Basic Probability

[TheBramble]

This is certainly NOT true in general. It is only true for disjoint events.

Of course it's disjoint. ANY two consecutive coins can either be both heads OR both Tails or one of each. Disjoint.

 

[Calinor]

lol, no no Bramble. If you are making the claim that they are the same, you would be required to provide the proof. I don't say they are the same. Why should they be? 🙂 It is not obvious that they are the same. If it is obvious to you, a proof should only take you a second or two.

A proof has already been given that the probability is 7/8. If you can find the error in that, then I think that will be enlightening to the thread. If you can't find the error and you have a different answer, then doesn't that suggest they are possibly not the same problem? Or does your method also give 7/8? It is definitely a tricky question.

Of course it's disjoint. ANY two consecutive coins can either be both heads OR both Tails or one of each. Disjoint.

So are you saying here that the probability of getting 2 heads consecutively in a snapshot of 4 coin tosses is not in any way affected by the possibility that those 4 coin tosses contain 2 consecutive tails...or 3 consecutives...etc.? Is there not a dependency issue here? Disjoint would imply that there is no outcome of positive probability that is in A and in B. Is this the case?

 

[TheBramble]

lol, no no Bramble. If you are making the claim that they are the same, you would be required to provide the proof. I don't say they are the same. Why should they be? 🙂 It is not obvious that they are the same. If it is obvious to you, a proof should only take you a second or two.

This could be fun, but it wont be….

I’m afraid the proof that they aren’t the same is upon you. I’m saying that a random 4 coin toss is EXACTLY the same as any random selection of 4 consecutive 4 coin tosses in an infinite string of random coin tosses. If you do not agree, I would be delighted (and surprised) to receive a convincing proof as to why they are not

A proof has already been given that the probability is 7/8. If you can find the error in that, then I think that will be enlightening to the thread. If you can't find the error and you have a different answer, then doesn't that suggest they are possibly not the same problem? Or does your method also give 7/8? It is definitely a tricky question.

How is it tricky? NO such proof has been offered, just a regurgitation of basic probability theory which I am refuting in this thread.

I’m going to be accused of word games, but for those with the wit to see it, it absolutely isn’t.

So are you saying here that the probability of getting 2 heads consecutively in a snapshot of 4 coin tosses is not in any way affected by the possibility that those 4 coin tosses contain 2 consecutive tails...or 3 consecutives...etc.? Is there not a dependency issue here? Disjoint would imply that there is no outcome of positive probability that is in A and in B. Is this the case?

What you’ve said doesn’t make sense. I’m saying the possibility of getting 2 consecutive heads in 4 coin tosses is not affected by what comes up before or after any two consecutive coin tosses.

While you mull that over:-

What's the probability of getting 2 consecutive Tails in 4 tosses of a coin?

 

[frugi]

The Probability of two consecutive Heads in 4 coin tosses is derived from the formula (Gardner, Berlkamp below)

1-(F(n+2)/(2^n))
where F(n+2) is the (n+2)th Fibonacci number

In our example the value is 0.1875 (3/16).

For the last question asked, the probability of two consecutive Heads OR 2 consecutive Tails the result is a Probabilistic OR which yields 0.375 (6/16).

Good morning Señor B,

It seems to me that pulling a random 4 flips from an infinite string is indeed what we're studying, as this is much the same as flipping a coin 4 times from cold. After all, that's what creates the string in the first place. I have no problem with that.

I've come to this thread late, but if I understand it correctly, you are saying that one of these series of flips is unlikely to look like it 'ought to' in a dusty textbook, because in real life coin flips don't really behave in a neat 50/50 way, e.g astoundingly long runs when one least expects them (even though over time the percentage will be drawn inexorably towards 50% of each). Therefore the 7/8 parroted by all the books is too generous. I have a nasty feeling I'm on the wrong track already?

If by some miracle I'm not, what I don't understand is how the difference between 7/8 [standard theory] and 3/8 [Gardner's] can be so large. If he'd said, say, 84% as opposed to 87.5% - no worries - but this difference is huge. Perhaps I missed a revision after Calinor suggested using a different start to the series? If so I apologise.

If the empirical formula is accounting for the lumpiness, such as 'unusually' long strings of heads, tails or anything that does not 'look like' an honest 1 in 2, which could of course easily mean one occasionally doesn't get a single consecutive HH or TT in, say, a whole five sets of 4 flips, then how does it account for when the lumpiness goes the other way?

What I mean is it is underestimating the standard 7/8 probability by such a large margin it must surely be wrong? I'd be really interested in why this Fib formula is more realistic [appropriate] than the dull textbook one as I just can't wrap my woolly head round it at all.

Though I think I understand where you are coming from. For instance my lucky coin currently has a 66% chance of coming up heads and thus it keeps making me short the euro, which is precisely what a lucky coin should be doing in this ephemeral moment of empirical testing. Yet soon caprice will skew its percentages t'other way for an indeterminate time, as my chairbound tail gets fatter. *shiver* I used to think I understood probability until you started reading about it. 🙂

 

[blackcab]

I'd like to take part in this thread more than I have, but simply haven't got time to read and digest it all so far. Really. So I'll reiterate and summarise my earlier reply:

If the question is "what is the probability of throwing two consecutive heads or two consecutive tails in a single four-coin sequence", then the answer as I understand it is 0.875.

If the question was at all different to that then it would help to have it restated as unambiguously as possible.

 

[frugi]

HHHH THHH HTHH
HHHT TTHH THTT
HHTT TTTH THHT
HTTT THTH HTTH
TTTT HTHT TTHT
HHTH

A brief clarification and summary would be welcome.

For instance, the chance of getting 2 consecutive heads (but not more) OR 2 consecutive tails (but not more) and NOT both would seem to be 6/16 or 3/8, but I don't think this is what you're getting at Mr. B?

 

[TheBramble]

(even though over time the percentage will be drawn inexorably towards 50% of each).

Not necessarily. My contention is that data do not regress to their statistical mean in reality. Much like your lucky EUR shorting coin...

(Therefore the 7/8 parroted by all the books is too generous.

Don't know about too genorous. There appears to be a wide range of disparity across the entire statistical spectrum. Absolutely they are out of whack with reality if current research in this area is accurate.

( I have a nasty feeling I'm on the wrong track already?

No, you're not.

But I need to get back later to do your comments justice. Somebody has just taken the other side of my extremely larger Long EUR position...

 

[TheBramble]

A brief clarification and summary would be welcome.

What is the probability of getting two consecutive heads or two consecutive tails from a 4 coin toss?

This is a 'real life' question, the sort where you don't get to question it for specificity or ask for completely unambiguous phrasing or anything else you can quite validly ask to do in the academic version of such trials.

 

[trendie]

What is the probability of getting two consecutive heads or two consecutive tails from a 4 coin toss?

This is a 'real life' question, the sort where you don't get to question it for specificity or ask for completely unambiguous phrasing or anything else you can quite validly ask to do in the academic version of such trials.

chances are 50%.
I thought this was agreed a couple of pages back.

EDIT: http://www.trade2win.com/boards/708626-post40.html

as per blackcabs post below. if you are only looking for "any" consecutive throw, whether heads or tails, then it is 14/16.

 

[blackcab]

What is the probability of getting two consecutive heads or two consecutive tails from a 4 coin toss?

This is a 'real life' question, the sort where you don't get to question it for specificity or ask for completely unambiguous phrasing or anything else you can quite validly ask to do in the academic version of such trials.

My answer is 0.875 then, the reason being that only 2 of the 16 possible outcomes do not contain any consecutive heads or tails - HTHT and THTH. 14 of the 16 (7/8 or 87.5%) do contain at least one instance of 2 consecutive heads or 2 consecutive tails and so satisfy the requirement.

I don't see what 'real life' vs 'academic' has to do with it, unless you're thinking of coins landing on their side, coins being biased, the thrower being conciously or unconciously biased, quantum fluctuations, etc.

As the question, as might normally be asked, has a simple answer, you seem to be exploring something other than the mathematics of basic probability.

 

[frugi]

This is a 'real life' question, the sort where you don't get to question it for specificity or ask for completely unambiguous phrasing or anything else you can quite validly ask to do in the academic version of such trials.

OK, so if the question is deliberately asked in an ambiguous manner (possible parallel with lack of luxury of specificity encountered in the market?) then the answer can be anything from 3/8 to 7/8. That could be interesting.

But on the other hand the answer depends entirely on how the question is asked, so it is only fair to demand some precision. We can do this together in the pub with real coins and Margaux for the winner - a real-life setting does not suddenly make clarifying the question unreasonable or unrealistic. Blackcab put it better above.

 

[Calinor]

Hmmm, one of my posts got deleted. I think that must have been accidental by the mods, because there was a 30-min flaming session afterwards, lol

Frugi, the 6/18 was incorrect and we corrected the number. It then came to that the probability of 2 H was 0.5 Probability of 2 tails was 0.5. So then Bramble was again asking what was the probability of 2H's or 2T's. Of course you can't just add the probabilities together to get the "OR", aside from it would give you probability 1, it is just wrong to do that here.

And you are on the right track, and 7/8 is the correct answer to that question. It is possible that the question Bramble is asking is a slightly different question which sounds the same but isn't, and that is why the confusion. Maths may not be the answer to all our trading dreams, but maths is not inconsistent. It will not give you 2 different answers to a question like this. So logically, either one answer is wrong, or we have two different questions, hence two different answers. In any case, Bramble, did you also have a formula for the "OR" part? That way we could easily check if we have 2 different answers or not.

I think it is important for us to have the numerical answer, otherwise we could be just wasting time.

 

[TheBramble]

It seems to me that pulling a random 4 flips from an infinite string is indeed what we're studying, as this is much the same as flipping a coin 4 times from cold. After all, that's what creates the string in the first place. I have no problem with that.

Good ho.

I've come to this thread late,

..you’re lucky…

but if I understand it correctly, you are saying that one of these series of flips is unlikely to look like it 'ought to' in a dusty textbook, because in real life coin flips don't really behave in a neat 50/50 way, e.g astoundingly long runs when one least expects them (even though over time the percentage will be drawn inexorably towards 50% of each). Therefore the 7/8 parroted by all the books is too generous. I have a nasty feeling I'm on the wrong track already?

Already responded to this, but yes, you’re right and no, you’re very much on the right track.

If by some miracle I'm not, what I don't understand is how the difference between 7/8 [standard theory] and 3/8 [Gardner's] can be so large. If he'd said, say, 84% as opposed to 87.5% - no worries - but this difference is huge. Perhaps I missed a revision after Calinor suggested using a different start to the series? If so I apologise.

Andy has already covered that error in my calculations.

If the empirical formula is accounting for the lumpiness, such as 'unusually' long strings of heads, tails or anything that does not 'look like' an honest 1 in 2, which could of course easily mean one occasionally doesn't get a single consecutive HH or TT in, say, a whole five sets of 4 flips, then how does it account for when the lumpiness goes the other way?

What I mean is it is underestimating the standard 7/8 probability by such a large margin it must surely be wrong? I'd be really interested in why this Fib formula is more realistic [appropriate] than the dull textbook one as I just can't wrap my woolly head round it at all.

The factor of difference has been covered. It was the difference in textbook classical probability theory and that which I encounter on a daily basis in the markets that led me down this path in the first place. I could not (still can not) reconcile what I see with what I am given as a theoretical example.

Initially concerned that I was mispricing commodity futures options premiums, I instinctively looked for flaws in the standard pricing models – there are many – models and flaws. But it’s more than that. Even when you tighten the focus to purely ATM (which most models are geared to generate toward) and use hypothetically valid IV, there is simply too much ‘play’ in the premium for there to be any correlation with theoretical probabilistic derivations based on historical data. I know traders who trade for their living know more about real prices than do theoretical models, so I am working on the basis that those that do this for a living know the real deal, even, or perhaps especially even, though it deviates from the theoretical models.

I am looking for an empirical basis to substantiate that which I experience on a daily basis in these areas to replace that which have been using from the classical probability school which does not work on a day-to-day basis in the markets.

Though I think I understand where you are coming from. For instance my lucky coin currently has a 66% chance of coming up heads and thus it keeps making me short the euro, which is precisely what a lucky coin should be doing in this ephemeral moment of empirical testing. Yet soon caprice will skew its percentages t'other way for an indeterminate time, as my chairbound tail gets fatter. *shiver* I used to think I understood probability until you started reading about it. 🙂

Not necessarily. In fact, quite unlikely in our non-Gaussian reality. Which is what this thread is all about.
 

I don't see what 'real life' vs 'academic' has to do with it, unless you're thinking of coins landing on their side, coins being biased, the thrower being conciously or unconciously biased, quantum fluctuations, etc.

No, nothing cute like that. Academic perspective can afford to be objective and hold all but one factor constant. Real life does not allow that luxury. You get the ‘whole thing’ and ‘in one go’ and you don’t get to examine it or test it or query it. You just get to make your trading decisions upon it – there & then.

As the question, as might normally be asked, has a simple answer, you seem to be exploring something other than the mathematics of basic probability.

Well, there’s nothing sneaky in the question. I’m just asking those who are interested if, instead of the standard classical probability for this, what you might want to suggest as a basis for exploring the difference between that and what we typically perceive in reality.

This has bearings not just ion options pricing, but on those who use Kelly for optimum position sizing, for those who incorporate risk as a critical factor in their trading and for reviewing our Pw and Pl on a predictive rather than a retrospective basis.

I’ve long held that few traders last long enough to produce a statistically valid set of data upon which to operate with Pw and Pl. Having the ability to do so with fewer data but greater accuracy seems like a good thing.

 

It is possible that the question Bramble is asking is a slightly different question which sounds the same but isn't, and that is why the confusion. Maths may not be the answer to all our trading dreams, but maths is not inconsistent. It will not give you 2 different answers to a question like this.

We’ve corresponded via PM on this and you will be aware I don’t necessarily agree with you on this point. If we took the opposite as a working hypothesis, would we perhaps get a clearer view of what I think we’re after?

I think it is important for us to have the numerical answer, otherwise we could be just wasting time.

Classical probability theory has an answer and it has been given, correctly by a number of poster son this thread. I’m not arguing with that.

I’m asking if you were to toss a coin, right now, 4 times, with the intent of getting either two consecutive heads or two consecutive tails, would your results tally, en masse, with the theoretical ideal?

I don’t think they would.

Look, I’ve been at this problem for some time and it’s not coming easily into wordage that makes much sense. For those that want to play along and either (a) prove me insane or wrong (that’s fine – I can live with that) or (b) help me find what the fluck it is I’m trying to empirically assess, let alone prove, I’#d be grateful for your assistance. I don’t have the brain power to resolve it alone.

I’ll come up with an instance of what alerted me to this issue and we can perhaps work with something more interesting and less ‘obvious’; than coin tosses.

 

[blackcab]

Classical probability theory has an answer and it has been given, correctly by a number of poster son this thread. I’m not arguing with that.

I’m asking if you were to toss a coin, right now, 4 times, with the intent of getting either two consecutive heads or two consecutive tails, would your results tally, en masse, with the theoretical ideal?

I don’t think they would.

No? Any idea why they wouldn't, or why you think they wouldn't? "Classical probability theory" says that you would expect the ratio of heads to tails from single coin tosses to converge on 1:1 over the long term (with the usual assumptions: fair coin, etc.). Equivalently, you'd assign a probability of a coin landing heads on any single toss as 0.5. But you wouldn't be surprised if you got five heads in a row. I assume you'd accept that, so why not accept the 0.875 probability given for the four-coin question? I'm clearly not 'getting it'.

Look, I’ve been at this problem for some time and it’s not coming easily into wordage that makes much sense. For those that want to play along and either (a) prove me insane or wrong (that’s fine – I can live with that) or (b) help me find what the fluck it is I’m trying to empirically assess, let alone prove, I’#d be grateful for your assistance. I don’t have the brain power to resolve it alone.

I’ll come up with an instance of what alerted me to this issue and we can perhaps work with something more interesting and less ‘obvious’; than coin tosses.

Please do, it's intriguing. We might get more illumination sticking to the coin toss problem first, though - if we can't clear that up we may not solve a more real-world, less-obvious problem.

 

[A Dashing Blade]

. . .
I’m asking if you were to toss a coin, right now, 4 times, with the intent of getting either two consecutive heads or two consecutive tails, would your results tally, en masse, with the theoretical ideal?

I don’t think they would.
. . .

Assuming that a statistically significant number of people did this experiment (>1050 for a 2SD level of confidence) + independence of outcomes (ie non-rigged coin) then the answer is yes, this would tally.

However, if you asked a question along the lines of . . . "if you tracked the log of daily changes in a market (any market, S&P, USD/JPY etc) would you expect these changes to statistically conform to a gaussian distribution?" . . . then my answer would be "no".

 

[TheBramble]

Assuming that a statistically significant number of people did this experiment (>1050 for a 2SD level of confidence) + independence of outcomes (ie non-rigged coin) then the answer is yes, this would tally.

However, if you asked a question along the lines of . . . "if you tracked the log of daily changes in a market (any market, S&P, USD/JPY etc) would you expect these changes to statistically conform to a gaussian distribution?" . . . then my answer would be "no".

It's the latter that interests me more deeply and why we generally assume we can fit these type of dispersal from the mean into classical theory. Even acknowledging fatter tails and skewness, it doesn't address the glaring discrepancies.

 

[TheBramble]

No? Any idea why they wouldn't, or why you think they wouldn't? "Classical probability theory" says that you would expect the ratio of heads to tails from single coin tosses to converge on 1:1 over the long term (with the usual assumptions: fair coin, etc.). Equivalently, you'd assign a probability of a coin landing heads on any single toss as 0.5. But you wouldn't be surprised if you got five heads in a row. I assume you'd accept that, so why not accept the 0.875 probability given for the four-coin question? I'm clearly not 'getting it'.

I didn’t either which is what prompted the initial query from me. Using even a basic MC on this simplified problem, using genuine random seeding, you get an increase in divergence away from not toward the mean. In classical theory you ‘should’ converge and we’ve come to get comfortable with that notion over the years, but ‘things’ don’t – otherwise they’d pretty much be where they started. While that statement might be a little off the wall, more pragmatically, in my trading I note that using classical probability does not give me the results that I experience on a daily basis in the markets. More importantly, these differences are driven by others setting prices so we’re all operating in a universe which although it acknowledges classical theory as ‘useful’ does not actually operate on that basis on a day-to-day level

Please do, it's intriguing. We might get more illumination sticking to the coin toss problem first, though - if we can't clear that up we may not solve a more real-world, less-obvious problem.

Good point. I think we’ve already lost those that aren’t going to hang on regardless of the set chosen.

 

[trendie]

I didn’t either which is what prompted the initial query from me. Using even a basic MC on this simplified problem, using genuine random seeding, you get an increase in divergence away from not toward the mean. In classical theory you ‘should’ converge and we’ve come to get comfortable with that notion over the years, but ‘things’ don’t – otherwise they’d pretty much be where they started. While that statement might be a little off the wall, more pragmatically, in my trading I note that using classical probability does not give me the results that I experience on a daily basis in the markets. More importantly, these differences are driven by others setting prices so we’re all operating in a universe which although it acknowledges classical theory as ‘useful’ does not actually operate on that basis on a day-to-day level

Good point. I think we’ve already lost those that aren’t going to hang on regardless of the set chosen.

why do you get a divergence from the mean?
I think I can see why a random distribution might diverge from the norm.
random coin-tossing isnt like the markets. are you saying they should be treated as such?

 

[TheBramble]

why do you get a divergence from the mean?

Divergence from the mean. The constantly changing values in any series define the mean. Bit like price in relation to MAs and Bollies. It is within classical probability, under specific circumstances; you expect regression to the mean. And it’s valid for a subset of data and applications, but not all. I think the problem is that we tend to use a one-size-fits-all approach to probabilistic theory and while that’s going to be just fine for most things, if you’re getting deeply into the real life aspects on any event or series of events for which we need to use an estimation of likelihood (or unlikelihood), to work with an inappropriate model is worse than no model at all.

random coin-tossing isnt like the markets. are you saying they should be treated as such?

I’m saying that the probabilities we use to asses events related to coin tossing are typically assumed to fall within classical theory, but even this application appears not to necessarily fit. We should, I’m suggesting, be looking for a model that supports what we perceive about chance in any specific endeavour at any given time, rather than rely on standard models.

I want to go into subjective probability to cover this, but it’ll be later on tonight.

 

[A Dashing Blade]

It's the latter that interests me more deeply and why we generally assume we can fit these type of dispersal from the mean into classical theory. Even acknowledging fatter tails and skewness, it doesn't address the glaring discrepancies.

Mandelbrot : The (Mis) Behavior Of Markets should deffo be your next port of call. Very accessible, non-technical, should give you a few ideas, it did for me.