Re: Just-Dice.com : Invest in 1% House Edge Dice Game

Is it somehow clear?

Yes, and it's simpler than that: every bet has an expected return of 99%, so the sum of all bets has an expected return of 99%, however you bet.

I'm not suggesting that Rick has found a winning strategy; I don't believe there is one.

I was just asking whether anyone can find the probability of his strategy busting, given that he can afford to double N times.

To make it more abstract:

a. start with x=0
b. add 1 to x (with probability p) or subtract 1 from x (with probability (1-p))
c. if x is negative, stop - you've won
d. if x is N, stop - you've lost
e. go to b.

You will eventually stop at c. (won) or d. (lose). What is the probability of stopping at d? Show your working!

It's easy enough to run a simulation and find an approximation for a particular p and N. I'm looking for an analytical answer.

Shouldn't that be "add 1/p to x"? Otherwise you induce drift and the mean hitting time will no longer be infinite.

Nah, dooglus' model is a random walk starting at the origin where you step left (winning a bet) with probability 1-p and step right (losing a bet) with probability p. The walk ends if you end up 1 step left of the origin (= you won a roll at your base bet, at which point the strategy resets) or if you end up at N steps left of the origin (you've busted).