Citadel · Probability & Brainteasers

Probability and Combinatorics Quickfire Quiz

TrueInterview

July 18, 2026 · 2 min read

Requirements

A non-interactive, timed multiple-choice or numeric-entry quiz hosted on HackerRank or a comparable service. It contains 15 questions with 30 minutes allowed in total. There is no programming; the material is entirely probability and combinatorics that can be worked out on paper.

Examples

The following questions are quoted exactly from a single reported attempt.

Six red cards and four black cards are randomly shuffled. What is the expected number of neighboring card pairs whose colors match?
Five people independently and uniformly choose one of two locations. What is the expected population of the location with fewer people?
X and Y are independent random real values selected uniformly from 0 through 100. What is the probability that their difference in absolute value is between 50 and 80?

The other 12 questions belong to the same probability-puzzle category, covering expected values, conditional probability, elementary combinatorics, and geometric probability in the unit square or cube.

Notes

For the expected number of adjacent pairs with matching colors, introduce indicator variables. Evaluate P(same color) separately for each of the 9 neighboring positions, then add the results. With 6 red and 4 black cards, a pair matches with probability (C(6,2) + C(4,2)) / C(10,2) = (15 + 6) / 45 = 21/45 = 7/15. Thus the expected number is 9 * 7/15 = 21/5 = 4.2.
For the expected number at the smaller of two locations when 5 people choose between them, use symmetry and the partitions (0,5), (1,4), and (2,3). Their binomial probabilities on either side are 1/32, 5/32, 10/32, or 1/16, 5/16, 10/16 after combining the symmetric outcomes. The smaller population is min(k, 5-k), so the weighted calculation is $2 \cdot (0 \cdot 1/32 + 1 \cdot 5/32 + 2 \cdot 10/32) = 2 \cdot 25/32 = 50/32 = 25/16 \approx 1.5625$ .
For a uniformly selected pair satisfying 50 <= |X - Y| <= 80, consider the portion of the [0,100] x [0,100] square lying between the two lines |X - Y| = 50 and |X - Y| = 80. Find the area where |X - Y| <= 80 by subtracting the two corner triangles from the square, then subtract the corresponding area for |X - Y| <= 50. Divide the band’s area by 10000 to obtain the probability.
Manage the pace at roughly 2 minutes per question. Move on and come back rather than spending too long on one item. Most problems favor quickly establishing an indicator sum or symmetry calculation instead of carrying out a lengthy Bayesian derivation.
A frequent mistake is to list every possible case rather than apply linearity through indicator variables. Expected-value problems generally simplify almost immediately with linearity.

Preparation

Practice the indicator-variable method until using it requires no deliberate thought. It applies to about half of the probability questions commonly seen in Citadel or quantitative-trading interviews.
Work through a standard collection of probability puzzles from beginning to end, such as 50 Challenging Problems in Probability, Heard on the Street, or A Practical Guide to Quantitative Finance Interviews. Rapid online assessments draw extensively from this established set of problems.
Rehearse symmetry techniques and drawing conditional-probability trees by hand at a 90-second tempo. This shortcut is consistently faster than rebuilding the argument from first principles.
Review geometric probability in [0,1]^2 and [0,1]^3, including uniform pair or triple differences, probabilities involving the triangle inequality, and Buffon-like integrations. These topics recur often.

Examples

The following questions are quoted exactly from a single reported attempt.

Six red cards and four black cards are randomly shuffled. What is the expected number of neighboring card pairs whose colors match?

Five people independently and uniformly choose one of two locations. What is the expected population of the location with fewer people?

X and Y are independent random real values selected uniformly from 0 through 100. What is the probability that their difference in absolute value is between 50 and 80?

Notes

For the expected number of adjacent pairs with matching colors, introduce indicator variables. Evaluate P(same color) separately for each of the 9 neighboring positions, then add the results. With 6 red and 4 black cards, a pair matches with probability (C(6,2) + C(4,2)) / C(10,2) = (15 + 6) / 45 = 21/45 = 7/15. Thus the expected number is 9 * 7/15 = 21/5 = 4.2.

For the expected number at the smaller of two locations when 5 people choose between them, use symmetry and the partitions (0,5), (1,4), and (2,3). Their binomial probabilities on either side are 1/32, 5/32, 10/32, or 1/16, 5/16, 10/16 after combining the symmetric outcomes. The smaller population is min(k, 5-k), so the weighted calculation is

2 \cdot (0 \cdot 1/32 + 1 \cdot 5/32 + 2 \cdot 10/32) = 2 \cdot 25/32 = 50/32 = 25/16 \approx 1.5625

For a uniformly selected pair satisfying 50 <= |X - Y| <= 80, consider the portion of the [0,100] x [0,100] square lying between the two lines |X - Y| = 50 and |X - Y| = 80. Find the area where |X - Y| <= 80 by subtracting the two corner triangles from the square, then subtract the corresponding area for |X - Y| <= 50. Divide the band’s area by 10000 to obtain the probability.

Manage the pace at roughly 2 minutes per question. Move on and come back rather than spending too long on one item. Most problems favor quickly establishing an indicator sum or symmetry calculation instead of carrying out a lengthy Bayesian derivation.

A frequent mistake is to list every possible case rather than apply linearity through indicator variables. Expected-value problems generally simplify almost immediately with linearity.

Preparation

Practice the indicator-variable method until using it requires no deliberate thought. It applies to about half of the probability questions commonly seen in Citadel or quantitative-trading interviews.

Work through a standard collection of probability puzzles from beginning to end, such as 50 Challenging Problems in Probability, Heard on the Street, or A Practical Guide to Quantitative Finance Interviews. Rapid online assessments draw extensively from this established set of problems.

Rehearse symmetry techniques and drawing conditional-probability trees by hand at a 90-second tempo. This shortcut is consistently faster than rebuilding the argument from first principles.

Review geometric probability in [0,1]^2 and [0,1]^3, including uniform pair or triple differences, probabilities involving the triangle inequality, and Buffon-like integrations. These topics recur often.