When searching for the optimal stopping point one could achieve a 1/3 chance of finding the best option by using Euler’s number. Simply divide the number of possibilities by Euler’s number, and then after checking that many possibilities one should keep going until a possibility better than any of the currently checked options is found.

For example:

100 possibilities
100 / e ≈ 37
Check 37 possibilities
Continue checking until an option is found which is better than those first 37 possibilities