The first and best answer I have received is from Dave Montana. Congratulations, Dave! His answer appears below. A correct answer was also supplied by Marshall Brinn
The Ants on a Stick applet was viewed by 54 distinct ip addresses. The first viewing was 7 minutes after I announced it and 12 more the first evening including one at 1:37 am, a customer of mpowercom.net and probably in or near Glendora, CA.
The answer is (100!) / ((50!)(50!)(2100)), which is approximately
8%. This is the probability that Alice has exactly 50 ants on each
side of her at the start.
Proof: Let each ant be carrying a different color token at the beginning.
If they exchange tokens each time they meet, the tokens travel in a
continuous path, changing directions only when they hit the wall.
Therefore, each token at the end of one minute will end up in a position
on the stick reflected about the center of the stick. Alice's initial token
will end up back in the middle of the stick. The question then becomes
whether Alice will be holding that token. While the ants remain in the
same order, the tokens reverse order because of the reflection.
Therefore, Alice will be holding the token only if she were in the middle
of all the ants at the beginning (and therefore remains in the same
position upon reversing the order).
To derive the formula for the probability, observe that each ant placed
on the stick is to the left or right of Alice with probability 0.5, essentially
a coin flip. So, the probability that Alice is the middle of the 101 ants is
equivalent to the probability that a coin flipped 100 times lands on heads
50 times. To calculate this probability, observe that the number of
possible positive outcomes is 100 choose 50, which is (100!)/((50!)(50!)).
The total number of outcomes is 2100. So, the probability is the
ratio of these two quantities.