This one is rather easy to pose, and can be difficult to solve, even with a proof sitting right in front of you. The result is a bit surprising as well, and gives some insight into the structure of the integers and prime numbers.
Given two random integers, what is the probability that they have no common factors besides 1?
or, how it is usually posed:
What is the probability of two random integers being coprime?
Where “coprime” means “the greatest common divisor is one”.
Adapted from Parade magazine, April 15th, 1990:
There are 1,000 tenants and 1,000 apartments. The first tenant opens every door. The second closes every other door. The third tenant goes to every third door, closing it if it is open and opening it if it is closed. The fourth tenant goes to every third door, closing it if it is open and opening it if it is closed. This continues with every tenant until the 1,000th tenant opens the 1,000th door. Which doors are open?
Bonus question: Can you be sure ahead of time that the 1,000th tenant closes the 1,000th door rather than opens it? How can you be sure?
Another problem from Parade (same column, same year) about doors:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?