Another interesting ant probability problem There are n ants of weight 1 uniformly distributed on a stick. numbered as 1, 2, 3, ..., n from left to right. At the beginning, each ant is moving in random directions with the same speed. When two ants meet each together, they will merge together, become a bigger ant with weight being the sum of ants' original weights. Then, one of the following two cases happens.
1) if they have different weights, the merged ant will head into the same direction as the heavier one.
2) if they have the same weight, the merged ant will head to left.
The speeds of ants remain constant and the ants will turn around when they reach the end of the stick. They keep moving until only one ant is left.
The first question: what is the probability that ant n survives?
The second question: when n is big enough, which ant has the highest probability of survival?
 A: The first thing that needs to be answered is can the ants move in any direction or only along the stick ? 
If suppose this is true then , 
Part 1 : 
Lets take n=1 
We have the total movements possible as 2 (left+right) and possible pass cases is 2 hence here probability is 1.
Lets take n=2 or 3 , if the ants turn around after reaching the end , there will be no nth ant left and by left I mean u cannot look at the ant on the stick and say that's the nth ant . They all will be eventually a big ant .
This by rule of mathematical induction n=k , will hold true if u place one more ant . Thus only possibility is with n=1 . So if we are finding the probability of value of n for which the ant survives , the answer can be 1/n . 
If suppose ant is moving in any random direction and its not restricted to the stick assuming the stick is on the floor with walls on either side at the ends, then the possibility for nth ant to survive is only if it moves up or down . No angular movement as it will eventually turn around from the end and meet some ant moving at (180-nth's angle) in degrees value. Hence total pass cases is 2 . Now to find total , each ant can move at some angle which lies between 0 and 360 . Assuming it only moves at an integral angle, we have total ways as 360^n . Hence probability here will be : 2/360^n .
Part 2: 
The possibilities here is no matter how big n is except for n=1 , the ants will eventually merge but if it is random in any direction not just along the stick , the answer would be same for all n as the up and down motion is same for each . 
PS The part which is still a bit confusing is u haven't mentioned anything regarding the weights . A possible usage of this can be if say n=3 and ants 1 and 2 merge and start moving left , the stick will be unbalanced and they will fall
PS : This is my understanding of the problem based on some assumptions as the problem statement isn't proving all the required information . 
