather than unordered ones, and reaching for the complement when "at least one" appears.
Coin runs and sequences
Independent flips have no memory, but sequences of flips have structure. The number of heads in n flips is binomial, and its probabilities come from counting which flips were the heads. Waiting times are different: how long until a pattern first appears depends on the pattern, not just on its length. HT arrives faster than HH on a fair coin — four flips against six — because a failed attempt at HT (a second head) is already the start of the next attempt, while a failed attempt at HH (a tail) throws everything away. The two habits this rewards are reaching for the complement when the question says "at least", and setting up a first-step equation when it says "how long until".
Expected number of draws
If something happens with probability p on each try, you expect to wait 1/p tries for it. That single fact — the geometric mean — answers most "how long until" questions, and it has a property people find hard to accept: it does not shrink as you wait. After ten failures the expected remaining wait is still 1/p, because independent trials carry no memory. The same idea chains: the wait for the m-th success is m waits of 1/p, and the wait to collect every one of N types is a sum of geometric waits whose success probabilities fall as the set fills up, which is why the last few coupons take longer than all the others combined.
Combinatorics
Most probability questions are counting questions in disguise: how many outcomes are there, and how many of them are the ones you want. The single decision that matters is whether order counts. Picking a committee does not care who was named first; assigning titles does. Get that right and the rest is arithmetic — and a surprising number of "hard" questions collapse once you notice you can count the complement, or fix one person in place and count the rest, or notice that a randomly chosen committee contains any particular person with probability k/n by pure symmetry, without counting anything.