Project Euler/901
From charlesreid1
Problem Statement
A driller drills for water. At each iteration the driller chooses a depth d (a positive real number), drills to this depth and then checks if water was found. If so, the process terminates. Otherwise, a new depth is chosen and a new drilling starts from the ground level in a new location nearby.
Drilling to depth d takes exactly d hours. The groundwater depth is constant in the relevant area and its distribution is known to be an exponential random variable with expected value of 1. In other words, the probability that the groundwater is deeper than d is exp(-d)
Assuming an optimal strategy, find the minimal expected drilling time in hours required to find water. Give your answer rounded to 9 places after the decimal point.
Convos
https://grok.com/chat/d00d2362-5f18-42e7-88ae-b669691d0656