Letters From Darwin - Understanding Probability With A Universe In A Box
How likely is one thing to happen as opposed to another? Is there a way for the human mind to easily understand probability beyond simple 50/50ish type chances?
If we give an infinite amount of monkeys, an infinite amount of time; will one of them randomly recreate every single Shakespeare play ever written?
You may have caught my recent article: Letters From Darwin - An Evolutionary Coin Flip. In that article I try and show via a thought experiment, that some things when viewed in isolation look magical. However once we take a step back or two, and view the bigger picture, we realise that the probability of our event taking place was near 100%.
However some of the answers I got to that post, especially the brilliant response from @prydefoltz made me realise that I hadn't made my point as succinctly as I intended.
So here we go with attempt number two, again we will use a thought experiment to try and illustrate our point.
Google A Googol
When Larry Page was considering what to call his internet search engine, he decided upon the word Google, which he adapted from the real word; Googol, which is a 1, with 100 zeros after it.
To put that into perspective, the amount of atoms in the whole universe is a 1 with 80 zeros after it, and if you filled the universe with grains of sand, you'd get to roughly 1 with 90 zeros.
A googolplex, is a 1, with a googol amount of zeros after it! It is a very, very large number, in fact it is the largest number with a name.
So if the universe is 1 googolplex metres across; there is definitely an exact copy of you living out their own separate life, somewhere in the universe.
How can I be so sure? Let's look at boxes to help us.
The Box Experiment
Let us imagine for one second that we are going to create dummy universes, using empty cardboard boxes, and six sided dice.
You are standing in a big featureless hall, at your feet is a cardboard box, and in your hand is a dice.
You place the dice in the box and close it; you then pick up the box and throw it across the room. Then you take another empty box, and put a dice in it, and throw it across the room.
Now you are looking at the two boxes, and of course you don't know what each one says. However you do know, that the probability of the second dice being the same as the first, is 1 chance in 6.
You open both boxes and you see that the first was a 2 and the second was a 5. You quickly realise that in order to guarantee that there are at least 2 dice which agree with each other, you are going to have to throw 7 boxes each containing one dice.
So you do, you place just one dice into each box and repeat this 7 times. This now gives you a universe whereby at least two numbers agree with each other.
However you want to make sure there are copies of every single number, so you work out that you need to throw 42 boxes. You figure that if seven boxes gives you one guaranteed double, then 6 x 7 boxes will give you six guaranteed doubles.
Imperfect Probability
So now you have tossed your 42 boxes each containing one dice. You check the boxes and see that you haven't got doubles of every single number. Remarkably, you have only thrown one 2. So you carry on throwing for a while until you get a two.
As it turns out, it takes you another 10 boxes to roll a two. You start the experiment again, you make sure that the numbers 1-6 are represented in your first set of boxes, and then you start rolling.
You realise that there is still a probability of not throwing doubles if you don't throw enough boxes, maybe 52 was lucky.
So instead you throw 36x36x36x36x36x36(x2) = 532 boxes.
Now you are happier, you know that there are 216 possible combinations when throwing 6 dice together, this is because each die has six faces, and when thrown with another die, you times 6 by 6 to get the number of possibilities.
You realise the room you are in isn't big enough, so you go next door to a room that is ten times as big. Now you can throw thousands of boxes if you want to. So you do, you throw five thousand boxes, now you know without a doubt, that every single number is represented at least twice.
This is because the probability of there only being one 5, for instance is almost zero, seeing there are only five other numbers that aren't five, and you have thrown so many dice.
From Boxes To Universes
Now we go back to the universe, and instead of dice in a box, we are looking at the human body; which is roughly 1 cubic metre.
Then instead of dice, we are using atoms, in fact we look at all the atoms in your body and realise that just like the dice, there are a finite way to arrange those atoms
For a volume the size of your body the number of possible arrangements of atoms is 10 x 10 to the power of 70, which is a 1 with 70 zeros after it. Which is far less cubic metres than would exist if our universe was a googolplex metres across.
So in the same way, as long as you had a big enough room like in our thought experiment and you kept on throwing boxes (rearranging atoms), you would eventually get a copy of every single dice. So too would you get a copy of every single person in a googolplex universe!
What's The Point?
My point is, that on a long enough time line, everything will eventually happen. You will toss 100 heads in a row, or type out the complete works of Shakespeare by accident.
In fact, you wouldn't need an infinite amount of monkeys; you would just need one monkey with an infinite amount of time and stamina. Because there are a finite amount of combinations, those typewriter keys can be hit; therefore given enough time, all combinations will be typed.
This is the essence of chance and probability, it is mundane, because it says just do something enough times, and the desired outcome will happen.
It's not very magical, or spiritual, so sorry about that; it is however real, and of course very useful; try, try, try again, and you will succeed.
Further reading & study:
Surviving A Post Truth World - Fighting A War Between Reason And Superstition
Numberphile Gogoolplex Universe
AS EVER, LET ME KNOW BELOW WHY YOU DISAGREE WITH ME!!!
I agree it could , but not would no matter how much time went by.
There would be no other choice but to :-)
Cg
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This was an amazing read @crytpogee. But if we are considering alternative version of ourself or planet, it is also fair to consider an alternative universe possible where the Big Bang never occurred.
And relating to your dice in a box illustration, it implies that the alternative versions of us are exactly like us. I watch alot of SCI FI and it has made me understand that different versions of ourselves are formed from choices we make, so I'd say from your dice illustration that those alternate versions implies may not necessarily be the same. They make look and behave like us but there may also be changes. I'm just implying that they may not necessarily be the same as us
Absolutely! There could be an infinite number of universes and this could be the first one that has produced organic life, or indeed intelligent life.
Only in appearance, of course an exact copy of you would have lived his own life , with his own hopes and dreams.
So yeah, you definitely filled in the blanks for me there! :-)
Cg
Very interesting topic and post! Made me think that maybe there is an exact copie of me for a short period of time but in the long run it seems impossible: surroundigs and conditions we live in affects us and our appearances (causes e.g. Oxidative stress). We would have to live in perfeclty copied surroundings with my copy and experiece exactly same things so that we would remain the same. Do you think this makes any sense?
Yeah damn agree with your lect like rain happen or not its also probability
And wha you say if i ask match win or lost or may be drawn
What you say is there a probability or not???
Reesteem. Nice to see the infinite monkey theorem applied here. But i like to disagree :D
Im not sure if the borel cantelli lemma isnt more binary then we biologists think it is
only IF there is infinite amount of time (wich is possibly not the case) THEN anything will happen
if there is no infinite amount of time, then most things are practicaly impossible (like life originating by random bruteforcing)
but however since evolution uses an intelligent algorithm (with selection + mutation) it does not depend on pure randomness (the monkeys)
it is like bruteforcing a SHA-256 generated private Key, not possible in practice even with quantum computation because it needs 256² iterations or something like this. But with an intelligent search alogorithm even the MIT cracked the Key of IOTAs Curl-Algorithm (which most people dont know) in no time. https://github.com/mit-dci/tangled-curl/blob/master/vuln-iota.md
*"intelligence" not in the sense of intelligent design (the blind watchmaker)
This post was really educative, I remember back then in college when probability theory was one of My best topic. It's really not magical
It isn't, but it does seem magical when viewed from a particular angle :-)
Cg
Hmmmmm, interesting piece this @cryptogee. Am amazed at your power to carefully analyse a scenario like this. Although no two persons are the same but I will agree with you on another basis which I found to be amazing, sometimes you will feel like you have done something you are doing right now. That moment will feel magical to you. Like the way it was scripted on the past is the exact way it is happening again. I think at that moment we witness a self probability of same events, so if there is a self probability, I believe there will be a cross probability as well. Although that scope is wide. But your analysis certainly shed some light on the matter
This is an interesting theory, and if we were just a collection of atoms, of course, there would be a copy of you out there.
Fortunately, we are all unique, in all of space and time.
The ramifications of the double slit experiment is that the universe doesn't hold superfluous information. Meaning, if there was actually two of you, it would just be you, in two places.
Further, what we call scientific laws are not the same galaxy to galaxy. In your analogy, that means this galaxy uses 6 sided dice in its boxes, and andromeda uses 7 sided dice (with different symbols), so there can never be a match.
Yes, your statistics are correct, but the assumptions are invalid.
I am always a day behind on these! I mentioned the monkeys in my comment on your last post lol. A day late and a dollar short as we say here.
It seems to me there is very little magic in the end. With science we continually get a peek at the real Oz or the man behind the curtain...it all boils down to math and physics. I have been challenged on this by math guys that argue it is ALL math, but I say why squabble.
Instead of learning statistics, acknowledging this fact will make life easier.
A Nice and very insightful post