PROBABILITY INTERVIEW QUESTIONS

Work these with a timer and a pen. The gap worth measuring is not whether you know the method, but how long it takes you to notice which method applies.

These are original representative questions, not real or leaked firm questions. If you want the underlying methods first, the probability shortcuts guide covers the five techniques these questions lean on.

Counting and complements

1. You roll a fair die four times. What is the chance you see at least one six?
   Show worked solution

   Count the opposite. No six on a single roll has probability 5/6, so 1 − (5/6)⁴ = 1 − 625/1296 = 671/1296, a little over 51%. Listing every way to roll a six takes longer and makes double counting easier.

2. Two cards are drawn from a standard deck without replacement. What is the chance both are hearts?
   Show worked solution

   (13/52) × (12/51) = 1/17, close to 5.9%. The second factor is where people slip; the deck has changed.

3. In a room of 23 people, is it more likely than not that two share a birthday?
   Show worked solution

   Yes, marginally: about 50.7%. The complement is a product of 364/365 × 363/365 × …, and it crosses one half at 23. Worth memorising as a reference point, since interviewers use it to see whether your intuition updates.

Conditional probability

4. A disease affects 1% of people. A test catches 90% of true cases and returns a false positive 5% of the time. You test positive. What is the chance you have it?
   Show worked solution

   (0.9 × 0.01) / (0.9 × 0.01 + 0.05 × 0.99) = 0.009 / 0.0585 ≈ 15.4%. The base rate dominates. Interviewers like this one because the intuitive answer is wildly wrong and the correction is one line of arithmetic.

5. A family has two children. At least one is a boy. What is the chance both are boys?
   Show worked solution

   As usually stated, 1/3: the equally likely cases are BB, BG, GB, and the condition removes only GG. The better answer flags the ambiguity, because if you learned this by meeting a specific child the answer becomes 1/2. State how the information was obtained before choosing an answer.

6. Three doors, one prize. The host knows the prize location, always opens a different losing door, and always offers a switch. You choose uniformly at random. Do you switch?
   Show worked solution

   Switch. Your first pick wins 1/3 of the time, so switching wins 2/3. The host's knowledge is what makes the extra information real; if the door had been opened at random, switching would gain nothing.

Expectation

7. How many rolls of a fair die do you expect before the first six?
   Show worked solution

   Six. For a repeated trial with probability p, the expected wait is 1/p. If you want the one-line derivation: E = 1 + (5/6)E, so E = 6.

8. Uniformly shuffle n numbered cards into n numbered slots, with n at least 1. How many land in their own slot on average?
   Show worked solution

   Exactly one, for any n. Each card has probability 1/n of matching, and expectations add regardless of dependence, so the total is n × 1/n = 1. The independence trap is the whole point of the question.

Symmetry

9. You flip 21 independent fair coins and I flip 20 independent fair coins. What is the chance you get strictly more heads than I do?
   Show worked solution

   One half. With one extra coin, exactly one of two things happens: you get strictly more heads, or you get strictly more tails, never both and never neither. Those two outcomes are symmetric, so each has probability 1/2. No summation required.

How to use these

Time yourself, and separate two failure modes. Not knowing the method is a study problem. If you know the method but freeze, try shorter timed sets and review where you hesitated. For a broader mix across mental math, sequences, and logic, see the full practice question set.

State the sampling model

The dice and coins here are fair and independent; the deck is uniformly shuffled. Birthday calculations assume independent, equally likely birthdays across 365 days and ignore leap years. The two-child example uses an idealized model with independent, equally likely boy/girl categories and an observation that filters families by the stated condition. These are mathematical models, not demographic claims.