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#91
07-18-2005, 02:27 PM
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Re: A Less Obvious Martingale Fallacy
[ QUOTE ]
They should be able to understand that if the Martingaler continues to play, and never quits when he loses, then they are almost certain not to win. [/ QUOTE ] "Almost" certain isn't certain. The less likely the sequence of straight losses, the greater the penalty, in proportion (plus a bit). If your single hypothetical gamber were replaced by countless simultaneous gamblers on independent tables, that's pretty much the same thing as one gambler gambling for endless duration, isn't it? But at no time could you expect the sum total of those gamblers' results to be showing a net profit. When you start a sequence, you don't know for certain in advance that it won't be the worst sequence ever--or even that it will eventually win. Just as you don't know that there won't be one player in that countless field of players who won't lose every single bet. And if he does, he will lose far, far more than his positive counterpart who might be winning every single bet. 
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#92
07-18-2005, 02:36 PM
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Re: A Less Obvious Martingale Fallacy
[ QUOTE ]
...and so on. Are you trying to say that at some point: Bet 1. Lose. -1 Bet 2. Lose. -3 ... Bet x. Win. <not +1> [/ QUOTE ] No, what they are saying is that there will be a series which does not end with a W. In addition, this series will occur with a 100% probability. I was saying that the opposite will occur as well. That is you will have a series WWW...W through infinite. The problem is, that if you ignore the aspect of infinity +1 = infinity and pretend that they are finite. What will happen is that the losing streak will be infinitely larger than the winning streak. Not only that, but because of the house edge, you will be "more likely" to experience the losing streak than the winning streak, even though they will both actually occur with 100% frequency. LOL Looking at this kind of makes me think of trying to use Newtonian physics on Quantum particles. hmm...
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#93
07-18-2005, 02:41 PM
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Re: A Less Obvious Martingale Fallacy
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And if he does, he will lose far, far more than his positive counterpart who might be winning every single bet. [/ QUOTE ] Yes, I get it now. I just explained it to my 'partner' here. I think I was trying to have it both ways in terms of infinity. Still, the question remains, I guess, that since the math concepts say that infinity +/- 1 = infinity, then the whole problem is pointless, because no one can actually have any edge at all, even on a 0 probability event. One quick question I guess should answer it for me. Solve for x: infinity - infinity = x
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#94
07-18-2005, 02:53 PM
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Re: A Less Obvious Martingale Fallacy
Whoa, you're saying that in an infinite series of rolls, he will necessarily have an infinitely long losing streak?
I'm pretty sure that's not true. The Martingaler's bankroll will never change. But we can still measure his net gain/loss. Every single series that isn't an infinitely long losing streak ends with +1. The money he bets is immaterial. It might as well not even be money - it is Monopoly money, or bananas, or whatever. What the Martingaler is interested in is his net gain/loss, which 1 unit per series completed. EDIT: Basically, the whole thing comes down to whether or not there can be a series of infinite losses. I need a probability theorist to answer that. Common sense and my once somewhat-formidable knowledge of math both say that in an infinite series of bets of any kind, there is a 0% chance that there will be an infinitely long chain of only one result.
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#95
07-18-2005, 03:04 PM
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Re: Infinite Hotel and Infinite Bus
If the casino had an infinite number of patrons and each one left as soon as he won one series, then each leaving patron would leave with $1. But at any point in time, the casino will be ahead. This shouldn't be too surprising. After all, making something out of nothing is easy with infinities.
A bus carrying an infinite number of passengers arrives at a hotel with an infinite number of rooms. Problem is, every room in the hotel is occupied. So the hotel manager announces over the PA that if you are in room n, you should immediately move to room 2n. Suddenly, all the odd rooms are empty and he can accomodate the passengers on the bus.
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#96
07-18-2005, 03:14 PM
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Re: A Less Obvious Martingale Fallacy
[ QUOTE ]
Whoa, you're saying that in an infinite series of rolls, he will necessarily have an infinitely long losing streak? [/ QUOTE ] Exactly [ QUOTE ] I'm pretty sure that's not true. [/ QUOTE ] It is true. Maurile started a table earlier in this thread that showed that your probability of winning each successive bet in the series grows geometrically, but never reaches 100%. Of course, there is also a series of infinite wins. How can this not be true? [ QUOTE ] Basically, the whole thing comes down to whether or not there can be a series of infinite losses. I need a probability theorist to answer that. Common sense and my once somewhat-formidable knowledge of math both say that in an infinite series of bets of any kind, there is a 0% chance that there will be an infinitely long chain of only one result. [/ QUOTE ] Where's Pzhon when you need him/her?
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#97
07-18-2005, 03:49 PM
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Re: A Less Obvious Martingale Fallacy
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What's weak is your refusal to look at it from the Casino's perspective. PairTheBoard [/ QUOTE ] The thing is that I can choose to stop after I win my series - but the casino can't make me stop at a time ideal for them (at least we are assuming this, I think).
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#99
07-18-2005, 03:57 PM
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Re: A Less Obvious Martingale Fallacy
[ QUOTE ]
[ QUOTE ] The thing is that I can choose to stop after I win my series [/ QUOTE ] No you can't. In this game the Martingaler plays forever, [/ QUOTE ] Ok, but isn't the whole point of the theory that it is broken into series and you can stop after a win? I'm not sure it is supposed to be valid if you aren't given that ability.
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#100
07-18-2005, 04:11 PM
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Re: A Less Obvious Martingale Fallacy
I'm pretty sure that an infinitely long losing streak doesn't exist. Wouldn't that imply that the EV of an even money bet is negative? That doesn't seem to make sense.
While I haven't thought this through completely, it would seem that the Martingaler can expect a win just through even money bets, but only because he can stop when he chooses. And that is the edge he has over the house, who has to play on demand.
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