TL;DR
Negative Binomial: Waiting for the k-th Success: A canonical quantitative trading interview question at intermediate difficulty. Commonly asked at Jane Street, SIG, Two Sigma, Citadel, HRT.
By Valenke Exam Prep Team·Last updated 2026-06-01
intermediateExpected Value & Variance
Negative Binomial: Waiting for the k-th Success
Asked at: Jane Street, SIG, Two Sigma, Citadel, HRT
Problem
A biased coin has . You flip until you get the 5th head.
(a) What is the expected number of flips?
(b) What is the variance?
(c) What is the probability that the 5th head occurs on exactly the 15th flip?
(d) A gambler pays per flip and receives upon the 5th head. Is this a good bet?
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