Shag: **septimus **gave the mathematical formal answer as you accept. What **Chronos **says here is (IMO) the key to a qualitative understanding.
As one continues to play the vigorish slowly wears down the bankroll. So as time progresses, the needed excess of wins over losses in the next little while becomes ever larger. And hence ever less likely. e.g. Tossing a fair coin 10 times and coming up 7 to 3 is pretty plausible. Tossing a fair coin 1000 times and it coming up 700 to 300 beggars belief; that isn’t going to happen.
We’ve done other statistical games like this: you either win real soon, or you dig yourself a hole that’s unclimbably deep.
Imagine the same game but with different odds. Each time you draw:
you have 0.0001% chance of winning $1
you have 99.9999% chance of loosing $1 million
Suppose that you already lost $10 billions. Play 10 billion times and there is a non-zero chance (0.000001^10000000000) that you will win all bets and come back positive. So you do that and in all likeliness you loose about 10 billion millions!
Now you need to bet 10 billion million times and win all bets in order to come back positive. There is a non-zero chance (0.000001^10000000000000000) that it happens, so you play 10 billion million times and of course it is very likely that you loose about 10 000 billion billions, etc…
Now it should be intuitive. Infinite time gives you a non-zero probability of scoring an (infinitely big) series of wins. But infinite time makes you loose money faster than it gives you a chance of getting it back.
You will also note that, in my example, the chance of getting back positive (0.000001^10000000000 + 0.000001^10000000000000000 etc) will not converge to 1 even if you re-iterate indefinitely.
I’m thinking that the fastest method of getting out would be to double your bet, every round.
You then have a 48.65% chance of getting out, every single hand
I believe I understand the point- while it is statistically possible with infinite spins to put together an arbitrarily long run of wins to climb out of a very deep hole, it is simultaneously statistically true that your very deep hole will get even deeper faster than you are (likely) going to be able to put that run together. And so on…
The thing that tickles my brain, is this sounds just a teensy bit like someone’s discussion of Hawking Radiation, and why it happens so rarely. I’m not sure who to credit (possibly Stranger, Chronos, or Asymp Fat?), but it was in an explanation of the “negative” mass falling into the black hole, a week or so ago. Interestingly, the term “debt” was used, and of course so was the word “hole”.
Is there any relationship or analogy here? Or is my fevered brain just picking up on superficial similarities?
Well, there’s a relationship between everything in mathematics… but no more of one here than anywhere else. One big difference between this and Hawking radiation is that Hawking radiation has a clear endgame: Assuming that you can keep any more pesky matter from falling in (which isn’t too hard; just wait a paltry few quadrillion years for the Universe to cool down enough), any event of Hawking radiation will decrease the mass of the hole, and any decrease in mass of the hole will increase the rate of Hawking radiation, so it gets faster with time, not slower.
I understood the question, but his setup includes a house percentage loss (a 0/00 hit) regardless of which way the bettor goes on the bet. House percentages are why it’s essentially impossible to win in the long run in casino gambling; the PC or rake will nibble away at the winning percentage and drive the player’s stake to zero. For most players, this is in no more than a few hundred rounds or bets, even when very large sums are involved. (And especially in casino poker, where there is a loss from the pot on every single hand.)
So while the OP’s question inspires some interesting statistical gymnastics, the gist here that the player has lost once his losses reach a certain fairly trivial level (-$100 or fifty bets) is the most important finding. Yes, you could be down half a quadrillion dollars and a universe-negating run of luck could take your string of wins back to the quadrillion+one level, but the $2 bet level largely negates the size of your stake here. Once the player is down more than about ten to twelve winning bets, the odds - and that house rake - are going to keep him down.
It’s an aside, mostly, to the basic question but it’s not irrelevant. The HP means that once in every 30 or so spins, there’s going to be an irreparable loss not related to the two-valued bet. NIbble, nibble, nibble… to death.
In an even more general sense, both John Scarne and Darwin Ortiz conclude that the only way to win at roulette is to never get within fifty feet of the table.
