Sandbox
It is interesting to discuss the Martingales that Jordan wrote about in terms of the modern version of Martingales. In current probability a Martingale is a stochastic process s(0),s(1),s(2),.... with the expected value of (s(n+1) = s(n), ie. if it be considered a fair game. Jordan's coin tossing problem corresponds to two such Martingales, one when you have a finite amount of money and another when you have and unlimited amount of money.
Given a Martingale stopping time as a random variable T giving the time top the game. It is assumed that the decision to stop at time T = n depends only on outcome at or before time n In In other words no clairvoience.
The Father of Martinales Joeseph Doob proved that If S(0),(1),(2) ,.. is a bounded martingale and T is a stopping time, then the expected valure of S(T) = S(0). In other words you can expect to make money on a bounded Martingale.
Additional Reading
Slate piece on martingales, expected value, and the bailout.