0

This is a follow-up on a previously asked question: Random Ants problem

Quoting the problem:

500 ants are randomly put on a 1-foot string (independent uniform distribution for each ant between 0 and 1). Each ant randomly moves toward on end of the string (equal probability to the left or the right) at constant speed of 1 foot/minute until it falls of a t one end of the string. Also assume that the size of the ant is infinitely small. When two ants collide head-on, they both immediately change directions and keep on moving at 1 foot/min. What is the expected time for all ants to fall off the string?

My question: Does the answer change if each ant moves to the left with a probability $p$, and to the right with a probability $1-p$?

Going by the approach of this answer, the expected sizes of sets $S_l$ and $S_r$ will change, but this won't affect the maximum of $S_l\cup S_r$. So, I think that the answer won't change. Is this correct?

And if this is indeed correct, it would be nice to know any alternative ways to deduce the result.

muser
  • 368
  • The ants are so tiny that I can't tell them apart. When you say two ants change directions, to me it looks like they just pass through each other. So the last ant to fall off is the one farthest from the end it sets out for. With that many ants, it's almost guaranteed that there will be one very far from one end who heads toward the other end, so takes nearly a full minute to drop off. And the value of $p$ has no effect on this. – Gerry Myerson May 30 '22 at 06:19
  • @Gerry I agree with this reasoning, but the original question was to find the (exact) value of the expected time. In the original question, this value turns out to be $500/501$ minutes (which indeed is close to your conclusion). Can we have such quantitative result for this question too? – muser May 30 '22 at 06:31
  • If each ant is located randomly and oriented randomly, I think that amounts to each ant being a uniformly random distance from its doom, so you're asking for the expected value of the maximum of $500$ numbers distributed uniformly at random between zero and one. Should be a simple exercise to come up with the answer, and a well-known one at that. – Gerry Myerson May 30 '22 at 06:37
  • So, if I'm understanding correctly, the answer remains the same ($500/501$ minutes). – muser May 31 '22 at 03:33
  • I haven't taken the time to figure it out, but it shouldn't be hard to do the calculation or to find some place where it has been done for you. – Gerry Myerson May 31 '22 at 03:52
  • 1
    Yes. The PDF of max of $500$ uniformly distributed random variables (in $[0,1]$) is $p_X(x)=500x^{499}$, and thus the expected value is $E[X]=500/501$. – muser May 31 '22 at 04:03

0 Answers0