There is a line of length $$$T$$$ ($$$1 \leq T \leq 10^8$$$), divided into $$$T$$$ consecutive empty slots of length 1. There are N blocks ($$$1 \leq N \leq 10^5$$$). The i-th block has length $$$x_i$$$ ($$$1 \leq x_i \leq 10^3$$$). The sum of all block lengths is not greater than $$$T$$$. The $$$N$$$ blocks have to be placed over the line. This is done sequentially and randomly, and there is a chance that the process may fail: for the i-th block, we choose one of the $$$T$$$ slots uniformly, and then we find the first empty slot from there to the right. If no empty slot is found, we fail. If we find an empty slot, then we check if we can place the block (we need at least $$$x_i$$$ consecutive empty slots starting from the empty slot we just found). If there is not enough space to place the block, we fail. Otherwise, we place the block, move on to the next and repeat. If we never fail and manage to place all the blocks, we succeed. What is the probability of succeeding? (link to original problem from NAC 2020: https://open.kattis.com/problems/allkill)