Out of Pod Experience

Introducing a New Term

If "almost certain" means that something is so likely to happen that it presumed that it will happen (although it is not definite that it will) then "within reason" will be defined as what is observably the most common and repeatable outcome per expanse of time. The idea of "within reason" will also come with a tiny measure of doubt, just like it’s component term almost certain, but like almost certain it will be presumed to be so likely to occur that it is presumed that it will occur a vast proportion of the time.



The Likelihood Of The Observer Scoring A Hit (Simplified)

A streak of 100 once in 602 billion trillion years
A streak of 50 once in 535.5 million years
A streak of 20 in roughly 6 months
A streak of 15 in roughly 5.6 days

Since flipping a coin takes 5 seconds and since flipping 100 coins in a row (be it a streak of heads or any other component set) would take 500 seconds (or 8.3 minutes), in order to be reasonably certain (there is that term again) of flipping 100 heads in a row, it ends up being a moment in time spanning 8.3 seconds on a timeline of 602 billion trillion years.

That is an incredibly small instance on the timeline.

For 50 it is 4.1 minutes in roughly 535.5 million years
For 20 it is 1.6 minutes in roughly 6 months
For 15 it is 1.25 minutes in roughly 5.6 days

What this means only starts to make sense when you visualize the observer. First you create a timeline that is the length of 602 billion trillion years and then you cross reference that with the lifespan of the observer. It is very likely that 1 observer with a lifespan of 100 years will be standing there during those crucial 1.25 minutes in 5.6 days that …almost certainly" yields a streak of 15 heads, tails heads to tails or tails to heads.

But do this again for the streak of fifty? You have a timeline consisting of 535.5 million years and an observer that can only live for 100 years. He has only 4.1 minutes in 535.5 million years to be "almost certain" of observing the sequence of 50. He would be one of 5,355,037 observers who would live and die consecutively in order to see every instance of this incredibly long timeline. In other words, there is no reason for you, being observer 1, to presume within reason (there is that new term) that you will be that lucky single observer. It is theoretically possible that you will roll 100 heads in a row during the very first 100 attempts, but doing so is not "within reason". Remember the definition of within reason as we continue on.




Now We Do It For Smaller Number


Now, in order to make any of the above make sense we have to do all of this again only with much smaller groupings. I will try and streamline this as much as possible. The new goal will be a streak of a repeating sequence of 1 heads and 1 tails (or) 1 tails and 1 heads (or) all heads (or) all tails 2 flips long, 4 flips long and 10 flips long.

For a streak of 2: each possibility comes up
TT -- HH -- TH --HT

The odds of getting any of these combinations are 1:4, in other words you are absolutely guaranteed to get one of these combinations every time you flip twice. The observer will see this transpire at a fixed rate of once every 10 seconds.


For a streak of 4: each possibility comes up 4 times out of 16 flips.

[TT-TT], TT-HH, TT-TH, TT-HT
HH-TT, [HH-HH], HH-TH, HH-HT
TH-TT, TH-HH, [TH-TH], TH-HT
HT-TT, HT-HH. HT-TH, [HT-HT]

The odds of getting any of these combinations are 1:16. In other words the observer is almost certain (there is that term again) to see one of these target combinations every 1.3 minutes.

Now this outcome is not guaranteed, it is after all ONLY …almost certain". But even in the absence of those desired combinations you are still guaranteed to always turn up one of these four combinations TT -- HH -- TH --HT every time that you flip twice.


For a streak of 10 = 1:1024 (1:1024)

The odds of getting any of these combinations is 1:1024. In other words the observer is almost certain to see one of these target combinations occur once every 85.5 minutes (or 1.42 hours). In the absence of those 4 outcomes you are still almost certain to get the desired outcome of 4 in a row once every 1.3 minutes and in the absence of that you are STILL guaranteed to get the sequence of two every time that you flip twice.

The Casino

For this example we will use a roulette table. In place of black and red we will instead use heads and tails in it’s place for the sake of continuity. This imaginary roulette table will have pictures of coins on it instead of numbers. Half of those pictures are of heads and the other half are pictures of tails. All of the same rules of roulette will apply. The only thing that will change will be the rate in which the roulette table spins ultimately reducing the amount of "flips per unit of time". But we will ignore that because it is ultimately irrelevant. For simplicities sake we will presume that we are rolling a complete spin in 5 second intervals.

Whether it is coins, roulette tables or binary outputs all of the same statistical rules apply. So when one is confronted with the question "what or when will be the first target sequence of 10 that I will encounter"" one can now begin to anticipate the outcome WITHIN REASON. Remember the definition of within reason. But before we do that we need to understand how long we will be staying at the casino. Remember, for all this to work the observer needs a lifespan. In this example the lifespan is not the actual life of the observer, but the duration he will be observing the coins flipping. We will presume 8 hours of gambling (or) one single shift in the above example. So we will have a chance to observer 5,760 rolls of the imaginary roulette table.



We know how often we can expect to see the sequence of 2 (once in 10 sec), 4 (once in 1.3 minutes) and 10 (once in 1.42 hours). All of these outcomes are entirely unpredictable because we cannot possibly anticipate the order in which the will occur. We can only be "almost certain" of how often they will occur per consecutive flips, and thus, per unit if time (perspective of the observer)

However in order to get 15 the temporal value climbs to 5.68 days which is much longer then the time that we will be spending at the casino. The notion of seeing the vale of 50 would require 535.5 million years.

So we can conclude within reason that we will probably encounter any value up to 10 during our visit, but we will not encounter the value of 15 or higher MOST OF THE TIME that we visit casinos for 8 hour stretches. (nothing is 100%)


Within Reason
We will expect to see large repetitions of the values of 2 and 4 quite frequently per unit of time, and every value up to 10 have a good chance of occurring within reason during our stay. It however is NOT within reason to presume that we will see an outcome of the value of 50 or above during our stay.

It is theoretically possible…
But it is not "Within Reason" that we will.


Now we can begin to anticipate the stretches of any repeating sequence above the values represented in the sets of 2 and 4 despite the fact that each coin toss (of our fair coin) has a static probability of 50:50.


This is how you plug relativity into statistics.

I would not advice you enter a casino unattended, a fool and his money are soon parted.

Nothing stridently wrong (just occasionally mildly inaccurate) until this time with anything you've said in the first 4 posts currently containing text out of 5 starter posts (except that very last sentence in post 4), and depending on the percentage chance needed for something to happen to call it "within reason" (and you also somewhat misuse "almost certain", which normally means something closer to "p=1 for infinity tries"), you might even be "close enough to not warrant very angry rebuttals" overall.
So, you're not quite accurate with all your numbers, but at least you got the orders of magnitude about right-ish. Either way, good enough for the time being.

However, that very last sequence, "This is how you plug relativity into statistics", it's a doozie... I don't know which of the possible definitions you personally use for the word "relativity", but I am almost certain that the one you selected for what the word "relativity" means in this context is not the same as the one most people would think of first.

Also, everything you said is not a new concept for statistics.
It's just that you only started understanding it (or admitting you started understanding it) more accurately.
And as a side-note, it doesn't have anything to do with Einstein's theories of relativity (i.e. special relativity nor general relativity), which pertain to something else entirely.

Why, the very first definition in the dictionary, of course, what else ?

relativity [ˌrɛləˈtɪvɪtɪ], noun
1. (Physics / General Physics) either of two theories developed by Albert Einstein, the special theory of relativity, which requires that the laws of physics shall be the same as seen by any two different observers in uniform relative motion, and the general theory of relativity which considers observers with relative acceleration and leads to a theory of gravitation
2. (Philosophy) Philosophy dependence upon some variable factor such as the psychological, social, or environmental context See relativism
3. the state or quality of being relative

You're using a form of the third.


Yes.
Those nitpicky wordings are what have the potential to separate "correct answers" from "inaccurate, confusing or misleading answers".

It's actually all fine up to the point where you call it "relativity".




You still don't get what the gambler's fallacy is all about.
Uppercased and bolded the RELEVANT parts of the quote.
You have already admitted more than once that the very next flip is still a 50:50 flip.

Yes, a longer streak is less likely than a smaller streak, but such streaks do happen every now and then.
As "an observer" you have no way of knowing the future, thus no way of knowing if you are at any given time inside a larger lucky streak that's not about to end just yet, or near the very end of a streak, the odds are always even on the very next throw.
A person suffering from the gambler's fallacy is more likely to bet as if the odds would be in his favour instead of the odds still being even.
And betting with the wrong odds in your mind is a surefire way to lose more money than you can afford to lose.


Sorry, not even remotely new.

Akita, I am very pleased to see you giving such earnest refutations to EP's musing. It is very apparent you are an expert in this field and he is lucky to have your attention in these threads (which are fun to read xD).

Also, EP, using the term "relativity" is confusing. While it is justifiable to use, it just hurts your position, especially to those familiar to statistical theory (which is begging for a lot of physics nerds to jump in)

You're still wrong about that. You can't play the odds of the set against the individual coin toss. HHHHHHHHHT and HHHHHHHHHH have precisely the same odds of occurring. *Your* fallacy comes in assigning greater significance to one combination over the other. Let's talk about the odds of each of these combinations:

HHHHHHHHHH
TTTTTTTTTT
HTHTHTHTHT
HHTTHHTTHH
HHTHTTHTTH
HHHHHTTTTT
THTHTHTHTT
HHHHHHHHHT
TTTTTTTTTH

They all have IDENTICAL odds of occuring, wouldn't you agree? Then why should HHHHHHHHHH be less likely to occur than HHHHHHHHHT?


Not new, merely dismissed as inaccurate.

Akita T, why don't you tell us all why:


1. if you flip a certain amount of coins per minute
2. And if you flip a certain amount of coins per hour



Why that would not TEND to generate certain set lengths (within reason) during an 8 hour stretch at the casino tomorrow?

Yea I get that now



But it applies IMO. If you throw a ball on a train then that ball is traveling 1. at a certain speed verses someone who is not on the train 2. at a certain speed verse someone on that same train and 3. at a fixed speed verses the speed of light.


Now I do not want to argue relativity, but who is to say that the same idea cannot be applied to infinite time? 1. Any stretch of time is a certain size (either larger or smaller) when compared to the lifespan of the observer (but) 2. it is always the same size verses infinity.


The ideas seem different, but they are not really so different.

HHHHHHHT and HHHHHHHHH are both sequences. That's what you seem to be completely missing. Every sequence of coin flips, no matter the length or order, has the EXACT same odds of occurring. You're saying the gambler's fallacy has some merit because HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH is so very unlikely, but it's no less likely than any other sequence of the same length.

Pick a sequence of sufficient length, and you're no more likely to see it occur as you are the same length of consecutive heads. You CANNOT base your expectation of a single coin toss on the previous results. It doesn't matter what temporal values you assign to the relative magnitudes to avoid a quantum field inversion cascade in the osmotic flux inhibitor, the simple truth is that you're assigning special value one of thousands of possible combinations and then saying that it is less likely than any other.

Unless you want to go back to that first thread of yours and actually assert that the odds of the coin toss change based on previous tosses, what exactly are you trying to claim here?