Does martingale really works?

[dooglus]

Earlier in the thread I left a comment about my own particular conundrum on this topic and I'm honestly a little surprised that no one followed up with me given the smart statistical heads that hand around this forum.  Here's my issue, we know that, as has been said above, in these sorts of games each trial is independent.  However, I think we also know that increasingly long streaks are increasingly unlikely.  How do we resolve these two (both valid, I think) intuitions?

I responded here:

This, exactly.   Because each roll is an independent event you never know if you're in the middle of a 100 loss streak.  However, some part of me thinks it also has to be true that rolling consecutive losses into infinity is also highly improbably and that longer and longer streaks of losses have to be more and more improbable.  So I always have a hard time rectifiying these two facts: the independence of each roll and the increasing unlikelyhood of increasingly large streaks.

Well, look at it this way:

Streaks of length 20 are twice as common as streaks of length 21, which are twice as common as streaks of length 22, etc.

And streaks of length 20 are exactly as common as streaks of length 21 or more (assuming a 50/50 game).

To see that last point, suppose streaks of length exactly 20 occur with probability p. Streaks of length 21 will occur with half that probability (p/2), Length 22 with probability p/4, and so on. So the probability of a streak of length 21 or more is p/2 + p/4 + p/8 + ... = p(1/2 + 1/4 + 1/8 + ...) = p.

By pre-rolling until you hit 20 losses in a row, you've probably had to make a million or so rolls. And now you're wondering whether the streak will end at 20, or go on to be more than 20. Well as we've just seen, the probability of those two things is the same. probability p it will end at 20, and probability p*(1/2+1/4+1/8+...) = p that it goes to length 21 or more.

Here's another way of looking at it:

Losing 10 in a row is about a 1-in-1000 chance.
Losing 10 more in a row after that is another 1-in-1000 chance.
Those are independent, so losing 20 in a row is a 1-in-a-million chance.

Where's the paradox here?

Or, to make it even simpler instead of two streaks of 10 making a streak of 20, consider two 'streaks' of 1 making a 2:

The first bet loses 1-in-2 times
The second bet loses 1-in-2 times
They both lose 1-in-4 times.

We had a bad case of this "I just lost 10 in a row so I must be due to win soon" syndrome in Just-Dice last night:

[Bardman]

Earlier in the thread I left a comment about my own particular conundrum on this topic and I'm honestly a little surprised that no one followed up with me given the smart statistical heads that hand around this forum.  Here's my issue, we know that, as has been said above, in these sorts of games each trial is independent.  However, I think we also know that increasingly long streaks are increasingly unlikely.  How do we resolve these two (both valid, I think) intuitions?

I responded here:

This, exactly.   Because each roll is an independent event you never know if you're in the middle of a 100 loss streak.  However, some part of me thinks it also has to be true that rolling consecutive losses into infinity is also highly improbably and that longer and longer streaks of losses have to be more and more improbable.  So I always have a hard time rectifiying these two facts: the independence of each roll and the increasing unlikelyhood of increasingly large streaks.

Well, look at it this way:

Streaks of length 20 are twice as common as streaks of length 21, which are twice as common as streaks of length 22, etc.

And streaks of length 20 are exactly as common as streaks of length 21 or more (assuming a 50/50 game).

To see that last point, suppose streaks of length exactly 20 occur with probability p. Streaks of length 21 will occur with half that probability (p/2), Length 22 with probability p/4, and so on. So the probability of a streak of length 21 or more is p/2 + p/4 + p/8 + ... = p(1/2 + 1/4 + 1/8 + ...) = p.

By pre-rolling until you hit 20 losses in a row, you've probably had to make a million or so rolls. And now you're wondering whether the streak will end at 20, or go on to be more than 20. Well as we've just seen, the probability of those two things is the same. probability p it will end at 20, and probability p*(1/2+1/4+1/8+...) = p that it goes to length 21 or more.

Here's another way of looking at it:

Losing 10 in a row is about a 1-in-1000 chance.
Losing 10 more in a row after that is another 1-in-1000 chance.
Those are independent, so losing 20 in a row is a 1-in-a-million chance.

Where's the paradox here?

Or, to make it even simpler instead of two streaks of 10 making a streak of 20, consider two 'streaks' of 1 making a 2:

The first bet loses 1-in-2 times
The second bet loses 1-in-2 times
They both lose 1-in-4 times.

We had a bad case of this "I just lost 10 in a row so I must be due to win soon" syndrome in Just-Dice last night:

Jesus he went crazy, he could have started betting 1 btc not that much

[tspacepilot]

Aha, sorry I missed your reply, dooglus.  You basically seem to be emphasizing the same point as Bardman, but in different words.  I think I see better now the insight you guys are providing.  It's good to have strong statistical minds around like you guys to point the more easily confused of us towards TRUTH! 

[dooglus]

Why do you think that multiplying the probabilities of two independent events causes the events to stop being independent?

I don't argue with your examples, I know what you say perfectly well myself. But you give examples which are not the same as successive rolls. If you pick a queen of hearts from a deck, will the probability of picking another queen of hearts from the same deck (provided there is only one queen of hearts in the deck) be of the same kind as the probability of picking next a king of spades?

I guess you need to define what you mean by a "kind" of probability.

I'm using this definition: "probability: n. the extent to which something is probable; the likelihood of something happening or being the case."

It's a number between 0 (impossible) and 1 (certain).

I don't see any scope for "kinds" in there.

The probability of picking the queen of hearts out of a deck from which you already removed the only queen of hearts is zero.
The probability of picking a king of spades from a 51 card deck which contains a single king of spades is non-zero (it's 1/51).

Those are different numbers. 0 isn't equal to 1/51.
But they're just different numbers.

Maybe you can say that 0 is a different "kind" of number than 1/51, but it wouldn't be clear what you meant. Maybe zero is a different kind of number than all the non-zero numbers, but I'm not sure that it helps to make that distinction. And I'm not sure what picking a 2nd card without replacement has to do with anything here anyway, since we're generally talking about independent events when looking at martingale bets.

You say that picking a single card in turn from two different decks isn't the same as rolling a number on two different dice? How is it different? The two card selections are independent of each other, and the two dice rolls are independent of each other.

[dooglus]

And more specifically, if we have 20 losing bets in a row, then the probability of the next 20 losing bets will be the same, right? But in this case these probabilities are different from the probability of 40 losses in a row, which seems paradoxical. Thus, in a sense, previous outcomes affect the probability of future rolls...

I don't see a paradox here:

If A and B are independent events then P(A then B) = P(A) times P(B).

A is "lose first 20 bets"
B is "lose next 20 bets"
P(A) is equal to P(B)

P(losing 20 bets) * P(losing 20 more bets) = P(losing 40 bets)

I see your point but it is not strictly the same. We know the probability of each loss in a row of 40 successive losses (so we know the area of a square beforehand), but with each roll made the probabilities for rolls yet to be made change, right? Therefore, it turns out that the outcome of each previous roll changes the probability of the next roll...

I don't find a fallacy in this logic, do you?

Before you start, it's 2^40 against that you will lose 40 times in a row.
If you lose your first bet, you only have 39 more to lose to reach your 'goal', so it's 2^39 against that you will lose 40 times in a row.
You've already 'made progress' and so have doubled your chances of 'success'.

Maybe reading about Bayes' theorem will help:



A = "we lose all 40 bets"
B = "we already lost the first 20"

The probability that (we lose all 40 bets), given that (we already lost the first 20) is:

P(A|B) = P(B|A).P(A)/P(B)
    = (probability that we lose the first 20 given that we lost all 40) * (probability that we lose all 40) / (probability that we lose the first 20)
    = 1 * 0.5^40 / 0.5^20
    = 0.5^20

We had a bad case of this "I just lost 10 in a row so I must be due to win soon" syndrome in Just-Dice last night:

Jesus he went crazy, he could have started betting 1 btc not that much

He was betting CLAMs, not BTC, so it's about 200 times less bad than it looks.

[deisik]

Why do you think that multiplying the probabilities of two independent events causes the events to stop being independent?

I don't argue with your examples, I know what you say perfectly well myself. But you give examples which are not the same as successive rolls. If you pick a queen of hearts from a deck, will the probability of picking another queen of hearts from the same deck (provided there is only one queen of hearts in the deck) be of the same kind as the probability of picking next a king of spades?

I guess you need to define what you mean by a "kind" of probability.

I'm using this definition: "probability: n. the extent to which something is probable; the likelihood of something happening or being the case."

It's a number between 0 (impossible) and 1 (certain).

I don't see any scope for "kinds" in there.

The probability of picking the queen of hearts out of a deck from which you already removed the only queen of hearts is zero.
The probability of picking a king of spades from a 51 card deck which contains a single king of spades is non-zero (it's 1/51).

Those are different numbers. 0 isn't equal to 1/51.
But they're just different numbers.

Maybe you can say that 0 is a different "kind" of number than 1/51, but it wouldn't be clear what you meant. Maybe zero is a different kind of number than all the non-zero numbers, but I'm not sure that it helps to make that distinction. And I'm not sure what picking a 2nd card without replacement has to do with anything here anyway, since we're generally talking about independent events when looking at martingale bets.

You say that picking a single card in turn from two different decks isn't the same as rolling a number on two different dice? How is it different? The two card selections are independent of each other, and the two dice rolls are independent of each other.

Actually, I wasn't going to expand more on this (was ready to call it a day really), but since you asked, hereby I reply

No, I didn't mean just numbers. 0 is not equal to 1/51, but sheer numbers are in no case enough to say that the first probability (0) is being of some other kind than the second (1/51). I meant something different which is related to randomness of this world (or lack thereof). Say, we have a time machine and can rewind events to a given previous state in the past which is absolutely the same (therefore we wouldn't even know that we moved back in time). So we have two choices in respect to future rolls, that is the outcomes will be absolutely the same (as in the previous future) and the outcomes will be different. We could also "witness" a situation where outcomes for one set of events will be the same (even with non-zero probabilities) while outcomes for another set of events will be different (future changers). In this manner, we should necessarily distinguish between the probabilities of these sets. That's what I meant by different types of probabilities...

[dooglus]

No, I didn't mean just numbers. 0 is not equal to 1/51, but sheer numbers are in no case enough to say that the first probability (0) is being of some other kind than the second (1/51). I meant something different which is related to randomness of this world (or lack thereof). Say, we have a time machine and can rewind events to a given previous state in the past which is absolutely the same (therefore we wouldn't even know that we moved back in time). So we have two choices in respect to future rolls, that is the outcomes will be absolutely the same (as in the previous future) and the outcomes will be different. We could also "witness" a situation where outcomes for one set of events will be the same (even with non-zero probabilities) while outcomes for another set of events will be different (future changers). In this manner, we should necessarily distinguish between the probabilities of these sets. That's what I meant by different types of probabilities...

OK, I see what you mean.

Most provably fair dice games are a good example of where the outcomes would be the same.

On Just-Dice, your client and server seed stay the same until you ask for them to be changed. Once they are set, your entire list of future dice rolls is pre-determined.

I could look on the server right now and see what your next roll will be. 12.3456 say. So there's a 1.0 probability that you roll 12.3456 next, and a 0.0 probability that you roll anything else next. But since you don't know that, you think the probabilities are different. As far as you're concerned 12.3456 is a one-in-a-million chance, just like any other number between 0 and 100 to 4 decimal places would be.

It seems like betting on a site like that is similar to betting on the result of a sporting event that already happened, but which you don't know the result of. You can watch a recording of the event, and it feels just as if it was happening live. The fact that the outcome is already decided doesn't affect the bet so long as you and the person you're betting against don't know the result already.

[tspacepilot]

No, I didn't mean just numbers. 0 is not equal to 1/51, but sheer numbers are in no case enough to say that the first probability (0) is being of some other kind than the second (1/51). I meant something different which is related to randomness of this world (or lack thereof). Say, we have a time machine and can rewind events to a given previous state in the past which is absolutely the same (therefore we wouldn't even know that we moved back in time). So we have two choices in respect to future rolls, that is the outcomes will be absolutely the same (as in the previous future) and the outcomes will be different. We could also "witness" a situation where outcomes for one set of events will be the same (even with non-zero probabilities) while outcomes for another set of events will be different (future changers). In this manner, we should necessarily distinguish between the probabilities of these sets. That's what I meant by different types of probabilities...

OK, I see what you mean.

Most provably fair dice games are a good example of where the outcomes would be the same.

On Just-Dice, your client and server seed stay the same until you ask for them to be changed. Once they are set, your entire list of future dice rolls is pre-determined.

I could look on the server right now and see what your next roll will be. 12.3456 say. So there's a 1.0 probability that you roll 12.3456 next, and a 0.0 probability that you roll anything else next. But since you don't know that, you think the probabilities are different. As far as you're concerned 12.3456 is a one-in-a-million chance, just like any other number between 0 and 100 to 4 decimal places would be.

It seems like betting on a site like that is similar to betting on the result of a sporting event that already happened, but which you don't know the result of. You can watch a recording of the event, and it feels just as if it was happening live. The fact that the outcome is already decided doesn't affect the bet so long as you and the person you're betting against don't know the result already.

I have an anecdote which is very similar.  Although I wasn't betting, I used to go to a sports bar in the afternoon which replayed UEFA champions league games in the afternoons so that those who couldn't watch during the live event (which happens at midday in my timezone) could have a change to see the matches.  There was a very strict rule there about no discussing results.  If the bar staff heard that you were discussing the result of a game which was about to be replayed, they would throw you out.  My friends and I used to purposely disconnect from any news sources on that day to ensure that we were in "the bubble" of not knowing what had happened.

I can't forget one day when I thought I had overheard someone in front of me saying the result before the game began.  I, of course, didn't tell anyone because even though "my bubble had been burst" there was no need to do this damage to others.  Wow, what happened was that the result I expected to occur from what I thought I overheard became more and more unlikely.  Into stoppage time it seemed that one of the teams was about to score 3 goals or something like this.  In the end, whatever I thought I overheard obviously was either wrong or was regarding something else because a completely different result occurred from what I thought I heard.  At this moment I realized how the perception that you know the outcome feels quite empowering but can also be misinformed.  I dunno that I have anything really to say about this other than it was a very interesting moment for me.

[deisik]

It seems like betting on a site like that is similar to betting on the result of a sporting event that already happened, but which you don't know the result of. You can watch a recording of the event, and it feels just as if it was happening live. The fact that the outcome is already decided doesn't affect the bet so long as you and the person you're betting against don't know the result already.

There is another phenomenon which is the reverse (in a sense) of what you say here, and you most certainly know about it. It is called a hindsight bias. In respect to rolls, it reveals itself in the tendency to consider the outcome of a roll as having been fully predictable beforehand. Apparently, there is no objective ground for being able to predict it, but you still feel frustrated at yourself that you have failed to see how "obvious" the outcome of that roll has actually been...

[itod]

It seems like betting on a site like that is similar to betting on the result of a sporting event that already happened, but which you don't know the result of. You can watch a recording of the event, and it feels just as if it was happening live. The fact that the outcome is already decided doesn't affect the bet so long as you and the person you're betting against don't know the result already.

There is another phenomenon which is the reverse (in a sense) of what you say here, and you most certainly know about it. It is called a hindsight bias. In respect to rolls, it reveals itself in the tendency to consider the outcome of a roll as having been fully predictable beforehand. Apparently, there is no objective ground for being able to predict it, but you still feel frustrated at yourself that you have failed to see how "obvious" the outcome of that roll has actually been...

I never had a feeling that the outcome of the roll was obvious after it happened. Always seems it could have gotten the other way, win or lose, although I know it's predetermined considering the way rolls are calculated.

[deisik]

It seems like betting on a site like that is similar to betting on the result of a sporting event that already happened, but which you don't know the result of. You can watch a recording of the event, and it feels just as if it was happening live. The fact that the outcome is already decided doesn't affect the bet so long as you and the person you're betting against don't know the result already.

There is another phenomenon which is the reverse (in a sense) of what you say here, and you most certainly know about it. It is called a hindsight bias. In respect to rolls, it reveals itself in the tendency to consider the outcome of a roll as having been fully predictable beforehand. Apparently, there is no objective ground for being able to predict it, but you still feel frustrated at yourself that you have failed to see how "obvious" the outcome of that roll has actually been...

I never had a feeling that the outcome of the roll was obvious after it happened. Always seems it could have gotten the other way, win or lose, although I know it's predetermined considering the way rolls are calculated.

Well, when I lose (and lose big at that), I always have a feeling that they know in advance (or even deliberately play against me when stakes are high) what my decision will be. So, knowing this, I try at second-guessing as if I really knew what the outcome would be. But, of course, failing miserably in the end whether I'm second-guessing, third-guessing, or whatever...

[dooglus]

Well, when I lose (and lose big at that), I always have a feeling that they know in advance (or even deliberately play against me when stakes are high) what my decision will be. So, knowing this, I try at second-guessing as if I really knew what the outcome would be. But, of course, failing miserably in the end whether I'm second-guessing, third-guessing, or whatever...

I've seen it happen a few times were people are playing with a 1% chance of winning, they'll hit a bad losing streak, lose like 400 times in a row, and then give up and change back to 49.5% chance of winning on the very bet that would have been a win for them at 1%. It's uncanny how lucky or unlucky some people can be. If it wasn't for provable fairness I'm sure we'd see an awful lot more accusations that the dice sites are cheating.

[deisik]

Well, when I lose (and lose big at that), I always have a feeling that they know in advance (or even deliberately play against me when stakes are high) what my decision will be. So, knowing this, I try at second-guessing as if I really knew what the outcome would be. But, of course, failing miserably in the end whether I'm second-guessing, third-guessing, or whatever...

I've seen it happen a few times were people are playing with a 1% chance of winning, they'll hit a bad losing streak, lose like 400 times in a row, and then give up and change back to 49.5% chance of winning on the very bet that would have been a win for them at 1%. It's uncanny how lucky or unlucky some people can be. If it wasn't for provable fairness I'm sure we'd see an awful lot more accusations that the dice sites are cheating.

Absolutely the same happens in trading. You buy some investments, then the price goes down, you wait (or may even average down), the price goes further down, you think WTF... Then you you start panicking, and just as you close your position ("cut your losses short", yeah), the price all of a sudden reverts...

[pecintapantai]

now my turn to ask does martingale absolutely not works?
no, sometimes its works.

[CoinCidental]

now my turn to ask does martingale absolutely not works?
no, sometimes its works.

way to get your post count up ,congrats son

[shoaibBTC]

well martingale really doesnt work if you have a small bankroll and start with big bets

[freeyourmind]

I haven't done the math yet, but I think making periodic withdrawals while winning is like having insurance.  At the end of the day, it's very possible and likely that a series of future bets will be continuous losses until zero.  The question is whether you can withdraw more than your initial principal before that happens.

From my own experience, I have been able to double my bitcoin more often than I hit a losing streak and lose everything.  Definitely not fail safe, but that is a bit less risky to me than regular martingale.

[dooglus]

From my own experience, I have been able to double my bitcoin more often than I hit a losing streak and lose everything.  Definitely not fail safe, but that is a bit less risky to me than regular martingale.

Any strategy that doubles more often than it busts is a winning strategy. You've almost certainly either been lucky or have mis-counted.

[freeyourmind]

From my own experience, I have been able to double my bitcoin more often than I hit a losing streak and lose everything.  Definitely not fail safe, but that is a bit less risky to me than regular martingale.

Any strategy that doubles more often than it busts is a winning strategy. You've almost certainly either been lucky or have mis-counted.

Well it hasn't been a winning strategy for me, because when I hit that eventual losing streak, it doesn't matter how many times I've doubled my funds, because it goes back to zero.

To your point, perhaps it has involved luck, but I don't think it's very odd for people to double their money before losing it all when following martingale.

[dooglus]

To your point, perhaps it has involved luck, but I don't think it's very odd for people to double their money before losing it all when following martingale.

No, it's not unusual. It just has to happen less than half the time on average, due to the house edge. Otherwise this simple strategy would be profitable:

* start with X; martingale until doubling; quit

If you successfully double Y% of the time, your expected profit would be:

    (Y*X - (100-Y)*X) / 100
    = X(2Y - 100) / 100

which is positive exactly when 2Y > 100, ie. when Y > 50%