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Version 1 of the Ant Problem

            Assume you have 101 ants on a yardstick, randomly distributed on the stick with no two ants at exactly the same place; one of the ants, we'll call her Alice, is exactly at the midpoint.  Every ant moves at the rate of one yard a minute, and each has her own random starting direction, either left or right.  When two ants bump into one another, they switch directions without slowing down; an ant will also change direction when she reaches either end of the yardstick.

            Question:  What is the probability that at the end of one minute, Alice is back where she started at the midpoint?

Version 2 of the Ant Problem

            Instead of a yardstick, we now have a hoop with circumference of one yard.  Instead of 101 ants, we have only 100, again with random and unique starting locations and random individual starting directions, either clockwise or counter-clockwise.  We still have all ants moving at the speed of one yard per minute, the bouncing off rule. We still have one favorite ant named Alice.

            Question:  What is the probability that at the end of one minute, Alice is back where she started?

Hints for the Ant Problems

1.  In both cases, think about what happens if there is only one ant.  Then think about two ants.  Then think about three ants.  With any luck, you may see a pattern that lets you answer the bigger problems.

2.  As you might guess, given the main topic of this website, the answers have something to do with the binomial coefficients.

3.  If you need more hints, or the whole answer, e-mail us.