Credit for this problem goes to The Riddler. Thanks!

Highest Card

This is a slighly modified version of this question from the Riddler.

You will be given N numbers in sequence. Each number is drawn from a uniform random distribution from 0 to 1. After each number, you have the option to stop. Your goal is to stop after receiving the highest number out of all N numbers. If you do that, you win, otherwise, you lose.

What is the strategy that optimizes your chances of winning (and what is your probability of winning)?