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 $1/29$.     

You've identified three outcomes for the second person:


*

*Person 2 doesn't draw

*Person 2 draws and doesn't get the X

*Person 2 draws and gets the X


and you're interested in the probability that the last of the three outcomes happens.
You then divided the outcomes into two categories:


*

*Person 2 doesn't draw

*Person 2 draws


You've even (nearly) correctly computed the probabilities


*

*Assuming person 2 doesn't draw, he has probability 0 of getting the X

*Assuming person 2 does draw, he has probability 1/29 of getting the X


But where you went off the rails is that you forgot your original goal was "What is the probability he gets the X?", and neither of the calculations you did were to compute that. They're similar-sounding questions, but similar does not mean the same. This is especially true for questions about probability, counting, and statistics, where sloppy reasoning often leads to wildly incorrect results.
If you are to use the work you've done, the problem now is how to correctly combine the two probabilities. This can be done with an additional piece of information:


*

*The probability that person 2 doesn't draw is 1/30

*The probability that person 2 does draw is 29/30


And thus we can compute


*

*The proportion of samples that person 2 doesn't draw is 1/30


*

*Of those, the proportion where person 2 gets the X is 0


*

*The total proportion of this case is $\frac{1}{30} \cdot 0 = 0$



*The proportion of samples that person 2 does draw is 29/30


*

*Of those, the proportion where person 2 gets the X is 1/29


*

*The total proportion of this case is $\frac{29}{30} \cdot \frac{1}{29} = \frac{1}{30}$




and then adding up all of the different cases gives $0 + \frac{1}{30} = \frac{1}{30}$.
