A two-digit number is written down at random. What is the probability that its two digits add up to 9?

The two-digit numbers run from 10 to 99, so there are 90 of them. Those whose digits add to 9 are 18, 27, 36, 45, 54, 63, 72, 81 and 90 — nine numbers — giving 9/90 = 1/10. Counting 09 as well, which is not a two-digit number, gives 10/90 = 1/9. Taking the pool as 100 numbers gives 9/100, and taking it as the 99 numbers from 1 to 99 gives 9/99 = 1/11.