Probability interview questions: twelve classics and how to solve them

Trading and quant interviews return to the same family of probability questions. Learn the techniques behind these twelve and most new ones become variations.

Conditional probability and Bayes

1. A family has two children and at least one is a boy. Probability both are boys? Equally likely cases: BB, BG, GB. 1/3.

2. A test for a condition affecting 1% of people is 99% accurate both ways. You test positive. Probability you have it? Of 10,000 people, 100 have it and 99 test positive; 9,900 do not and 99 test positive. 99 / 198 = 1/2.

3. Three cards: one red on both sides, one blue on both, one red and blue. You see a red side. Probability the other side is red? Three red sides are equally likely, two of them on the red-red card. 2/3.

Counting

4. Probability of at least one six in four rolls? 1 − (5/6)⁴ ≈ 0.518.

5. Probability that 23 people all have different birthdays? About 0.493, so a shared birthday is slightly more likely than not.

6. Probability three dice sum to 10? 27 of 216 outcomes: 1/8.

Expectation

7. Expected number of rolls to see a six? 6.

8. Expected rolls to see every face? 6 × (1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6) = 14.7.

9. Expected flips to get two heads in a row? Let E be the answer. E = 0.5(E + 1) + 0.25(E + 2) + 0.25 × 2, so E = 6.

Symmetry and tricks

10. 100 passengers board a plane. The first sits at random; each later passenger takes their own seat if free, otherwise a random one. Probability the last passenger gets their own seat? By symmetry between seat 1 and seat 100: 1/2.

11. You break a stick at two random points. Probability the three pieces form a triangle? No piece may exceed half the length: 1/4.

12. Two points are chosen at random on a line of length 1. Expected distance between them? 1/3.

Techniques to recognise

• Complement: "at least one" means 1 minus "none".
• Condition on the first step and write an equation for the unknown.
• Count equally likely cases carefully.
• Use symmetry before calculating.
• Linearity of expectation for counts.

In the interview

Say your approach before computing, check small cases, and state the answer as a fraction when it is exact. For more, see trading brainteasers.