I've encountered this question and found two "contradictory" answers to it. of course they are not really contradictory, but I'm having some trouble explaining why not.
Say there is a frog standing at $0 \in \Bbb Z$ at time $0$. At each step it jumps either one leaf to the left or to the right.
I have a flashlight that can light one leaf $\in \Bbb Z$ at each step. Is there a way I can make sure that eventually I'll find the frog?
First answer: No
I can not find the frog for sure, since whatever my strategy is, eventually what I'm really doing is building a sequence $a_n \in \Bbb Z$, and it is possible that the frog's moves happens to be "always to the left, unless this leaf is litten in the next step". If this happens to be the case, of course I'll never find the frog.
Second answer: Yes
At time $n$, there are $n+1$ possible leaves where the frog might be. Say I choose one randomly(uniformly) and lit it up.
$P($Frog never found$) \le P($Frog not found till step $n) = \displaystyle\prod_{i=1}^n P($Frog not found in step $i) = $
$$= \frac 12 \cdot \frac 23 \cdot \frac 34 \cdot ... \cdot \frac{n-1}{n} = \frac 1n$$
So, $P($Frog never found$) \le \frac 1n$ for every $n$, then it has to be $0$. The frog must be found.
Now, I understand that this two answers are not really contradictory - it's just a different approach to what it means to "Make sure the frog will be found". But I'm having a really hard time to explain the difference to people that know only basic probability. It is especially confusing because the question itself has no probability in it. I'd be really glad to hear your input about this and how you'd explain this to others.
If I wanted to explain this "paradox" intuitively, I would say that in your first explanation, the frog knows what you are doing with the flashlight, and this allows him to avoid it. In the second explanation, the frog's moves are random, and so he is eventually (with probability one) bound to run into your light. Even though the possibility of him happening to move in just the right way to avoid it is still there, it is a measure-zero event.