Interviewers probe a candidate's foundational understanding of probability theory, their ability to apply concepts like conditional probability and Bayes' Theorem to real-world scenarios, and their grasp of random variables and common distributions. They look for clear logical reasoning and the capacity to articulate probabilistic thinking.
17 questions (5 easy · 6 medium · 6 hard), each with what a strong answer covers and where people lose the point. Free to read, no account.
6.What is the probability that the sum of two dice is 8, given that the first die rolled a 3?
Core
What a strong answer covers
Define the events: A = sum is 8, B = first die is 3.
Identify the reduced sample space given that the first die rolled a 3.
From this reduced sample space, identify the outcome(s) where the sum is 8.
Calculate P(A|B) as the ratio of favorable outcomes in the reduced sample space to the total outcomes in the reduced sample space.
Where people lose the point
×Not correctly identifying the reduced sample space after the condition is given.
×Confusing P(A|B) with P(A ∩ B).
×Incorrectly applying the general conditional probability formula P(A|B) = P(A ∩ B) / P(B) without first simplifying the problem by reducing the sample space.
8.A rare disease affects 1 in 1000 people. A test for the disease is 99% accurate (correctly identifies positive and negative cases). If a person tests positive, what is the probability they actually have the disease?
Core
What a strong answer covers
Define the events: D = has disease, D' = does not have disease, T+ = tests positive, T- = tests negative.
State the given probabilities: P(D), P(T+|D), P(T-|D'), and infer P(T-|D) and P(T+|D').
9.You play a game where you roll a fair six-sided die. If you roll a 6, you win $10. If you roll a 1, you lose $5. Otherwise, you win nothing. What is the expected value of playing this game?
Core
What a strong answer covers
Identify the possible outcomes (die rolls) and their associated monetary values (X).
Determine the probability of each outcome (P(X=x)).
Apply the formula for expected value of a discrete random variable: E[X] = Σ x * P(x).
Calculate the sum to find the expected value.
Where people lose the point
×Incorrectly assigning probabilities to the outcomes.
×Making arithmetic errors in the summation.
×Misinterpreting 'win nothing' as a non-outcome or incorrect value.
12.Explain the Monty Hall problem and its solution. Why is it counter-intuitive for many people?
Hard
What a strong answer covers
Clearly describe the setup of the Monty Hall problem (3 doors, car behind one, goat behind others, contestant chooses, host opens empty door, contestant can switch).
Explain the optimal strategy: always switch doors.
Provide a clear probabilistic argument for why switching doubles the probability of winning (e.g., by considering the initial choice's probability and the host's action).
Discuss why it's counter-intuitive, often due to the 'equal probability' fallacy after the host's action.
Where people lose the point
×Incorrectly stating the optimal strategy (e.g., saying it doesn't matter if you switch).
×Failing to provide a rigorous probabilistic explanation for the solution.
×Not adequately explaining the source of the counter-intuitiveness.
13.You have a population with a mean of 50 and a standard deviation of 10. If you take a sample of 100 observations, what can you say about the distribution of the sample mean?
Hard
What a strong answer covers
State the Central Limit Theorem (CLT) and its conditions (large sample size, independent observations).
Explain that, due to the CLT, the distribution of the sample mean will be approximately normal.
Specify the mean of this sampling distribution (equal to the population mean).
Calculate the standard deviation of the sampling distribution (standard error) using the formula σ/√n.
Where people lose the point
×Confusing the distribution of the sample mean with the distribution of the population.
×Incorrectly calculating the standard error.
×Failing to mention the 'approximately normal' aspect or the conditions for CLT.
16.For a Binomial distribution B(n, p), what are its expected value and variance? Derive or explain the intuition behind them.
Hard
What a strong answer covers
State the expected value E[X] = np for a Binomial distribution.
Provide an intuitive explanation for E[X] = np (e.g., if you flip a coin n times with success probability p, you expect np successes).
State the variance Var[X] = np(1-p) for a Binomial distribution.
Provide an intuitive explanation for Var[X] = np(1-p) (e.g., relating it to the variance of a single Bernoulli trial and summing variances for independent trials).
Where people lose the point
×Incorrectly stating the formulas for expected value or variance.
×Failing to provide a clear intuitive explanation for the formulas.
×Attempting a full mathematical derivation without being asked, or providing an incorrect derivation.
A question a Probability panel actually asks, answered out loud, scored on what you said and how you said it. Under two minutes, and nothing to sign up for.
“What is the probability of getting exactly two heads in three coin tosses with a fair coin?”
We never store the audio. Your answer is deleted within 24 hours unless you save the result.
How Probability answers get judged
The weights a Probability interviewer is holding, whether or not they say so out loud. Round Zero scores your practice answers against exactly these, and quotes your own words back as the evidence for each.
Correctness
40%
The accuracy of definitions, formulas, calculations, and final answers. Are the probabilistic statements and numerical results precise?
Conceptual Depth
30%
Demonstrated understanding of the underlying probabilistic principles, assumptions, and implications. Goes beyond rote memorization to explain 'why'.
Problem-Solving Approach
20%
The logical steps taken to break down and solve the problem. Is the method chosen appropriate and efficient? Are assumptions clearly stated?
Communication
10%
Clarity, conciseness, and organization of the explanation. Can complex ideas be articulated simply and effectively?
You have read what strong Probability answers contain. The next thing that moves the needle is producing one under time, out loud, and finding out where it falls apart.
What Probability interview questions should I practice?
Start with the core areas Probability interviewers probe: What is the probability of getting exactly two heads in three coin tosses with a fair coin; What is the probability of rolling a sum of 7 with two fair six-sided dice; What is the probability of drawing an Ace from a standard 52-card deck. This page outlines strong answers and common mistakes, and the scored path drills each one with follow-ups.
Is the Probability practice free?
Yes. The Probability path runs free inside Round Zero: lessons, practice questions and flashcards. Drills are unlimited on every plan, free included. So is the full scorecard. Free also covers 3 complete scored interviews, no card.
How is this different from a Probability question list?
A static list gives you questions with no feedback. Round Zero runs a live scored practice that probes your actual answers, rotates difficulty, and tells you exactly what to fix, grounded in a Probability rubric.
How should I prepare for a Probability interview?
Learn the concepts, drill the questions until answers come fast, then prove it in a scored mock. Round Zero sequences all three so you know you are ready, not just that you read about Probability.
How is a Probability answer scored?
Probability answers are scored on correctness, conceptual depth, problem-solving approach, communication, with evidence quoted from what you actually said, so feedback is specific instead of generic praise.
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