## Is this bathroom occupied?

After a brief hiatus from Riddling, I’m back! This Riddler problem is about probability and bathroom vacancy.

There is a bathroom in your office building that has only one toilet. There is a small sign stuck to the outside of the door that you can slide from “Vacant” to “Occupied” so that no one else will try the door handle (theoretically) when you are inside. Unfortunately, people often forget to slide the sign to “Occupied” when entering, and they often forget to slide it to “Vacant” when exiting.

Assume that 1/3 of bathroom users don’t notice the sign upon entering or exiting. Therefore, whatever the sign reads before their visit, it still reads the same thing during and after their visit. Another 1/3 of the users notice the sign upon entering and make sure that it says “Occupied” as they enter. However, they forget to slide it to “Vacant” when they exit. The remaining 1/3 of the users are very conscientious: They make sure the sign reads “Occupied” when they enter, and then they slide it to “Vacant” when they exit. Finally, assume that the bathroom is occupied exactly half of the time, all day, every day.

1. If you go to the bathroom and see that the sign on the door reads “Occupied,” what is the probability that the bathroom is actually occupied?
2. If the sign reads “Vacant,” what is the probability that the bathroom actually is vacant?
Extra credit: What happens as the percentage of conscientious bathroom users changes?

Here is how I solved the problem:
[Show Solution]

## 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:
[Show Solution]

## 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:

• 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]