Probability Interview Practice Set
30 fixed-order questions with layered hints, expanded worked solutions, memory tips, checks, and further reading.
Instructions
- Every attempt uses all 30 questions in the original section order.
- The order is fixed because later problems extend methods introduced earlier.
- Assessment mode allows 90 minutes and withholds hints until submission.
- Practice mode is untimed and reveals up to three hints in sequence.
- Exact fractions are preferred. No calculator is required.
- Questions may be flagged and revisited before submission.
- After submission, every problem includes a detailed derivation, a verification check, a memory tip, a common-trap warning, and relevant Wikipedia reading.
Method map
- Structured counting: condition on an arbitrary first object, then count legal choices for the next object.
- Modular sums: reserve a final variable that is uniform modulo the target modulus, or convolve residue counts when uniformity fails.
- Alternating games: define the value at the start of a turn and write a recurrence after one full cycle.
- Fair randomness from biased devices: pair outcomes that have equal probability by symmetry and reject the unpaired outcomes.
- Conditional geometry: classify latent object types, multiply prior counts by observation likelihoods, then normalize.
- Broken sticks: represent piece lengths as a uniformly distributed point on a simplex and compare areas or volumes.
- Pattern races: track only the longest current suffix that is also a prefix of a target pattern.
Random selections are uniform unless stated otherwise. Repeated tosses or rolls are independent unless stated otherwise. Region boundaries have probability zero in the geometric-probability questions.
Keyboard: 1–4 selects an answer; left/right arrows move between questions; F flags; H reveals the next hint in practice mode.
Results
Final Review Checklist
- Was the sample space conditioned correctly after an observation?
- Was a symmetry claim justified rather than merely asserted?
- In an alternating game, does the recurrence return to exactly the same state?
- In a debiasing method, do the paired accepted events have equal probability under the stated assumptions?
- In a painted-object problem, were prior counts multiplied by observation likelihoods?
- In a broken-stick problem, was the correct simplex dimension used for scaling?
- In a pattern problem, does each transition retain the longest suffix that is a target prefix?