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 $1/29$.     

Actually, it looks like its fair to me.
The probability that person 1 chooses an "X" is $\frac{1}{30}$, since only one of the 30 lots has an "X" on it.
The probability that person 2 chooses an "X" is $\frac{29}{30}*\frac{1}{29} =\frac{1}{30} $, since Person 1 must have not chosen a lot with an "X" on it.
The probability that person 3 chooses an "X" is $\frac{29}{30}*\frac{28}{29}*\frac{1}{28} =\frac{1}{30}$
You can show that this is true for every successive person, and to further prove this, we can look at the 30th person.
The probability that person 30 chooses an "X" is the probability that everybody else did not choose an "X", which is $\frac{29!}{30!} = \frac{1}{30}$
So it is indeed fair for every person.
