Colorful balls puzzle

This Riddler puzzle about an interesting game involving picking colored balls out of a box. How long will the game last?

You play a game with four balls: One ball is red, one is blue, one is green and one is yellow. They are placed in a box. You draw a ball out of the box at random and note its color. Without replacing the first ball, you draw a second ball and then paint it to match the color of the first. Replace both balls, and repeat the process. The game ends when all four balls have become the same color. What is the expected number of turns to finish the game?

Extra credit: What if there are more balls and more colors?

Here is my solution to the first part (four balls):
[Show Solution]

Here is my solution to the general case with $N$ balls:
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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]

The lonesome king

This Riddler puzzle is about a random elimination game. Will someone remain at the end, or will everyone be eliminated?

In the first round, every subject simultaneously chooses a random other subject on the green. (It’s possible, of course, that some subjects will be chosen by more than one other subject.) Everybody chosen is eliminated. In each successive round, the subjects who are still in contention simultaneously choose a random remaining subject, and again everybody chosen is eliminated. If there is eventually exactly one subject remaining at the end of a round, he or she wins and heads straight to the castle for fêting. However, it’s also possible that everybody could be eliminated in the last round, in which case nobody wins and the king remains alone. If the kingdom has a population of 56,000 (not including the king), is it more likely that a prince or princess will be crowned or that nobody will win? How does the answer change for a kingdom of arbitrary size?

Here is my solution:
[Show Solution]