I’m trying to convince someone that if you’re going to gamble to win, the best way to do it is to bet your whole stake at once in a game with a low house edge. For this particular case let’s consider the following set up. *

You have a bankroll of $1,000 that you are willing to loose.

You want to walk away with $2,000.

You’ve decided to play craps and bet on the pass line, which has a house edge of 1.414%.
4a. You decide to double your money by betting all $1,000 at once. Your chance of winning works out to 49.2929… %.
4b. You decide that you can’t stand the sight of loosing $1,000 all at once. Instead you decide to try betting $25 at a time, each individual bet with the same odds as above.

What are the odds that you will reach $2,000 with the 4b method before you go broke?

I understand that the best bet is not to gamble. The question here is how much better is the second best method compared to trying to grind it out?

I just read a book (John Haigh’s Taking Chances) which covered this kind of problem in detail.

Per one of the equations (on page 197) the chance of success (i.e. getting to $2000 without going broke first) is:

((x^n)-1)/((x^L)-1)

, where n is the initial stake (in units of the size of your bet - so 1000/25=40), L is the goal (again in units of the bet size (so 2000/25=80), and x is the difference between that probability that you win an individual bet, and the probability that you lose an individual bet.

Assuming that the goal is to double your initial bankroll, and assuming that all outcomes that do not reach that goal are all equivalent, and assuming that the “best” method is the one that has the highest chance of success, then your conclusion is correct. But all of those assumptions could go other ways, leading to other conclusions. For instance, if you want to maximize your odds of winning something, but don’t care how much, and don’t care what happens if you don’t win, then you want a Martingale.

Well I assume you’d want a submartingale (a stochastic process in which the expected value increases rather than remains the same), but casinos don’t offer them – or martingales. They only offer supermartingales (or submartingales from their viewpoint).