Theoretical probability question

[Corax]

Theoretical probability - fun, right?

I think it kinda is though

Also in instance also a part of lexicology, which was where my interest in this question originates from - the reckless* way in which "certain" gets thrown about. My understanding is that *certain* and **impossible* are not valid probabilistic concepts, strictly speaking. It's very very lots-of-verys likely that if I walk into a wall the result will be me bumping into a wall. The likelihood is, I'd guess 99.9% recurring. But 99.9% recurring ends up being resolved into 100%, meaning that it's *certain* - and my argument is invalid

But what if it's not actually recurring, it's just more 9s than you can count then ending in "...4362" or something? My GCSE teacher taught us that mathematical probability took this to always be the case, so in that respect neither 1% nor 0% were possible. In my wall example that must be true, because it's unlikely but technically feasible for it to happen at a quantum scale* if everything lines up perfectly, so therefore must be at a macro scale, albeit reeeaaally unlikely it's still possible

So this isn't actually probabilistically certain, right? In which case, my example was right in the first place

Just walked into a wall to test it. Bit bruised. Only another 99.9 recurring attempts to go. Except that's not how probability works (another misunderstanding that I see happen a lot) so it could be the next one...



A different smiley at the end of each paragraph - do I win a prize?

* Interesting etymology by the way. Nothing to do with "wreck" but a closer relationship to "regard"...
** IIUC - and if I don't, I'm sure there are other examples that can be applied

 

[Puddy_Tat]

 

[Pickman's model]

mrs quoad

 

[mrs quoad]

I think all you’re doing is asking how often you won’t hit a wall if you walk into it, but are making it look more sciencey with numbers.

?

Edit: and then you’ve made up a number that you’ve attached to walking into walls (without any attendant workings / parameters / assumptions that would support that number being used), and pointed out that that number is effectively meaningless.

 

[NoXion]

Theoretical probability - fun, right?

I think it kinda is though

Also in instance also a part of lexicology, which was where my interest in this question originates from - the reckless* way in which "certain" gets thrown about. My understanding is that *certain* and **impossible* are not valid probabilistic concepts, strictly speaking. It's very very lots-of-verys likely that if I walk into a wall the result will be me bumping into a wall. The likelihood is, I'd guess 99.9% recurring. But 99.9% recurring ends up being resolved into 100%, meaning that it's *certain* - and my argument is invalid

But what if it's not actually recurring, it's just more 9s than you can count then ending in "...4362" or something? My GCSE teacher taught us that mathematical probability took this to always be the case, so in that respect neither 1% nor 0% were possible. In my wall example that must be true, because it's unlikely but technically feasible for it to happen at a quantum scale* if everything lines up perfectly, so therefore must be at a macro scale, albeit reeeaaally unlikely it's still possible

So this isn't actually probabilistically certain, right? In which case, my example was right in the first place

Just walked into a wall to test it. Bit bruised. Only another 99.9 recurring attempts to go. Except that's not how probability works (another misunderstanding that I see happen a lot) so it could be the next one...



A different smiley at the end of each paragraph - do I win a prize?

* Interesting etymology by the way. Nothing to do with "wreck" but a closer relationship to "regard"...
** IIUC - and if I don't, I'm sure there are other examples that can be applied

I'm pretty sure that our current understanding of quantum mechanics means that, given enough attempts, you will be able to walk through that wall at least once. The positions of particles are probabilistic, and there is a non-zero probability that all of the particles that make up your body could change position from one side of a barrier to the other, in a process akin to quantum tunnelling. This page goes into more detail.

The issue being that the probability of you being able to walk through a wall is so ridiculously low that you would have to be making attempts for absolutely ludicrous stretches of time - I'm not sure of the exact figures and I have no idea how one would calculate them, but the length of time is something on the order of many times greater even than the most optimistic projections for how long the familiar world of baryonic matter is expected to last until overtaken by universal heat death.

So this means that for all practical purposes, it's impossible for you to simply walk through a wall unaided, even if you have a non-zero possibility of success in theory.

However, if the Many-Worlds Interpretation of quantum mechanics reflects reality, then all physically valid configurations of matter and energy are in existence, which means there are an infinite amount of universes where you jump through walls all the time, as well as a much larger infinite amount of universes where you just end up with a bloody nose. This video is relevant and informative, and really hammers home the weirdness involved:

 

[mrs quoad]

Probablity of Walking through a Wall

^^^ lots of threads from physicists there.

 

[2hats]

There are so many elementary particles in your body that it would take an incomprehensible number of universe lifetimes for you to experience quantum tunnelling through a nearby wall. The chance of it happening is finite, does exist, but is so mindbogglingly small that you have a better chance of winning every lottery in the world every time it is played whilst simultaneously being struck by lightning (far less than 1 in 10^(10^30) - that’s a number that dwarfs the number of atoms in the observable universe).

 

[Mumbles274]

Big numbers are weird

A googol... 1 followed by 100 zeros is more than all the atoms in the observable universe.

Infinity is an even weirder number, i think there is still a documentary on iPlayer about Infinity

 

[Mumbles274]

Yep... It's

BBC - Horizon, 2009-2010, To Infinity and Beyond

 

[High Voltage]

But if you were to run at the wall, full tilt and, say, lead with your face, I'm sure THAT would work. Give it a go and report back.

 

[NoXion]

Big numbers are weird

A googol... 1 followed by 100 zeros is more than all the atoms in the observable universe.

Infinity is an even weirder number, i think there is still a documentary on iPlayer about Infinity

Infinities aren't numbers. 1 is a number, as is Pi. Infinities, while they can vary in size relative to each other, are uncountable, thus not numbers. There is an infinity of positive integers above 0, and another infinity of negative integers below 0. Positive and negative integers together constitute a larger infinity than either separately.

 

[Mumbles274]

Infinities aren't numbers. 1 is a number, as is Pi. Infinities, while they can vary in size relative to each other, are uncountable, thus not numbers. There are an infinite number of integers about 0, and another infinity of negative integers below 0. Positive and negative integers together constitute a larger infinity than either separately.

Hence why they are weird

 

[NoXion]

Hence why they are weird

Infinities are weird for sure, but they are certainly not numbers. You can't count an infinite number of objects, because as soon as you stop counting, you reach a finite quantity that can be counted to.

 

[TruXta]

Hence why they are weird

You wanted weird infinity?

Hilbert's paradox of the Grand Hotel - Wikipedia

 

[NoXion]

Here's a video about infinities:

 

[NoXion]

You wanted weird infinity?

Hilbert's paradox of the Grand Hotel - Wikipedia

Damn your eyes. Infinitely.

 

[Mumbles274]

In the BBC documentary ^

 

[squirrelp]

My understanding is that *certain* and **impossible* are not valid probabilistic concepts, strictly speaking.

They are. I think most mathematicians would be happy to agree that the probability of rolling a six with a die that has no sixes on it is zero.

Practically speaking, if there is always a slim chance, however slim, of a rule being broken, then we are living in a universe with no absolute rules. Is that quite certainly the case?

 

[TruXta]

In theory there's nothing odd about a probability of zero or 100%. But OP is on about ontology rather than probability.

 

[squirrelp]

Almost surely - Wikipedia

 

[littlebabyjesus]

Infinities aren't numbers. 1 is a number, as is Pi. Infinities, while they can vary in size relative to each other, are uncountable, thus not numbers. There is an infinity of positive integers above 0, and another infinity of negative integers below 0. Positive and negative integers together constitute a larger infinity than either separately.

Infinities are a bit weirder than that tbh. The infinity of positive integers is the same size as the infinity of all integers. They're both countable, meaning that you can line them up next to one another in a particular order that doesn't miss any out. Then there are the so-called uncountable infinities such as the set of real numbers (including irrational numbers with rational numbers). Irrational numbers are another concept with a slightly fuzzy edge to their properties.

Cantor's diagonal argument proves the uncountability of real numbers.

 

[littlebabyjesus]

In theory there's nothing odd about a probability of zero or 100%. But OP is on about ontology rather than probability.

It's always possible to set up an impossible condition. You can use maths to do it, so eg you pick two integers a and b at random and divide a by b. Now you square the result. What is the probability that your answer will be exactly 2? Precisely zero.

 

[mrs quoad]

Another one for anyone who’s keen to get another 43 minutes closer to death whilst listening to something about infinity: BBC Radio 4 - In Our Time, Infinity

 

[squirrelp]

Infinities are a bit weirder than that tbh. The infinity of positive integers is the same size as the infinity of all integers. They're both countable, meaning that you can line them up next to one another in a particular order that doesn't miss any out. Then there are the so-called uncountable infinities such as the set of real numbers (including irrational numbers with rational numbers). Irrational numbers are another concept with a slightly fuzzy edge to their properties.

Cantor's diagonal argument proves the uncountability of real numbers.

Perhaps reassuringly though, infinities have turned out to all be the same size
Mathematicians Measure Infinities, and Find They're Equal

 

[Corax]

I'm pretty sure that our current understanding of quantum mechanics means that, given enough attempts, you will be able to walk through that wall at least once. The positions of particles are probabilistic, and there is a non-zero probability that all of the particles that make up your body could change position from one side of a barrier to the other, in a process akin to quantum tunnelling. This page goes into more detail.

The issue being that the probability of you being able to walk through a wall is so ridiculously low that you would have to be making attempts for absolutely ludicrous stretches of time - I'm not sure of the exact figures and I have no idea how one would calculate them, but the length of time is something on the order of many times greater even than the most optimistic projections for how long the familiar world of baryonic matter is expected to last until overtaken by universal heat death.

So this means that for all practical purposes, it's impossible for you to simply walk through a wall unaided, even if you have a non-zero possibility of success in theory.

However, if the Many-Worlds Interpretation of quantum mechanics reflects reality, then all physically valid configurations of matter and energy are in existence, which means there are an infinite amount of universes where you jump through walls all the time, as well as a much larger infinite amount of universes where you just end up with a bloody nose. This video is relevant and informative, and really hammers home the weirdness involved:

I've only just started reading the replies to this thread, but the above is brilliant, and answers my question beautifully.

So, theoretically, the next time I walk into a wall I could pass straight through it.

At which point I'd tell anyone and everyone I could about it, post up YouTube videos of me failing to repeat it, be ridiculed to the ends of the earth, and ultimately be institutionalised and heavily sedated for my own protection

Or, as the same probabilities would apply to each attempt, I'd be able to walk through walls for the rest of my life, and be venerated globally, unite peoples of all faiths and none, end all wars, redistribute the wealth of the earth equally, and be able to influence the future destiny of all aspects of mankind.

 

[kabbes]

Perhaps reassuringly though, infinities have turned out to all be the same size
Mathematicians Measure Infinities, and Find They're Equal

The article doesn’t say that infinities have turned out to all be the same size.

It says that two specific infinities have turned out to be the same size. Specifically, two infinities that both measure ways of dividing up the natural numbers:

“Some problems remained, though, including a question from the 1940s about whether p is equal to t. Both p and t are orders of infinity that quantify the minimum size of collections of subsets of the natural numbers in precise (and seemingly unique) ways.

The details of the two sizes don’t much matter. What’s more important is that mathematicians quickly figured out two things about the sizes of p and t. First, both sets are larger than the natural numbers. Second, p is always less than or equal to t. Therefore, if p is less than t, then p would be an intermediate infinity—something between the size of the natural numbers and the size of the real numbers. The continuum hypothesis would be false.”

It turns out that p and t are equal in size, which is what the article is about. Note that BOTH are larger than the infinity of natural numbers, however. But as the article points out, the infinity of real numbers eclipses the infinity of natural numbers. And, in fact, there are an infinity of infinities greater than this, even.

 

[Corax]

I'd have thought that some infinities had to be larger than other infinities. The set of whole numbers is infinite. But for every whole number there are ten decimals, so that set is also infinite but bigger than the first one.

Seeing that simple thing explained is something that turned my non-mathematician brain to sludge.

 

[Corax]

They are. I think most mathematicians would be happy to agree that the probability of rolling a six with a die that has no sixes on it is zero.

Practically speaking, if there is always a slim chance, however slim, of a rule being broken, then we are living in a universe with no absolute rules. Is that quite certainly the case?

But under many worlds quantum weirdness I'd have thought it possible that the atoms may spontaneously rearrange themselves in a very specific way on one of the faces - so still not *impossible*...

 

[kabbes]

I'd have thought that some infinities had to be larger than other infinities. The set of whole numbers is infinite. But for every whole number there are ten decimals, so that set is also infinite but bigger than the first one.

Seeing that simple thing explained is something that turned my non-mathematician brain to sludge.

There are more decimals (ie real numbers) than whole numbers... but not for the reason you gave.

For example, there are precisely as many whole numbers as there are numbers that are divisible by 10. So your reasoning is erroneous.

 

[Corax]

There are more decimals (ie real numbers) than whole numbers... but not for the reason you gave.

For example, there are precisely as many whole numbers as there are numbers that are divisible by 10. So your reasoning is erroneous.

You've lost me!