Put 10 balls in the jar then randomly take 1 out. Do it infinitely many times. Find the probability of resulting in an empty jar The original discussion (in Chinese): https://www.zhihu.com/question/58702489
The original problem was from an probability theory exam. The problem is translated as:

Assume an infinitely large jar and infinitely many balls $b_1, b_2, b_3, \dots$ outside the jar. At 1 minute before 12:00 a.m., put $b_1\sim b_{10}$ in the jar, then randomly take 1 ball out from the jar.(assume that these operations take no time) At 1/2 minute before 12:00 a.m., put $b_{11}\sim b_{20}$ in the jar, then randomly take 1 ball out from the jar. At 1/4 minute before 12:00 a.m., put $b_{21}\sim b_{30}$ in the jar, then randomly take 1 ball out from the jar. And so on. Question: what is the probability of the jar containing no ball at 12:00 a.m.?

There are two simple but contradictory answers:


*

*Zero. The number of balls always increases by each step, so it can never be zero.

*One. For each ball $b_i$, the probability of being inside the jar at 12:00 a.m. is $p_i=\frac{9}{10}\cdot\frac{18}{19}\cdot\frac{27}{28}\cdot\dots =0$. Let $X_i$ be a 0-1 random variable indicating whether the ball is in the jar, we can obtain the expectation of number of balls in the jar, which is $E[\sum_i X_i]=\sum_i E[X_i]=\sum_i p_i=0$


From the original discussion, it seems that this paradox is not trivial at all. Most people argue that the problem was not well-defined, but some disagree (like this: https://zhuanlan.zhihu.com/p/26460906). What is your opinion?
 A: It is unclear to me what would happen in the physical world.  But this particular problem can be rewritten (formulated? interpreted?) as a purely mathematical problem as follows.   
Let $x = (x_1,x_2,\ldots)$ be a sequence of natural numbers with 
$x_n\in \{ 1,\ldots, 9n+1\}$.  
We define a sequence $(A_0, A_1,A_2,\ldots)$ of a finite subsets of $\{ 1,2,\ldots\}$.   (Depending  on $x$, so they should be written as $A_0^x$, etc.) 
$A_1 = \{1,\ldots, 10\}$, and $B_1 = A_1 \setminus \{k_1\}$, where $k_1$ is the $x_1$-th element of $A_1$. 
$A_2= B_1 \cup \{ 10n-9,\ldots, 10n\}$, and $B_2 = A_2 \setminus \{k_2\}$, 
where $k_2$ is the $x_2$-th element of $A_2$.
Etc.  Each $A_n$ has $9n+1$ elements, and $B_n$ has $9n$ elements. 
There is a naturally defined limit of these sets $A_n$ (although it is unclear to me whether this limit has anything to do with the "real world". 
For any such sequence $(A_n^x: n =1,2,\ldots)$, the "lower limit" $\bigcap _n \bigcup_{k\ge n} A_k^x$ and the "upper limit" $\bigcup_n \bigcap_{k\ge n} A_k^x$  are the same set, which I will call $A^x_\infty$. 
Now the question can be rephrased as follows:


*

*What do we know about the set $S$ of all $x$ such that $A^x_\infty = \emptyset$?

*What do we know about the set of all $x$ such that $A^x_\infty$ is your favorite property here?  (Is finite. Has positive density. Contains only prime numbers. etc.)


You already argued that set $S$ has measure 1, if we use the usual product measure (with equidistribution on all factors) on the set of all possible $x$.  It seems also clear to me that this set is co-meager (=residual). 
In conclusion: there is no paradox, but you have to explain what "randomly" means. I gave one possible interpretation. 
