Quant Interview Probability Roadmap
A quant interview probability roadmap from sample spaces and counting through Bayes, expected value, and random walks.
Candidates who need a probability-first path for quant interview prep.
Start with sample spaces
Probability prep starts with defining outcomes, not with famous problems. Before Bayes or random walks, four questions have to be automatic: are outcomes ordered or unordered, are draws with or without replacement, are the events independent, and what information has already been revealed. Nearly every early miss traces back to one of these. Two aces off the top of a deck is (4/52) x (3/51) = 1/221 because the deck changes between draws; if the first card were replaced and reshuffled it would be (4/52)^2 = 1/169. Same story, different denominator - and the denominator is the whole question.
Build the counting layer
Counting carries most interview probability, so drill complements, permutations, combinations, inclusion-exclusion, and symmetry until they are reflexes. The complement is the highest-return single tool: "at least one six in four rolls" is 1 - (5/6)^4, about 0.518, and counting it directly wastes minutes. Cards, dice, and urns are the right practice material precisely because they punish overcounting fast. If both C(52,5) and 52 x 51 x 50 x 49 x 48 show up in the same piece of work, you have mixed ordered and unordered counting, and the answer is off by a factor of 5! = 120.
Add conditioning and Bayes
Once counting is stable, add conditional probability and Bayes. The most effective single habit is converting percentages into counts. Take a disease with 1 percent prevalence and a test that is 90 percent sensitive and 90 percent specific, and imagine 1,000 people: 10 are sick and 9 of them test positive, while 990 are healthy and 99 of them test positive anyway. A positive result therefore means 9 out of 108, roughly 8 percent - not 90 percent. Candidates who work in counts get this right; candidates who reach straight for the formula usually do not. The question to keep asking is: given what was observed, which universe am I now counting inside?
Connect probability to expectation
Expected value is the next layer, because most probability questions become pricing or betting questions one follow-up later. Practice linearity, indicator variables, and simple stopping rules so a probability answer can turn into a fair value. Linearity is the workhorse: the expected number of sixes in four rolls is 4 x 1/6 = 2/3 immediately, with no distribution required. It is worth noticing that this same 2/3 is the union bound on "at least one six," which is why it sits above the true 0.518 - the bound double-counts rolls with more than one six.
Concrete roadmap example
A two-week probability block of eleven sessions: three on sample spaces, complements, and ordered-versus-unordered counting; three on cards, urns, and combinations; two on Bayes tables written in counts; two on linearity and indicator variables; and one mixed review that decides which method earns the next repair block. Retest the week-one misses cold at the start of week two rather than at the end, so the repair has somewhere to go.
Common mistakes
Candidates jump to famous hard problems before sample-space discipline is reliable, which produces memorized answers and no transferable method. The second common error is assuming independence by default: if the problem removes an item, reveals information, or lets the process stop early, the model has changed and the denominator has changed with it. The third is starting arithmetic before the setup is agreed, which turns a recoverable modeling error into five wasted minutes.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.