2.1.8 - Large Numbers in Probability

2.1.8

Larger Numbers in probability, Statistics, and Combinatorics

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The real world as we have learned is huge. Yet even these numbers pale in comparison when we begin to consider the kinds of numbers that can occur in combinatorics, or when considering the ways in which something can happen, or the number of ways that objects can be arranged. These kinds of numbers grow very rapidly. Are these numbers "real" ? They are in the sense that they are still attached to the way we understand our world, and in that they are related to real world phenomena. Let's explore this new avenue and see where it takes us...

Ways in which a coin can land

When we first learn probability the first illustration we are often given is the flip of a coin. We are told there are 2 possible outcomes and that they are equally likely. Thus the probability of getting "tails" is 1/2. It also means that there are 2 sides to the coin, and 2 ways it can land. In this way we are counting something.

Ways in which 2 distinct coins can land

An important rule in probability is the product rule. If 2 distinct events occur , and we want a specific result from both events, we simply multiply the probabilities. A common example is that in order to get 2 heads in a row, the first coin has to fall heads (1/2 chance), and the second has to fall heads as well (1/2 chance). Multiplying the probabilities we get 1/2 x 1/2 = 1/4. This also means that there is 4 different ways that 2 distinct coins can fall ( say a penny and a nickel for example).

Ways a dice can land

Most people are familiar with the common 6-sided dice. It is shaped like a cube with it's edges and vertices softened. One can conclude that there are 6 ways in which such a dice can fall.

Ways in which 2 distinct die can land

Many games involved rolling double dice (monolopy for example). The chances of rolling 2 ones would be 1/36 . Likewise, if the pair of die are distinct ( say they are different colors ) then there are 36 ways in which they can land.

720

Ways in which 6 objects can be ordered in a row

When I was a kid, there was this volume of 6 books on the old shelf. For some reason I became curious and wondered in how many ways the 6 books could be ordered on a book shelf. I came up with a method for counting out the orders, starting with 123456 and ending with 654321. I eventually realized that I could find the answer by multiplying 6 x 5 x 4 x 3 x 2 x 1. I reasoned as follows " in the first position I can choose any of 6 books, then for the second position there are only 5 to choose from, the third only 4 , and so on until there is only 1 book remaining ". I had rediscovered factorials. When one wants to multiply all the counting numbers from 1 to n together, it can be written compactly as n ! . Thus, 6 objects can be ordered in 6 ! ways. 6 ! = 6x5x4x3x2x1 = 6x5x4x3x2 = 6x5x4x6 = 6 x 5 x 24 = 6 x 120 = 720.

1024

Ways in which 10 consecutive coin flips can occur

Let's say you were flipping a coin repeatedly. Statistically speaking, as you flip, you should find that the number of tails and head remains relatively the same. What if you suddenly flipped 10 tails in a row ? would you conclude that the laws of probability are no longer at work ? It's true that such a sequence is unlikely, but it's far from impossible. In fact, it wouldn't surprise me if it occured to someone somewhere already, although one would be hard pressed to find any record of this. Alittle bit of searching for the longest known coin streak only turned up one relevant result. A random blogger admitted to getting 3 heads in a row once [1] . The probability of this is only 1/8 which isn't a very small probability.

1 : 576,000

Odds of being struck by lightning

Supposed odds of getting hit by lightning [2] .

1 : 649,740

Odds of getting a royal flush

In poker, if you get to draw five random cards from a standard deck (no jokers) your odds of getting one of the 4 possible royal flushes is 1 / 649,740

1 : 2,598,960

Odds of getting a royal flush in spades

In poker, if you get to draw five random cards from a standard deck ( no jokers ) your odds of getting a royal flush in spades is 1 / 2,598,960.

1 : 158,000,000,000,000

Chance that someone has an IQ of 220 or higher

By applying the bell curve to IQ scores, we can come up with estimates of the number of people of IQ's within certain ranges [3]. Doing so, I estimate that the chances of someone having an IQ of 220 or higher are about 1 in 158 trillion. As there are only 6.6 billion people on the earth, the chances are that no one has ever had an IQ this high. In fact, looking over records of the highest recorded IQ's, there is no occurance of an IQ of 200 or higher.

1,638,365,164,028,614,560

Number of years it would take for all planets to align within 4 degrees of percision

This might seem to be more about duration ( a quantity ) than about combinations. But it reflects the vast number of possible states the solar system can take. The solar system is complex enough that you would have to wait about a quintillion years ( roughly 100 million times the current age of the universe ) for all the planets to align within 4 degrees of each other. It is doubtful a planetary alignment this percise ever occured.

519,024,039,293,878,272,000

All Possible configurations of the rubrik's cube

The rubriks cube is a puzzle game consisting of a 3x3x3 cube where it's squares are coloured using 6 different colors, and one can twist the cube to change to position and orientation of the cubelets that make up the whole. The goal is to configure it so that on each face, only a single colour occurs. These were popular at various times. It turns out that this deceptively simple looking puzzle has over 519 quintillion possible configurations. The solution only represents a single configuration. This means that odds of solving the puzzle purely by chance are 1 /( 5.19x10^20), which is extremely unlikely.

1.26765 x 10^30

Number of ways 100 coins may land

What would be the odds of flipping 100 heads in a row ?! roughly 1 / (10^30). This is extremely unlikely. It is doubtful that this has ever occured in human history.

1.478088 x 10^118

Estimate of all possible Chess games

It has been said that there are roughly 10^120 possible games of chess. The estimate I have here is a simple one. Assuming that the average number of moves possible on every turn is about 30, and assuming that each player gets up to 40 turns, we simply need to compute 30^80 to get an idea of how many games might be possible. This produces the result of 1.478 x 10^118 possible ways the game could play out. Already this is past a Googol, and the total number of sub-atomic particles in the known universe !

9.332621 x 10^157

Number of ways 100 books can be ordered on a shelf

A hundred books can be ordered on a shelf 100 ! different ways. 100 ! = 100 x 99 x 98 x 97 x ... ... x 4 x 3 x 2 x 1 ~ 9.33 x 10^157

1.071508 x 10^301

Number of ways 1000 coins may land

The probability of flipping a coin repeatedly and getting a 1000 heads in a row on your first try are about 1 / 10^301 . The number of possible outcomes for 1000 flips is very close to a centillion.

1 : 6.449 x 10^284,265

Chance that a monkey would randomly type out Hamlet

Imagine a monkey, devoid of any sensibility, at a type-writer randomly pounding away without reason. Let's assume that the probability of hitting each of the keys is equal. Most of the time of coarse, the monkey will type out gibberish, but once and awhile the letters will fall into just the right place so that something intelligable is written. For example the chance of typing "cat" is roughly 1/19,683. Now imagine that this monkey were able to type for any indefinitely long period of time. What are the chances that the monkey would write out the entire manuscript for Hamlet ?

We can approximate it by making a few decisions. Hamlet would fill roughly 117 pages. Assuming each page contained about 1600 symbols, the entire work of Hamlet would be about 187,200 symbols long. The standard type-writer contains about 33 keys ( included the alphabet, the space, and puncuation marks ). Thus the probability would be roughly 1 out of 33^187,200 which is approximately 1 : 10^284,265. This number is absurdly huge !!! Compare it to 10^341, the hyper volume of space-time measured in planck units. This number is in a completely different class of numbers ! To put this in perspective somewhat, you would have to raise 10^341 to the 834th power to get a number as large as 10^284,265. This number is not a few order of magnitudes larger, it is an absurd number of magnitudes larger. In fact, if we divide 10^284,265 by 10^341 we get 10^283,924 which doesn't seem much smaller than the original value !

Already we are way way beyond what was discussed in the previous article ! But the numbers can get much higher still ...

3.622 x 10^12,041,199,826

Total number of possible Human Beings that could ever exist

It is almost certainly true that no one has ever had your DNA, even for the 4.5 billion years that life supposedly existed on earth. How can I be so sure ? Because the odds are incredably slim. Unless of coarse you are a member of twins , triplets , quadruplets , etc. or have been cloned, your DNA is VERY unlikely to arrise in another person purely by chance.

Most people assume that because everyone is unique, that this means that there is an infinite number of possible individuals that could exist. In otherwords, even if life were to continue indefinitely ( and we are talking ridiculously long time periods that could probably never occur ) that your DNA would never spontanteously reoccur in some other living thing. This however can't be true. Why ? Because DNA is broken into discrete units, a bit like a computer is broken down into "bits" of information. Our DNA consists of a series of chemical codes. There are 4 possible states of each discrete unit, commonly refered to as A , T , G , and C. DNA is severely curled up so that it can fit in a single cell, but if the DNA was completely unrolled it could stretch as much as 2 meters long ! That means that the DNA code may be composed of some 20 billion discrete units. Now imagine, each discrete unit can have any of 4 possible states, and there are 20 billion of them. There would be a massive number of possible combinations. One of these combinations of coarse represents you. Another someone else. In fact, every single person alive who has lived, or person who has ever lived,or will live would be one of these combinations. It has been said that somewhere around 100 billion humans have ever lived. The scary part is that, this would represent only the very tiniest fraction of all the possible combinations !

The number of combinations is staggeringly mind-bogglingly huge, but it is NOT infinite. It is simply part of a class of numbers I like to call "Hyper exponential". A hyper-exponential number is a number, that when expressed as a power of ten, has an exponent which itself is an exponential class number. In this case, the number of all possible combinations of human DNA is a number is about 12 billion digits. In otherwords, a number with 12 billion orders of magnitude ! Now consider, the number of all humans that ever lived is only 100 billion, or 10^11. That's a mere 11 orders of magnitude. As we saw in the last article, the universe may not be eternal, and even if it was, our chances of surviving as the universe continues to expand get slimmer and slimmer as time goes by. But even assuming humanity lasts for the roughly 10^34 years that the universe has ( in theory ) and assuming that the human race eventually colonized all of the universe ( which is unlikely if we can't go faster than light ) that would still only add a few measly orders of magnitude to the number of all living humans !! The bottom line is, you really don't have to worry about bumping into someone with exactly the same DNA who is otherwise unrelated to you.

10^2,860,989,078,790

Number of ways a single day on modern earth could play out

Consider this idea ... there are perhaps as many as 10^2,860,989,078,790 ways in which this very day on earth could "happen". How can we arrive at such a huge figure ?

Let's only consider the decisions that human beings make in this time frame. Let's assume that there are at least 2 choices that a person may make every minute of there lives. This is a generous underestimate. Surely we can think of at least 2, if not more, choices we could make within the next 60 seconds. So then, just how many choices do we make in a day ? Well a day is 1440 minutes, so we could say that we have 2^1440 ( approximately 10^433 ) choices of how we live out the day. Pretty impressive. But now consider, there are 6.6 billion people on the earth, and each also has 10^433 different options today. So all the possible ways that all humans could live out today would be (10^433)^(6,600,000,000). The resulting number has about 2.8 trillion digits !! Some of these possibilities could be earth shattering. Someone might choose to launch a nuclear war-head and initiate armageddon. But most possibilities would be quite mundane. After all, everyone in the world might just decide to sleep-in for the day.

10^(10^36)

Estimated odds of a person living at least a 1000 years

Robert Munafo explains that life insurance companys use a extrapolation formula to compute the odds of person living past a certain age. By applying this formula, Munafo made this estimate [4] .

10^(6.5446 x 10^343)

Promaxima : The total number of possible parallel universes

Let's now consider the total number of possible parallel universes. Actually the existence or non-existence of parallel universes is not important here. We merely want to know all the possible ways that the universe could have evolved from it's birth at the big bang to it's death as it fades away.

Specifically we want to create an upper bound on the number of possible universes. To do this , we will assume that as the universe expands it is always packed solid with sub-atomic particles, that each particle is distinct, and that they can be arranged in any manner. For example, the current volume of our universe is estimated at 10^184 cubic planck lengths. This means that our current universe has a maximum capacity of 10^184 subatomic particles. Now imagine that those particles can be arranged in any way (without overlapping). In this case there would be (10^184)! ways that the 10^184 particles could be assigned to the 10^184 spaces. We can then using sterlings approximation for factorials to get an estimate of this value. Sterlings approximation is ...

n ! ~ sqrt(2pi) x e^(-n) x n^(n+1/2)

Lastly we multiply all the possibilities for every planck time together for the entire history of the universe. I have taken alot of liberty with the calculations, but it does produce a figure which makes sense given the considerations. The upper bound on the number of possible universes should be roughly 10^(10^343). This number is much bigger than even the Googolplex, a mere 10^(10^100).

I've made this calculation several times, and gotten different values depending on how I set up the estimate [5]. I call this number "Promaxima" , meaning "probability maximum" [7]. It is meant to be a reasonable upper bound on the total number of possible ways the history of the entire universe could have played out. This is also the largest number that could possibly occur in science, because we have already considered every possible configuration of everything known to exist. In otherwords, this is the largest number which represents a tangible reality. Numbers larger than promaxima are beyond scientific application and are therefore "chimeras", numbers without real world counter-parts. I use promaxima as a dividing line between corporeal numbers ( counting numbers less than promaxima ) and ethereal numbers ( counting numbers greater than promaxima.)

I first coined promaxima back in 2004. However I was not the first to make such estimates. In fact, Robert Munafo lists a few estimates himself.

For example he says that "Dave L. Renfro" calculated by a different method the number of possible universes to be 10^(10^166). Munafo provides his own estimate of 10^(3.67 x 10^281) possible universes [4].

Beyond this, larger numbers only occur if we consider even more abstract and esoteric situations. The realities of the following examples is borderline between "real" and "purely hypothetical".

10^(5.5 x 10^405)

Robert Munafo's "Factorial of the Single-perturbation count"

In Munafo's own words this number is a " highly theoretical estimate of the number of different ways all the particles in the known universe could be randomly shuffled at each moment in time since the universe's creation". This sounds similiar to the way I usually compute upper bounds for promaxima. But he goes on to say that in "quantum mechanics, it is the number of universe timeline wave-functions that exist simultanteously from the viewpoint of an observer outside our universe" [4].

I'm not entirely sure that we are talking about the same thing, but this number does relate to reality in an abstract way. One reason this value is not realistic, is because it assumes that a particle can appear at any location in space and any given point in time. This means that there are cases where a sub-atomic particle could jump to opposite ends of the universe within a planck time, but this is impossible because the particle would need to be traveling many orders of magnitude faster than the speed of light. At best these numbers serve as upper bounds for actual probabilities. In this way it is just one step of abstraction beyond the largest numbers in combinatorics.

10^(10^(1.51 x 10^3,883,775,501,690))

Quantum Poincare recurrence time for quantum state of an extremely hypothetical black hole with mass of entire universe

This is the largest number I have ever seen related to physics !!! It is sited on Robert Munafo's large number list on page 21 (the last page of the list) [6] . This number represents a length of time, but it also represents an improbability.

The "Poincare recurrence theorem" states that a closed system will eventually return to any given state in the system an indefinite number of times given an indefinite amount of time. For example, imagine those lottery machines with the balls all being shuffled about. Imagine this was a closed system, and rather than some number being called the shuffling was allowed to continue indefinitely. Now imagine a single instance in time. Let's assume each ball is distinct ( perhaps each one has a unique number written on it). Now at some instance in time , call it t1 , each ball we be at some location within the mixer. Record the exact location of every single ball in the mixer. Now imagine the system being shuffled after this point in time. The balls get rearranged in a seemingly random manner, and pretty soon every ball is in a completely different place. Now the question is, will all the balls ever return to exactly the same configuration that they were in at t1 ?

Assuming that space is broken into discrete units, one would have to conclude that there are only a finite number of configurations. Therefore, given an indefinite span of time, any given configuration has to occur and indefinite number of times. In otherwords, some time in the future after time t1 , there will be another instance, call it t2 , in which the configuration will be identical ! The amount of time this recurrence could take could be extremely large however ! The amount of time between such recurrences is called the "Poincare recurrence time".

So basically the number shown above represents a recurrence time. Now in quantum mechanics, it is said that the void of space itself is unpredictable, and that it is possible for things to pop into existence from nothing, even though this is unlikely. In fact, our entire universe may have begun this way. The recurrence time in this case refers to the existence of a black hole with a mass of about the entire universe. In other words the number represents the amount of time it would take for the existence of a ultra massive black hole to recur if we were to treat the entire universe as if it were a closed system. This is only a rough idea of what the number is referring to though, based on my best understanding.

The number was first computed by Don N. Page, a gravitation physist. And it also appears in print in one of his published works. It could very well be the largest number that a physist has ever referenced in a published work to represent a "meaningful physical quantity". "Meaningful" is a bit ambiguous here, as even Don Page admits that this is all very hypothetical. In thermodynamics it is said that the entropy of a system always increases, and this is in conflict with the Poincare recurrence theorem. It may be that our universe is not a "simple closed system", and that the theorem can not be applied. Others argue however, that since the period of time is so vast, that perhaps the entropy law does not apply in the very long term. In any case we won't be around long enough to know the answer.

Here is the original quote of Don page to Frank Pilhofer where he explains this number ...

" I estimated a quantum Poincare recurrence time for the quantum state of an extremely hypothetical rigid nonpermeable box containing a black hole with the mass of what may be the entire universe in one of Andrei Linde's stachastic inflationary models and got 10^(10{10^[10^(10^1.1)]}] Planck times, millenia, or whatever. "

Don goes on to state that ... " So far as I know , these are the longest finite times that have so far been explicitly calculated by any physicist."

That seems likely. Physicists don't seem prone to just throw out large numbers carelessly, and probably would regard numbers without any physical representation as meaningless.

And that brings us to my most important point. There are limits to the numbers that could arrise in known physics. Recall that numbers go on "indefinitely". But the universe we know, seems to be finite. So there must only be a finite number of applications for numbers, and therefore must be a largest number with a physical meaning. But we have to be careful here. What do we mean by "physical meaning". After all, counting possiblities is not the same as counting objects. And Poincare recurrence times for real world systems is very theoretical. Rather than there being a sharp cutting off point between "real" numbers and purely "fictional" ones, there is a gradual blurring of the real and unreal. But eventually things would become so hypothetical that we would have to conclude that we are merely guessing or making things up with little emperical evidence. Such ideas would have to be treated as "pure abstractions" with no proven link to reality. Yet who is to say that our reality might no be much larger than we realize ? The truth is we can't be entirely sure where the boundary between the real and unreal lies, but even more frightening is that such a boundary may not exist at all. None the less, this too is speculative. The bottom line is that for all practical purposes things lieing beyond "known reality" are very remote and abstract to us. So one can say that certain numbers lie beyond all practical applications and only have limited meaning for us.

Numbers which are related to our understanding of the real world I call corporeal numbers. But beyond this there is an infinite array of larger numbers I call ethereal numbers. This distinction is important because I am really talking about large numbers as abstractions. We will soon be leaving the comfortable to domain of the corporeal numbers, to explore the titanic numbers far beyond them !

Before I get ahead of myself, let's get started on the next article. The next article goes into alittle more depth about my "number ranges"...

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Source Material :

[1] http://forumserver.twoplustwo.com/25/probability/most-heads-row-332174/ : This blog talks about longest known coin streaks and comfirms the fact that very few records of this exist. At the bottom of the page however a blogger named linkblaine offers this "I got heads 3 times in a row once".

[2] http://www.funny2.com/odds.htm : This page lists a number of "odds" including the chance of being struck by lightning somewhere in the 2nd part of the list.

[3] The formula for the bell curve is f(x) = e^((-x^2)/2)/sqrt(2 pi ) . In IQ scores, 100 is average ( z score = 0 ), and the standard deviation is 15 points. Therefore an IQ of 220 cooresponds to a z score of 7. By computing the area under the curve from -infinity to 7 and subtracting it from 1 you get the probability of IQ's 220 and up.

[4] http://www.mrob.com/pub/math/numbers-20.html : A page of Robert Munafo's big number site. Under the entry for 10^10^36 he explains where he got this value from. Also you can find the entries for 10^(10^166) , 10^(10^28) and 10^(10^405).

[5] My first computation of Promaxima, computed in 2004, gave me the value of 10^(10^245). Later in 2007 I revisited the calculation and made some improvements. I provided a range of values for promaxima between 10^(10^160) and 10^(10^270), which meant even a generous lower bound for the number of possible universes is still larger than a Googolplex. This latest calculation uses a model of the universe as the surface of a glome, and uses an "average radius" for the entire history of the universe because computing this value with the universe continuously expanding complicates computation.

[6] http://www.mrob.com/pub/math/numbers-21.html#page_alt_univ_count : This is page 21 of Robert Munafo's large number list, where he mentions the number 10^(10^(10^(10^(10^1.1)))). It is the largest number explicitly computed my a professional scientist to represent an "actual theoretical quantity".

[7] http://c2.com/cgi/wiki/?ReallyBigNumbers : This wiki page was the very first place that I ever made mention of "Promaxima" on the world wide web. The article I wrote can be found more than half way down the page and can be identified by the signature I left at the bottom of my entry.