# Expected falling time of all $500$ random ants

The random ant question is asked in this post. I reproduce it below for completeness.

Question: $$500$$ 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?

The question above is equivalent to asking the expected value of the maximum of $$500$$ IID random variables with uniform distribution between $$0$$ and $$1$$.

We know that the expected value of $$\max(X_1,...,X_{500})$$ where $$X_1,...,X_{500}$$ are IID, is $$\frac{500}{501}$$, as shown in another post.

However, the answer given to the random ant question is $$\frac{499}{500},$$ which I fail to decipher.

• On a rather unrelated tangent that may be interesting or completely coincidental, a problem on the silver level USACO (USA Computing Olympiad) competition of December 2019 had a very similar problem if the solution to the problem could be of any help: usaco.org/index.php?page=viewproblem2&cpid=967 – Soham Konar Jan 1 at 3:11
• physics.montana.edu/avorontsov/teaching/problemoftheweek/… is also concluding $N/(N+1)$ as mentioned mean time when initially there were $N$ ants. – dan_fulea Jan 1 at 3:27
• Could you please point out where exactly the answer $\frac{499}{500}$ is given? I could find it neither in the post you linked to, nor in the one that it links to. – joriki Jan 1 at 4:20
• Actually the answer that I posted above comes from 'A practical guide to quantitative finance interview'. The book has exactly the same question above and the answer that it gives is $\frac{499}{500}.$ – Idonknow Jan 1 at 4:27

The guide that gives $$\frac{499}{500}$$ as an answer is wrong. You're right that it should be $$\frac{500}{501}$$.
• What if the ants were moving at a different speed. Instead of $1 foot/min$, what if they were moving at $x foot/min$? Would the answer simply be $\frac{500}{501}x$? – David May 6 at 13:31
• Can't edit my first comment anymore, but I think it's wrong. It should be $\frac{\frac{500}{501}}{x}$ – David May 6 at 13:38