i know. i was pointing out how the erroneous view of ‘about 1 in 2048 flips’ made no sense to me as it cannot explain why a change of perspective changes the odds.
i never said that. the idea was that 11 bad flips will only happen about 1 in 2048 flips. certainly it can happen on the first flip, or it can happen 18 times within those 2048 flips. i did take ‘about 1 in 2048 flips’ literally and i did expect it to average out in the long run, (e.g. 180 streaks in 368640 flips) and that was my mistake.
Mathochist, what do you want explained about Martingales? If you want to know how they work, then I can’t help you, because they don’t. If you want to know how they’re supposed to work, I can help.
The idea is that I walk into the casino with a big stack of money, and I want to win a dollar. OK, first I bet $1 on a red/black bet, or some other similar nearly 50-50 bet. If I win, I walk away happy with my dollar. If I lose, I’m down $1, so now I need to win $2 to reach my goal. So now I bet $2 on red. If I win, I’m up a buck, and walk away happy. If I lose, then I’m now down $3. So then I bet $4 on red, and…
Lather, rinse, repeat, until I win, run out of money, or hit the table limit. You can see that I have a very good chance of walking away from the table with a buck more than I walked in with. But there’s a chance that I’ll lose all of my bets in a row, and since my bets are increasing in size exponentially, if that happens I’ll lose big. There are also trivial variations on this which will result in a large win, rather than a small one, but they require that you raise your bets even faster, so the catastrophe (if it comes) is even more severe.
Note that the house expects to get exactly as much money from me using this method, as if I bet each dollar individually, or if I went all in on one spin, or gambled my money in any other way. Nothing can change the house’s advantage.
There is no mathematical theory behind Martingalle. There’s nothing more than what has been stated by several posters. In the end, you change your basic gamble from a good chance of a small loss to a small chance of a very big loss. The overall expected gain/loss does not change. Nothing else changes. That’s it.
Not really a theory, but here’s how I came to understand Martingale. I read many years ago (I think I was in high school, which ended for me 26 years ago), of a situation where two people bet each other on the flip of a coin. The bet amount changes each time, and is always equal to half of the amount that Player A currently has. They play a bunch of times, and at the end it turns out that Player A won just as many times as Player B. So who now has more money? The surprising result is that Player B has come out way ahead.
I took this factoid and tried to fashion my own betting system. I figured that I should be Player B, so I would imagine that the casino had a certain amount of money, and I would bet a fixed percentage of what he has, and I would clean up! Not having the SDMB to query at the time, I simulated the situation with my fancy TI-30 calculator, and came up with a system that is essentially the Martingale.
Later, when I got my new HP-41C calculator, I played around with simulations, and found that it would pretty much work, but there was always this problem of a catastrophic loss. Then I realized that you can’t play a game where each bet has a negative expectation, and work it into a system with a positive expectation. I feel like I learned much more having gone through this myself.
Well this I know is wrong just from the fact that as I said before, it was due to be covered in the second semester of my graduate statistics course. Yes, I understand that it’s not a sound betting system, but there does exist a mathematical concept called a “martingale” which is a generalization of this one object.
Further, I’ll bet that that theory is what proves rigorously that the expected loss is exactly the same.
I attempted the Martingale system on some casino software and here’s my results:
Starting with $3000 I was placing $50 on red. If I lost I would double my bet till I won. Once I won I started over with the initial $50 bet. I was intent on working my way up to $5000.
I had some incredibly lucky streaks (such as 12 red spins in a row that got me $600). However, at about $4700 I hit a small streak of black/green (6 spins) that completely cost me my bankroll.
Like others have posted, Martingale system = Many small wins vs. one big loss
I’ve also found that red or black streaks are very common in roulette. When passing through a casino take note of the “spin history” electric signs at the tables to see what kind of streaks there are.
I’ve read, and I don’t have a cite to back me up on this, that adding those electronic signs to the roulette table is responsible for a resurgence in popularity in the game. Seems that folks walking by a game would notice that maybe red had shown up 5 or 6 times in a row and would decide to plunk down a bet.
The Martingale system can be counted on to work, but only if there is no table limit and no limit to your bank roll. As Hampshire noted, he only lost once he hit his limit.
Not true. The expectation remains zero. Over an infinity of trials, expectation is still zero. There is no theoretical situation in which the Martingale progression can be said to work.
Isn’t there only a single case where it will fail though – where you have an infinity of consecutive losses? In every other case, given an infinite bankroll and limit, you could stop as soon as you get that one win. Further, couldn’t you argue that an infinity of consecutive losses is impossible if the game has a stated non-zero probability of winning?
So then you have an infinitesimal chance of losing. But what if you do? Then you lose an infinite bankroll. The expectation, then, is 1 - infinity/infinity. We have here an indeterminant form, so the only sensible thing to do is to look at the limit of our expectation as our bankroll approaches infinity. And for any finite bankroll, the expectation is zero (or less, if the game we’re playing isn’t completely fair), so the limit is really easy to take.