Coin flip probabilities and relevance

[sol1]

Originally posted by Verasace
Finally, some discussion...



Once you can perceive probability as a waveform, the trend towards 50/50 despite long consecutive "favor" for either heads or tails is easy to understand.

I am not a mathmatician, formulas are admittedly not my strong point. Objective observation and interpretation outside of the box is my forte.

Again, LOOK at my graph to truly grasp what I am proposing.

mechanics gives electron waves structure
[/URL]


Quantum Well

Any congregation of energy is defined in the Quantum Harmonic Oscillator, and Zero point defines any particle state. From a fuzzy nature, energy is not real defined, yet, it is describing a particle? You see?

Sol

 

[chroot]

Originally posted by Verasace
Just curious, who out there has looked at the graph? Only 4 others besides me since I posted it til now

I looked at it. It's a one-dimensional random walk. It's nothing that any decent scientist or mathematician hasn't already seen a hundred times.

Should I assume that "basic well-accepted concepts of maths and physics" are to never be challenged?

No, you can certainly challenge them. But when someone responds to your challenge by explaining to you in very specific language how things really work -- to anyone's satisfaction but yours -- you become a crackpot.

Originally posted by sol1
Any congregation of energy is defined in the Quantum Harmonic Oscillator, and Zero point defines any particle state. From a fuzzy nature, energy is not real defined, yet, it is describing a particle? You see?

Jesus H. Christ, this thread is going downhill.

- Warren

 

[sol1]

Originally posted by jcsd
But you've not proved in any way that the 50/50 line must be crossed for any value of n.

I wasn't necssarily referring to you verasace, I think you have basically just misunderstood a vital element of probabilty.

I wasn't necssarily referring to you verasace



I guess that would be me then.

I am always open to corrections in light of the string issues. The graviton has helped explain things we had not understood before. If you take that position, then how has probabilty dterminations changed?

Consider me the child here then[b(] Boltzman is a considerable factor to think about in the issues of Quantuim mechanics yet there is some "joining" taking place. How far ahead are these math minds in QM to have undertsood something about GR? Big question?

Sol

 

[chroot]

Originally posted by sol1
I am always open to corrections in light of the string issuess. The grvaiton has help explain things we had not undrtsood before. If you takethat psotion then how aas probabilty dterminations changed?

I have a hard time believing that even YOU understand what the hell you say. Do you just string together all the scientific words you can think of, in hopes of one day arriving at a sensible sentence?

- Warren

 

[sol1]

Originally posted by chroot
I have a hard time believing that even YOU understand what the hell you say. Do you just string together all the scientific words you can think of, in hopes of one day arriving at a sensible sentence?

- Warren

Maybe you should reread my post yu linked. There has been some additions.

Without a vision of the gravity field(hyperdimensional space) you don't know what you are talking about either. What is supergravity? Do you understand? Maybe probability statistics can help you explain what is happening in a much more dynamical way?

The answer is, you can't without GR.

Sol

 

[jcsd]

Originally posted by sol1
I wasn't necssarily referring to you verasace



I guess that would be me then.

I am always open to corrections in light of the string issues. The graviton has helped explain things we had not understood before. If you take that position, then how has probabilty dterminations changed?

Consider me the child here then[b(] Boltzman is a considerable factor to think about in the issues of Quantuim mechanics yet there is some "joining" taking place. How far ahead are these math minds in QM to have undertsood something about GR? Big question?

Sol

I was referring to crackpots in general, your posts thopugh don't make any sense.

Probabilty is an abstract mathematical tool it has zilch to do with quantum mechanics, etc. though it can be applied in these areas if needed.

 

[sol1]

Originally posted by jcsd
I was referring to crackpots in general, your posts thopugh don't make any sense.

Probabilty is an abstract mathematical tool it has zilch to do with quantum mechanics, etc. though it can be applied in these areas if needed.

Maybe you should reread my post yu linked. There has been some additions.

Without a vision of the gravity field(hyperdimensional space) you don't know what you are talking about either. What is supergravity? Do you understand? Maybe probability statistics can help you explain what is happening in a much more dynamical way?

The answer is, you can't without GR.

Sol

 

[chroot]

Aye dios mio. Que lastima.

- Warren

 

[sol1]

Originally posted by chroot


Jesus H. Christ, this thread is going downhill.

- Warren

Originally posted by chroot
Aye dios mio. Que lastima.

- Warren

Before you go I was hoping you could correct this statement for me

Temperature is sure hard to explain when supersymmetry asks us to look at the early universe and the contiuity with which are are able to define this movement.

Yet we know that plasma effects and super gravity are closely associated?

Please stay and I will go. Hopefuly they will be an answer to the question?

Sol

 

[Hurkyl]

Sol: You really should be asking physics questions in one of the physics forums here (such as the Strings, Branes, & LQG forum) rather than in the mathematics forum. And you definitely shouldn't be asking in a thread on a different topic.

 

[sol1]

Originally posted by Hurkyl
Sol: You really should be asking physics questions in one of the physics forums here (such as the Strings, Branes, & LQG forum) rather than in the mathematics forum. And you definitely shouldn't be asking in a thread on a different topic.

I don't mean to be a pest...but the questions have to do with this math. Coin flips and such. If I cannot be straighten out then indeed I will remain in illusion.

It seems I find those who are gifted in maths might be deficient in other areas, so they make rude comments, like I do not undertand.

But they have to trust that I have spent considerable time looking at the issues, in order to understand the use of math(?) to question the viabilty of current thinking. If they cannot marry themselves, to current thoughts, then they will have remained as distinctive parts of, but have not understood their connection to a vast network.

They(the math) are part of the TOE

Sol

 

[Hurkyl]

Verasace:

The reason you are getting this response from everyone is:

(a) Your claims are very contradictory to established mathematics.
(b) You know full well your claims are very contradictory to established mathematics.
(c) You are trying to convince everyone you're right, but you are making no effort to prove you're right.
(d) You don't consider input from others.


Others have mentioned something called a random walk. Have you went to investigate what such a thing is and what properties they have?


For example, consider the experiment you have done; flip a coin 5000 times and let Χ be the number of times the running score is 0.

If you assume that each coin flip has a 50% chance of being heads (that is, there is no "probability pressure"), the statistical mean of Χ is approximately 56.4274.


You made an interesting observation, and you would probably have learned quite a bit and gained a deeper understanding of probability & statistics if you had buckled down and investigated why your observation might be true.

Instead, you felt the need to invent some strange, new idea, and are refusing to consider that established mathematics can explain this phenomenon. Why do you feel the need to do such a thing?


Anyways, if you're still dead set on convincing everyone that probability pressure are real, you should consider ways of actually proving it. You said you understood everything you learned in your statistics classes; you should recall discussing hypothesis testing. You need to design an experiment and derive a hypothesis test that can confirm with, say, 99% confidence that the results could not have been generated by the usual model.

 

[Verasace]

Hurkyl,

Thank you, and the others, for your thoughtful response.

I will indeed consider your helpful suggestions and keep all informed.

As for now, I'm off to New Orleans for an extended Halloween weekend.

Verasace

 

[phoenixthoth]

Originally posted by chroot
You absolutely should not. This is just evidence you have no idea what probability means. As a result, Njorl requested that you tell us where you've taken your graduate probability classes, so that we can all be sure to never take anyone from that school seriously.

- Warren

this is hilarious! the guy is making obvious "independence" errors yet more than one person is making the same error in assuming that everyone from that school thinks the same way!

there is no such thing as a "probability wave" like your mentioning. if the coin has been 10,000 heads in a row, there is no "pressure" making the probability of the next one being heads any different from 0.5. this is because the events are independent.

this reminds me of a famous anecdote regarding independence. a "statistician" thinks that while the probability of someone brining a bomb on to an airplane may be low, it is even lower that two people independently bring bombs on board. therefore, to lower the odds of there being a bomb, he brings his own bomb on board.

 

[O Great One]

Difference between ratio and absolute number of heads and tails

Verasace,
There are two things that are getting confused here and it doesn't look like anyone has pointed this out. It is true that the ratio (Heads:Total number of flips) will approach .50 as the number of flips increases. But, what is also true is that the difference between the number of heads and tails will get larger and larger as the number of flips increases. This is what you are measuring with your graph. If you increase the number of flips you will find that the graph won't return to zero. In fact, it will move away from zero. I hope this clears up the confusion.

 

[Hurkyl]

That is incorrect; the graph will eventually return to 0 with probability 1.

However, the maximum observed difference between the number of heads and tails will also diverge to infinty as the number of flips increases. (also with probability 1, I think)

 

[O Great One]

You're right.

I'm sorry, you are correct Hurkyl. I guess as the number of flips increases the graph will continue to cross the zero line but with less and less frequency?

 

[Hurkyl]

IIRC, when I did some calculations, it looked like the number of zero crossings was roughly proportional to the square root of the number of flips, which could be interpreted as saying the frequency of zero crossings goes down.

 

[FearsForLife]

But I am not talking about samples, or portions of a wave, I'm talking about an infinite ocean, and eventually from the start point you will return to 50/50

Go ahead and sweep (censor) me, or just don't respond if you wish.

Just curious, who out there has looked at the graph? Only 4 others besides me since I posted it til now

Actually V, you're first mistake was posting in a forum designed for physicists. I've met a few, they tend to be closed minded.

Second, being weak in physics, and...uh...also weak in general mathematics, and yet having still taken a few classes...Maybe think about it using discrete induction.

I like that you think outside the box, I really do, but if you hit this using induction, maybe it'll clear some stuff up. If the odds of a coin flip is 50% for n, and the odds of a coin flip is 50% for n+1, that should (if I've got my induction done correctly) prove that, to infinity (and beyond!), each coin flip should uniquely have a probability of 50%. Just a new way to look at it, suppose you were getting tired of being called a moron by the snoots.

Originally posted by Hurkyl
Sol: You really should be asking physics questions in one of the physics forums here (such as the Strings, Branes, & LQG forum) rather than in the mathematics forum. And you definitely shouldn't be asking in a thread on a different topic.

I don't mean to be a pest...but the questions have to do with this math. Coin flips and such. If I cannot be straighten out then indeed I will remain in illusion.

It seems I find those who are gifted in maths might be deficient in other areas, so they make rude comments, like I do not undertand.

But they have to trust that I have spent considerable time looking at the issues, in order to understand the use of math(?) to question the viabilty of current thinking. If they cannot marry themselves, to current thoughts, then they will have remained as distinctive parts of, but have not understood their connection to a vast network.

They(the math) are part of the TOE

Sol

Sol, if you were in a different forum, surrounded by my people (the unintelligent ones), we would berate you into the ground for even CONSIDERING Hijacking a thread. GO! Go post a new thread! I wasted 5 minutes trying to read through your posts to see if they were relevant to the topic at hand! Bad internet form!



I hope I didn't upset anyone...except for Sol.
It's not my fault you guys come off as snooty.

 

[D H]

Actually V, you're first mistake was posting in a forum designed for physicists. I've met a few, they tend to be closed minded.

Your biggest mistake is dredging up a thread that hasn't been active for *five* years.

 

[FearsForLife]

Wow. Awesome, talk about the biggest bump ever.

Well, I guess that guy gave up. I insulted somebody who hopefully is 5 years wiser about how to use the internet, and I'll shut up and go away now.

Smooches
FFL

 

[Kittel Knight]

... but if you hit this using induction, maybe it'll clear some stuff up. If the odds of a coin flip is 50% for n, and the odds of a coin flip is 50% for n+1, that should (if I've got my induction done correctly) prove that, to infinity (and beyond!), each coin flip should uniquely have a probability of 50%.

Does it make any sense using induction to prove that?
...infinity and beyond?

Where did you learn induction?
Where did you learn math?

Hmmm...I'm quite sure you and Verasace were classmates...

 

[regor60]

Bad form by a lot of people here...better off addressing misconceptions with facts than with belittling

 

[ssd]

I feel the urge to mention a thing or two in addition to all those said to counter op's notion of so called "probability pressure".

1/ Say, we get H in 1st toss. If there is any such pressure then after the 1st toss the pressure shall be towards T, to bring to 50-50. So we must get T in 2nd toss (since there is a pressure towards it and negative pressure towards H).
Therefore, under the pressure theory we must get alternate H and T. {Of course newer "pressure" or whatever theory has to be developed to counter the real life sequences}.

2/ I want to know that whether op really tossed a coin or used computer generated random numbers for his graph. If computer generated numbers are used, did he perform a (statistical) test of randomness? If yes, how did he perform the test (because his concept of pressure will affect again the distribution of any r.v.). So, he cannot relay on any existing statistical test.
Unless he tested the used numbers for randomness in a "logical' way, his graphs and findings do not remain valid.

 

[statdad]

the simplest explanation (other than the one that says the OP doesn't know what he's doing) is that he has confused empirical observation with theoretical probability (which goes back to not knowing what he's doing)

 

[Hanshananigan]

I saw this thread at the top of a Google search, so I thought it might be worth an illustration of why the OP's logic doesn't hold up.

I used Excel to create 104 sets of 400 "coin flips." By the OP's logic, if the first 200 flips in each set tended to be more heads or more tails, the second 200 flips should have "pressure" to reverse that trend to result in a closer to 50/50 split. Ignoring the few sets that had 100 "heads" in the first 200 flips (50/50 split, so there would be no "pressure") and those that had 100 "heads" in the second 200 flips, I counted how many sets had >100 heads in the first 200 flips and <100 heads in the second 200 flips and did the same thing for tails (perhaps providing evidence of "pressure"), and calculated a rate of how many trials supported the "pressure" hypothesis (ignoring those trials with 100 "heads" in the first or second 200 flips). To keep things simple, I ran each set of trials 10 times and calculated the rate for each set.

It was expected that the rate of trials supporting the "pressure" hypothesis would be significantly > 50%.

The results: 56, 49, 53, 53, 48, 39, 45, 48, 52, 37

That is, an average 48% of the time, if #heads > #tails in the first 200 flips, that trend would be reversed in the second 200 flips. Thus, no evidence of "pressure."

In terms of application on the roulette table, if you have enough cash for 40 spins, and you know the outcomes of the first 20 tended to be "black," you will not necessarily come out ahead if you bet on "red" for the next 20 spins.

Looking at it another way, for those trials in which at least 110 flips of the first 200 were heads, the 201st flip was tails only an average of 45.1% (36, 40, 56, 60, 64, 44, 58, 67, 50, 40). Although 45% seems like a decent spread for betting, the variance across trials is huge due to the infrequency of finding at least 110 flips among 400 trials, so the confidence interval will be wide. Again, even if the roulette table has tended to run "black," it does not affect subsequent spins. Not worth betting on.

These outcomes will be expected by most of those who posted on this thread. For anyone else, I hope this example provides a link between the OP's logic and the properties of probability offered by others.

 

[heyhay4062]

As others have not so politely stated get a clue. Just because there is a statistical anomaly there is no”probability force” that will make it correct itself. It may never correct itself, or more correctly stated it may take an infinate number of chances to correct itself. At every odd numbered toss you are guaranteed a statistical anomaly. After one toss it will be either heads or tails. You are literally saying that if you toss a coin twice if the first result is heads, then the second toss is bound to be tails. Which hopefully you understand isn’t the case...

 

[roryjester]

I'm not a mathmatician or physicist, though I have a couple of degrees in the sciences, the specifics of which I will leave out in case I will be belittled for irrelevance here. I did take one probability and statistics course at university a long time ago, an introductory one which I somehow passed and actually did ok at, though I came out not really getting 'it'. Like how much of it really can be applied for any 'specific' instance in life, being based usually on things like infinity or at least, a large enough sample size or population, as well as all kinds of manipulations of figurative mind and pencil, and predictions good only on a large numerical scale (although good for big entities like corporations predicting profits and armies predicting all sorts of things that are necessary for military success, et cetera). Being a gambler at heart and actuality, I've experienced more than I've ever thought, especially about things like flipping a coin, which may be a serious disadvantage in some people's thinking, but not so much in the minds of others who based most of their actions and decisions on the unpredictability of life and real experience. I've read thru the entire thread and though I don't understand a great deal of what's being used to argue each poster's particular points, I do find it interesting, at least from my narrow limited point of view and comprehension. Let's begin with my two bits worth. Since infinity is something almost impossible to grasp, except maybe abstractly, like mathmatical singularities, it might not be such a great idea to use it to argue more mundane things like the flipping of a coin. If I comprehend right, in an infinite sample size which of course means the inclusion of all flips or sets of flips ever performed or imagined in the universe from the infinite past (debatable) to the infinite future (again debatable), the number of heads and tails will (or have already) come out to a 1:1 ratio or 50/50. Just true randomness or unpredictabiltiy at work to give us a final predictable or non-random number, I guess. Now in any 'finite' sample of flips, anything can happen including a million flips in a row which result in all heads, let's say. This would be a great statistical anomaly, but funnier things than this have happened, like perhaps the greater improbability of human life (or any kind of 'life') actually starting up in this great big universe (yet that's what has happened). Now, the OP's question as I understand it, is: Is there pressure for the next million or so spins after the 'all-heads' sample to favour more tails than heads? I'd say for any individual flip after that, the probability would be 50/50 just as if the coin never knew it flipped heads a million times in a row before that (heads again, baby!?). But the OP's question is really bigger than that. He's saying in the infinite minus 1 millions spins after that, is there going to be more tails than heads? This is just my gambler's intuition speaking, but I would say yes, although not by much (the ratio will still be approaching 50:50 for all 'practical' matter). So if u got infinite amounts of money, time, and patience, it might not be a bad idea to bet on tails in the infinite time after u see the first million flips go all heads, although another question to be asked is: Were u there to see the previous million flips before the all-heads streak, cos u know, it might have been a 'million-all-tails' result set before that; then ur back where u started: 50/50 and no real or perceived 'pressure' to compensate for older statistical anomalies. Please inform me if my jerry-rigged gambler's intuition is wrong here somehow. By the way, does anyone here know if slot machines are truly random, or do I just have to stay with a cold machine until it is 'pressured' into becoming hot again (so I can get all my money back)? I understand however that the 'payout-percentage' programming (involving a 'truly' random number generator?) may not be based on infinite 'spins' though (maybe a million, a billion, even a gazillion, but not infinite). By universal law, it has to pay back a certain percentage of the finite money put into it in a finite time. It's just predicting those times (or the length of time before payouts) that's the 'infinite' problem, isn't it? Aah, what the hell am I talking about? Cmon, smile, be happy.

 

[CRGreathouse]

Since infinity is something almost impossible to grasp, except maybe abstractly, like mathmatical singularities, it might not be such a great idea to use it to argue more mundane things like the flipping of a coin.

Actually infinite numbers are pretty easy to pick up (depending on the type you pick); I'm not sure where this idea comes from, though it's common. But you're right that we don't need it; there are finitistic ways to thinking about it. Here's one:

For any positive percentage (say, 0.1%) and any certainty less than 1 (say, 99%), there is an M such that for all N > M,
the chance that N fair coin flips will fall between 50% - the percentage and 50% + the percentage (49.9% to 50.1% in this example) is at least the specified certainty.

Now, the OP's question as I understand it, is: Is there pressure for the next million or so spins after the 'all-heads' sample to favour more tails than heads? I'd say for any individual flip after that, the probability would be 50/50 just as if the coin never knew it flipped heads a million times in a row before that (heads again, baby!?). But the OP's question is really bigger than that. He's saying in the infinite minus 1 millions spins after that, is there going to be more tails than heads? This is just my gambler's intuition speaking, but I would say yes, although not by much (the ratio will still be approaching 50:50 for all 'practical' matter).

Those contradict each other! If each following coin flip is unbiased, then the collection of coin flips will also be unbiased.

So if u got infinite amounts of money, time, and patience, it might not be a bad idea to bet on tails in the infinite time after u see the first million flips go all heads, although another question to be asked is: Were u there to see the previous million flips before the all-heads streak, cos u know, it might have been a 'million-all-tails' result set before that; then ur back where u started: 50/50 and no real or perceived 'pressure' to compensate for older statistical anomalies. Please inform me if my jerry-rigged gambler's intuition is wrong here somehow.

Yes, intuition has failed you this time. It happens.

By the way, does anyone here know if slot machines are truly random, or do I just have to stay with a cold machine until it is 'pressured' into becoming hot again (so I can get all my money back)?

Slot machines are generally random, modulo concerns about their use of pseudorandom numbers rather than true RNGs (don't worry about it; it doesn't affect your question). But one slot machine need not be like another. It's possible to have one machine in a room that pays out more often than others in that same room -- and from what I hear, that's not uncommon. So within a machine, it's essentially random, but between machines I wouldn't expect similar long-term results.

 

[praeclarum]

Thanks for the unridiculed (almost) reply.
If there is no pressure to return to 50/50, then why doesn't one just flip heads indifinitely?

The fallacy in your reasoning is in making the assumption that because something is likely to happen, then it naturally tends to that. While this is true to some extent, it is only indirectly - it is a product of the fact that as n (number of tosses) approaches infinity, the heads to tails ratio approaches 1:1, simply because as n increases, it is increasingly improbable for you to keep up a streak of all tails and all heads that just happens to comply with the data.

Here's a mini-demonstration:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
...

This is Pascal's triangle, but it represents the relative probability of getting a certain number of heads out of the tosses. So if you flipped a coin 6 times, there is a 10 / 32 chance of it getting 3 heads, whereas there's only a 1 / 32 chance of it getting 0 heads.

Anyway - a little off-topic, I guess. Big picture - you're mistake is in saying that the relative higher chance is because of some "law of averages" that has a will of its own.

 

[Stephen Tashi]

I think these two qualitative statements need not be contradictory:

1) An imbalance in the first N tosses of a fair coin gives no information that improves our prediction of the result of the next M tosses of it.

2) As the sequence of tosses of a fair coin progresses it is likely that there will be times when the total numbers of heads has a big lead over the total number of tails or vice versa.

As I recall, one of Feller's books discusses 2) in a mathematically rigorous way. One can compute the probability of one result or the other taking a lead of a certain size. It's understandable that people who look at graphs of real or simulated sequences of coin tosses get the impression that swings one way are balanced by swings the other way.

It would be interesting to see the results from a model of coin tosses where the tosses are not independent and a specific formula is given to describe the dependence. For example suppose for toss i < 7 the probability of a head is 1/2 and for i > 6 the probability of a head is given by: 1/2 + (0.4)(3-K)/3 where K is the number of heads in the previous 6 tosses.

 

[yudiski4]

Forget "waves", there are none. Try to look at it this way. In an infinite number of tosses the heads/tail ratio will come out to very, very close to 50%/50%. Agreed? And in that infinite number of tosses, there will have been, almost assuredly, a streak of 1000 straight heads. Also agreed? But the wave theorist says, "Woah, after those 1000 heads, assuming it was running close to 50/50 up to then, there would have to be a "tail wave" for it to end up 50%/50% at the end".

But the fallacy is: in infinity, there is no "end". The intuitive force that makes the "wave" seem inevitable is tied up in the human brain's inability to conceive of or think in terms of infinity.

 

[Parahandy]

We are almost agreed on the very cose to 50:50 on infinite coin tosses (arguably it's exactly 50:50 but who's arguing?). But your 1,000 straight heads is a bit far fetched and that is the fundamental problem of infinity - it has been said (in some BBC programme) that if you attribute a number to all letters of the alphabet then the infinite progression of Pi would hold every book that has ever been written and every book yet to be written. I understand the logic, which interestingly becomes more philosophical than physical and therefore is never going to happen. This is where progressive logical steps end up in a ridiculous proposition and so is the idea of coin tosses producing a 1,000 heads in a row.

I haven't done it but I guess if you plot the incidence of coin tosses starting at 1 head, 2 heads in a row, 3 heads, e.t.c. you will end up with a front end skewed curve with a mean around 4 or 5. You might manage to throw 10 in a row but thereafter, at some point not too far away, your chance runs out to zero. I said it before, the concept of infinity must be flawed; it highilghts a deficiency in our ability to percieve our surroundings, i.e. there is something wrong with maths and numbers and our logic - irrational numbers are just that - irrational.

Coming back to the question, disagreeing with your 1,000 heads makes the point. It is a fact that the coin tosses will pass through the 50:50 and err for a while on the tails side as much as heads. They will switch from one result to the other sooner rather than later and one run can't carry on indefinitely (the laughable thing about infinity is that, if you believe in it, there will be, at some point, an infinite run of heads but, of course, the same would apply to tails - and to the coin landing on its edge).

The problem with probability is that there are some certainties about it but no discernable pattern that we can see. It is as if there is something hidden that is discoverable and would solve the problem. If that is true and someone discovers the solution, it would turn maths on its head and (here's a philosophical point) may destroy reality as we know it because we would have certainty of the future - and that's just not allowed!

The fallacy is the whole concept of infinity.

 

[yudiski4]

"The problem with probability is that there are some certainties about it ... The fallacy is the whole concept of infinity."

If you need to have certainties, and you outright reject the concept of infinity, then the study of probabilities is going to lead you to a stone wall.

 

[chiro]

Forget "waves", there are none. Try to look at it this way. In an infinite number of tosses the heads/tail ratio will come out to very, very close to 50%/50%. Agreed? And in that infinite number of tosses, there will have been, almost assuredly, a streak of 1000 straight heads. Also agreed? But the wave theorist says, "Woah, after those 1000 heads, assuming it was running close to 50/50 up to then, there would have to be a "tail wave" for it to end up 50%/50% at the end".

But the fallacy is: in infinity, there is no "end". The intuitive force that makes the "wave" seem inevitable is tied up in the human brain's inability to conceive of or think in terms of infinity.

You don't need to think of it necessarily in terms of infinity, but rather in terms of something "really large".

For many practical purposes the strong law tells us a lot about the kind of limiting probabilities for large enough sample sizes as it would for an infinitely large number of them.

To understand this its best to think of the derivative of 1/x. If x is big enough then any change thereafter is not going to have much of an effect if the observations up to that point reflect a mostly unbiased sample. If the sample is highly biased then we can't necessarily do this, but for most purposes "large enough" samples will provide a distribution that is good enough to represent the true distribution for "infinite" sample sizes.