Probability questions become manageable when you stop treating each story as new. Most reduce to a small set of structures: counting, complements, conditioning, expectation, symmetry and state transitions.

Core probability topics

CountingPermutations, combinations, arrangements and avoiding double counts.
Complements“At least one” and “none” events that are easier to calculate backwards.
Conditional probabilityUpdating after information, selection effects and dependence.
Bayes’ ruleBase rates, false positives and reverse conditioning.
Expected valueFair prices, linearity, indicators and repeated games.
Stopping timesWaiting for patterns, gambler’s ruin and hitting events.

Question patterns worth recognizing

At least one success

P(at least one six in n rolls) = 1 − P(no sixes)

Linearity of expectation

Break a total into indicator variables. Independence is not required for linearity, which makes it one of the most powerful interview tools.

Condition on the first step

For recursive games or waiting times, define states and write an equation from the next outcome. This is cleaner than enumerating every future sequence.

Exploit symmetry

Random orderings, shuffled decks and fair games often become simple once equally likely positions or labels are identified.

A reliable solving workflow

  1. Define the sample space and random variables.
  2. Ask whether a complement is easier.
  3. Check independence and replacement assumptions.
  4. Look for symmetry or linearity.
  5. Compute and test the answer against extreme cases.
  6. Explain how the answer changes when one parameter moves.

Frequent mistakes

  • Confusing mutually exclusive events with independent events.
  • Using the wrong denominator after conditioning.
  • Ignoring base rates in diagnostic-style questions.
  • Double-counting arrangements.
  • Assuming expectation describes the most likely outcome.
  • Brute-forcing a process that has a short state recursion.

Randomized probability practice

QuantPrep varies coin biases, dice sizes, urn compositions, stopping conditions, payoffs and state transitions. Repeated practice therefore tests whether you recognize the mathematical family rather than whether you remember a specific fraction.

Turn the guide into timed practice

Select the matching category or firm profile, then run a short session and review every miss.

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