In a room, there are 5 people who always tell the truth and 4 people who always lie. You ask one person if they are a truth-teller or a liar, and they say "I am a truth-teller." What is the probability that this person is actually a truth-teller?
The probability that this person is a truth-teller is 5/9.

Here's the reasoning:

1. We know that there are 5 truth-tellers and 4 liars in the room.
2. If you asked a truth-teller whether they are a truth-teller or a liar, they would honestly say "I am a truth-teller."
3. However, if you asked a liar the same question, they would also say "I am a truth-teller" because they always lie.
4. So, regardless of whom you ask, you will always get the answer "I am a truth-teller."
5. To find the probability that the person you asked is actually a truth-teller, we need to consider the total number of possible outcomes (in this case, the total number of people) and the number of favorable outcomes (the number of truth-tellers).
6. There are 9 people in total (5 truth-tellers + 4 liars), and 5 of them are truth-tellers.
7. Therefore, the probability that the person you asked is a truth-teller is 5/9.