1D Random Walk of people meeting

[frinny913]

Problem: Three friends do a 1D Random Walk with p=q starting at 0. Are they guaranteed to all meet again somewhere? So all the friends are starting at 0 at the same time and we want to know if they are guaranteed to meet again anywhere in the walk.

Hint: if they do meet, which places are likely?

So, so far I have found through calculations that they aren't guaranteed to all meet at zero and that four friends aren't guaranteed to all meet somewhere. The problem with the three people meeting somewhere random is that when i try to use the same technique i used to find if they all meet at 0 or to find if 4 people meet somewhere doesn't work for this one. I also don't really understand how to use his hint.
I have a feeling that the probability of three people meeting anywhere is greater than the probability of three people meeting at 0 but that doesn't really get me far either.

 

[Valkarie]

Answer: The probability of any three friends meeting at any point of their 1D Random Walk with p=q starting at 0 is not guaranteed, but it is highly likely. The most likely points for them to meet are the origin (0), their respective maximums, and their respective minimums.