I've come across this interesting little problem. I think that solving it is just a matter of thinking about it in the right way, but I'm stumped so far. Could someone please give me a hint in the right direction?
The Problem
Suppose you have $n$ logs (pieces of wood, not logarithms) of heights $X_1,\ldots,X_n$. $X_1,\ldots,X_n$ are independent, identically distributed random variables. Their common distribution is the exponential distribution with parameter $\lambda > 0$. The logs are to be stacked in two piles, one-on-top-of-the-other, according to the following procedure. First log 1 is stacked in a pile, then log 2 is stacked in a pile, then log 3, and so on. The i-th log is always stacked in the shortest of the two piles. That is, having stacked logs 1 to i-1, log i is stacked in the shortest of the two piles. What is the probability that log i ends up on top of the tallest pile?
What I've Tried
I've tried the problem with $n = 3$ and $n=4$, but couldn't see what to do. The main idea I've come up with so far is to consider $ \mathbb{P}(i \in A | \sum_{a \in A} X_a > \sum_{b \in B} X_b ) $ where $A \cup B = \{1,\ldots,n\}$. But I can't figure out how to work with it.
Thanks.