Probability Generating Functions in Interviews
A practical guide to probability generating functions in quant interviews, focused on dice sums, coefficients, counting intuition, and when the method is overkill.
Advanced candidates encountering sum distributions and combinatorics in quant interviews.
Coefficient intuition
A generating function packages counts into polynomial coefficients. Multiplying polynomials combines independent choices, and the coefficient of a power gives the count for that total.
Dice sums
For one die, x + x^2 + ... + x^6 encodes possible rolls. For two dice, squaring that polynomial gives coefficients for each possible sum.
Concrete example
The coefficient of x^8 in (x + x^2 + ... + x^6)^2 is 5, matching the five ordered pairs that sum to 8.
When useful
Generating functions are useful when repeated independent choices create a sum distribution and direct listing is tedious. They can clarify structure for dice, counts, and constrained sums.
When overkill
For simple two-dice or one-step questions, direct counting may be faster and clearer. In interviews, use generating functions only if you can explain them simply.
Common mistakes
Candidates sometimes introduce generating functions to look advanced and then lose the interviewer. The method should simplify the problem, not hide it.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.