Is this lot drawing fair?

Sorry for a stupid question, but it is bugging me a lot.

Let's say there are $30$ classmates in my class and one of us has to clean the classroom. No one wants to do that. So we decided to draw a lot - thirty pieces of paper in a hat, one of which is with "X" on it. The one who draws "X" has to do the cleaning. Each one starts to draw...

Is this kind of lot drawing fair or not fair? It looks to me like the first one's chances to get an "X" are equal to $1/29$, while the second one's chances would be equal either to $1/28$ (in case the first one didn't draw an "X") or zero $0/29 = 0$ (in case the first one drew an "X"). However, neither $1/28$, nor $0/29$ is equal to $1/29$.

P(first one draws $X) = \dfrac1{30}$
P(first one doesn't draw $X) = \dfrac{29}{30}$

The possibility of the second one drawing $X$ arises if and only if the first one doesn't, so
P(second one draws $X) = \dfrac{29}{30}\cdot\dfrac1{29} = \dfrac1{30}$ again !

You will find, if you continue in the same way, that each one will have the same probability, $\dfrac1{30}$, and you needn't compute at all !

Imagine all papers arranged randomly in a line.
The $X$ is as likely to be in the first place as the last (or any other)

It's fair. The easiest way to see that is to imagine the people drawing their papers but not looking until all have got theirs.

Each person's chance is $1/30$.

In your way of thinking about the problem the first person's chance of getting the X (and having to clean) is 1/30 (not 1/29 as you state in your question - those are the odds, not the probability).

If the first person doesn't get the X then the second person's chance is 1/29, but it wasn't 1/29 from the start - it only became 1/29 once you knew the first person didn't get the X.

To find the true probability for the second person you have to reason backwards from the 1/29 chance he sees, to take into account the fact that he wouldn't have to draw at all when the first person gets the X. That happens 1/30 of the time, so the second person has to draw just 29 times out of 30. So the probability (from the start) that the second person draws the X is $$ \frac{29}{30} \times \frac{1}{29} = \frac{1}{30} $$

This is the essential algebra in many of the other correct answers.

If by chance none of the first 29 people gets the X then the chance that the last person does is 1/1, which is no surprise. But it wasn't 1/1 from the start, of course. You can make the algebra show you that too.

Edit: To address the OP's question about odds.

Odds and probabilities are two different ways to express the same mathematics. In your problem the first person to draw has a probability of 1/30 to draw the X. That means he (or she) cleans 1 time out of 30. His odds are 1:29, which you read aloud as "1 to 29". The odds mean he cleans 1 time for each 29 times he doesn't. Odds aren't usually written as fractions, because that often leads to confusion.

You can figure out problems like yours working with probabilities or with odds, but it's easier with probabilities as all the answers show.

Your logical error about the odds is this.

If the first person doesn't get the X then the second person's odds are indeed just 1:28, as you say, but they weren't 1:28 from the start - they only became 1:28 once you knew the first person didn't get the X.

To find the true odds at the start for the second person you have to reason backwards from the 1:28 odds in his draw to take into account the fact that he wouldn't have to draw at all when the first person gets the X. That's what using probability instead of odds lets you do, as above.

You might be able to see the error in your logic if you imagine there are three people instead of 30. The first person has odds of 1:2. If he doesn't get the X then the second person has odds of 1:1 (even chance, which makes sense) but that isn't the case from the start since he actually has to draw only 2/3 of the time.

The draw is fair. It just seems counter-intuiutive because of the way you are thinking of it.

Suppose that all thirty pieces of paper were handed out simultaneously and all at once the papers were flipped and the person with the $X$ would clean. This would seem fair, right?

Well, drawing a paper one at a time is virtually the same thing, the only difference is that the draw stops once the $X$ is revealed. Everyone still has a $\frac{1}{30}$ chance of getting the $X$.