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  • 100 ants are placed randomly on a 1 meter stick.
  • Additionally, Alice ( also an ant ) is placed exactly at the middle.
  • They move at a constant speed of $1$ meter/min and are equally likely to initially be moving left or right.
  • If they collide ( with each other or the ends of the stick ), they bounce off each other.
  • What's the probability Alice returns to the center after exactly one minute ?. $\rule{10cm}{0.25mm}$
  • I'm thinking about the problem by treating collisions as if the ants are passing through each other but switching "identities" -- in this case, the only relevant distinctions are Alice and not-Alice.
  • The event where Alice returns to the center after exactly one minute is equivalent to the event that the ant that starts with the "identity" Alice ending with the "identity" Alice after $1$ minute ( as this is the only ant that will be exactly in the middle at $1$ minute ).
  • I've thought through some smaller cases $\left(N = 1, 2, 3\right)$, and it seems that for this to happen, there must be the same number of ants on both sides of Alice but I haven't been able to formalize this reasoning.
  • Assuming the initial locations the ants are i.i.d standard uniform, I then calculated the probability using the $$ {100 \choose 1/2}\ \mbox{pmf}: \operatorname{P}\left(A\right) = \binom{100}{50}\left(\frac{1}{2}\right)^{100} = 0.08$$.

Could someone help to verify if my approach is correct and/or formalize a solution to this question ?.

Felix Marin
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Richard Zhang
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1 Answers1

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Your intuition is right.

The stick can be represented as an open interval $(-\frac 12, \frac 12)$ and and the position of an can be represented by a number $x$ of this interval. If the position $x$ of an ant is circular shifted (around the center) its new position is $x-1$.

The number of ants left and right of Alice does not change, because their paths do not cross.

After one minute the positon of the ants changed: now all the of ants moved from one half of the stick to the other half of the stick, because as in the original "ants on a stick problem" , you can assume that the ants do not bounce when they meet but pass through each other and so the new position is the starting position circular shifted around the center of the stick. So if they bounce, after one minute exactly one ant is in the center of the stick and the number of ants left of it is now the number of ants that were initially right of the center of the stick and the number right of it is equal to the number of ants that were initially left of the center of the stick. So Alice is at the center of the stick if and only iff the number of ants that were initially left of the center of the stick is equal to the number of ants that were initially right of the center of the stick.

miracle173
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