A supreme court puzzle

This timely Riddler puzzle is about filling supreme court vacancies…

Imagine that U.S. Supreme Court nominees are only confirmed if the same party holds the presidency and the Senate. What is the expected number of vacancies on the bench in the long run?

You can assume the following:

  • You start with an empty, nine-person bench.
  • There are two parties, and each has a 50 percent chance of winning the presidency and a 50 percent chance of winning the Senate in each election.
  • The outcomes of Senate elections and presidential elections are independent.
  • The length of time for which a justice serves is uniformly distributed between zero and 40 years.

Here is my solution:
[Show Solution]

2 thoughts on “A supreme court puzzle”

  1. For your distribution of the number of vacancies, I don’t think the X_i are independent (e.g. if control switches from divided to aligned, all vacancies fill at the same time).

    1. I think you’re right. Doesn’t matter for finding the expected number of vacancies, but it does matter for computing their distribution. This problem turned out to be so much trickier than it seemed at first!

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