Murphy's Law

1. ## Murphy's Law

I remember hearing the law from Interstellar. The law states that "anything that can happen, will happen" or "anything that can go wrong, will go wrong".

This law makes the most sense when applied to, say a die. If you roll a die once, you only have a 1 in 6 chance of rolling a 6. However, if you repeatedly roll a die, you necessarily have a 100% chance of eventually rolling a 6. This can be computed directly. The probability that a die does not roll a 6 n times is given by the following formula:

${\color{White}(5/6)^n}$

If we take the limit as n tends to infinity, the probability dwindles to 0. Put differently, that means that "It's impossible for one to roll a die over and over until the end of time and never get a 6".

This is the same principle underlying the monkey-typewriter principle. It's very unlikely a monkey hits a random selection of keys and writes out the works of Shakespeare. But the probability that it doesn't happen, when powered by large enough n, will eventually dwindle to 0 too. This means the monkey will necessarily eventually write the full works of Shakespeare. Though it's very difficult, the monkey has literally all of eternity to get it done.

However, Murphy's Law does have its limitations. You can make up games where at every stage you could win but there's still a chance you'll never win. For example:

Suppose you play a game where, in the first round, you just need to flip a heads to win. If you lose, you decide to make it harder for yourself - in the second round, you demand you need to flip two heads in a row to win. If you fail this, then in the third round you'll demand you need three heads, and so on. It turns out that it's possible to never win this game, even though the probability of winning in just a couple rounds is greater than 60%.

How come? In the previous cases, though the probability of winning at any particular moment may have been challenging, it was at least fixed. This game gets harder the longer we play. Thus, after writing out the equation and taking the limit as n tends to infinity, one sees that even all of eternity may not be enough to win, if you're unlucky enough.

But is that really the case? What does that statement even mean - "it's possible to never win"? The statement about the previous games "you'll eventually win" was at least something that was possible to tangibly construct. You could get out a die, roll it over and over and verify that you do eventually get a 6 every time. The task of seeking a win will take finite time, even if you're not sure how much time exactly.

On the other hand, this new statement is another kind of monster. You can't verify that there exist games you'll never be able to win. You play this game for a hundred rounds and sure, you're probably in a game you'll lose no matter what because now you need a hundred coinflips in a row to win. But you may nevertheless still win the next round. And if not the next round, then possibly the one after that. It may be unlikely you'll win soon, but you can literally keep going until you win.

That's kind of where I'm going with this. This mathematical statement is, as an equation, at the very least logically consistent. But we will literally never be able to verify or falsify the statement in the real world. Hence, can the statement truly be trusted to reflect our reality? Perhaps our reality does respect our pretty little formula and will allow some games to last forever - or perhaps it doesn't. Perhaps it simply doesn't care.

If your instinct is to say "our reality will respect the formula because math is math and math is intrinsically correct", I ask you to question where this trust in maths comes from. Indeed, we trust maths so much precisely because we can verify that our calculations reflect our reality's behaviour through experimentation - without the experimentation, why should we still trust math?

2. ## Re: Murphy's Law

Interesting. This reminds me of sometimes when I can't remember, for example, a concrete sentence from a book. I then think how much time would take me to find the correct sentence, if not by memory then by mere chance.

How many years, centuries, millenia? Stupid thoughts, for sure, but quite similar to what you have talked about here.

3. ## Re: Murphy's Law

If you're right and that math, the most logical and "non-biased" way to approach the world, doesn't work, then it implies humans will never ever access the "actual reality", the objective truth, and that there's nothing to do. Experimentation itself is biased because we are humans and see the world with our eyes, not the eyes of an almighty being that knows everything exactly.

That being said, I think we should trust math. We've been trusting them, and hey, it's working! If maths were that bad, I don't think the technology humanity has at the moment would be possible to have.

As for the dice thing, I'd like to come back on the gambler's fallacy: even though you have 1/3 chance of rolling a 6 once when you roll a dice twice, you have 1/6 chance once the dice has been rolled once. Repeat that infinitely, and it's totally possible that you will never get that 6.

4. ## Re: Murphy's Law

But how do we know we have infinite possibilities?
Only so many chances have occurred in time thus far. As far as we know, the universe has only existed for a set amount of time. Will the universe continue to expand, bringing time with it? Or will entropy fall, and the universe ceases to move. Chances only exist, given the correct circumstance after all. And circumstances only exist for so long.

But why should we trust math?
Math is really just deduction, the basis of which rests on simple laws and abduction. We can apply deduction to simple calculations, run the experiment, and see that it works. Then by abduction, we no longer need to repeat the experiment, we just know it works. Amass the simple calculations, and there will be reason to believe that the complex calculations will run the same way. Alas, some things are more complex than what we can predict or perceive, but stressing over our trust brings just undue worry. It's likely that some theory will topple quantum physics, just as it had toppled Newtonian physics, but until that day comes, I'm content trusting that particles are entangled, and that everything is a wave. Oh, and that the sun will rise tomorrow even though it's not massive enough to force nuclear fusion.

5. ## Re: Murphy's Law

You're all confusing physics and the physical world with math. You absolutely can verify that you'll never win that hypothetical game because you've set up the rules for it and proved that it is potentially unwinnable. The game doesn't exist in reality, so it being experimentally unverifiable is completely irrelevant. Concepts like time don't exist in the example game you've defined. Does it actually reflect our reality? No, of course not, because that's outside the scope of the game you defined. You absolutely could, if you wanted to, calculate the real-world probability that you'll never win the game within constraints (a human lifetime, the lifetime of the universe) and make it as detailed as possible.

Also, yes the statement "math is math and math is intrinsically correct" is accurate. Any math defined within the axioms you accept is accurate. It doesn't even matter if it reflects the real world because math isn't an intrinsic property of the universe, it's just useful for describing certain things. Pure math, for example, is entirely disconnected from the real world and often has no connection or relevance to the physical world.

6. ## Re: Murphy's Law

Originally Posted by yzb25
I remember hearing the law from Interstellar. The law states that "anything that can happen, will happen" or "anything that can go wrong, will go wrong".

This law makes the most sense when applied to, say a die. If you roll a die once, you only have a 1 in 6 chance of rolling a 6. However, if you repeatedly roll a die, you necessarily have a 100% chance of eventually rolling a 6. This can be computed directly. The probability that a die does not roll a 6 n times is given by the following formula:

${\color{White}(5/6)^n}$

If we take the limit as n tends to infinity, the probability dwindles to 0. Put differently, that means that "It's impossible for one to roll a die over and over until the end of time and never get a 6".

This is the same principle underlying the monkey-typewriter principle. It's very unlikely a monkey hits a random selection of keys and writes out the works of Shakespeare. But the probability that it doesn't happen, when powered by large enough n, will eventually dwindle to 0 too. This means the monkey will necessarily eventually write the full works of Shakespeare. Though it's very difficult, the monkey has literally all of eternity to get it done.

However, Murphy's Law does have its limitations. You can make up games where at every stage you could win but there's still a chance you'll never win. For example:

Suppose you play a game where, in the first round, you just need to flip a heads to win. If you lose, you decide to make it harder for yourself - in the second round, you demand you need to flip two heads in a row to win. If you fail this, then in the third round you'll demand you need three heads, and so on. It turns out that it's possible to never win this game, even though the probability of winning in just a couple rounds is greater than 60%.

How come? In the previous cases, though the probability of winning at any particular moment may have been challenging, it was at least fixed. This game gets harder the longer we play. Thus, after writing out the equation and taking the limit as n tends to infinity, one sees that even all of eternity may not be enough to win, if you're unlucky enough.

But is that really the case? What does that statement even mean - "it's possible to never win"? The statement about the previous games "you'll eventually win" was at least something that was possible to tangibly construct. You could get out a die, roll it over and over and verify that you do eventually get a 6 every time. The task of seeking a win will take finite time, even if you're not sure how much time exactly.

On the other hand, this new statement is another kind of monster. You can't verify that there exist games you'll never be able to win. You play this game for a hundred rounds and sure, you're probably in a game you'll lose no matter what because now you need a hundred coinflips in a row to win. But you may nevertheless still win the next round. And if not the next round, then possibly the one after that. It may be unlikely you'll win soon, but you can literally keep going until you win.

That's kind of where I'm going with this. This mathematical statement is, as an equation, at the very least logically consistent. But we will literally never be able to verify or falsify the statement in the real world. Hence, can the statement truly be trusted to reflect our reality? Perhaps our reality does respect our pretty little formula and will allow some games to last forever - or perhaps it doesn't. Perhaps it simply doesn't care.

If your instinct is to say "our reality will respect the formula because math is math and math is intrinsically correct", I ask you to question where this trust in maths comes from. Indeed, we trust maths so much precisely because we can verify that our calculations reflect our reality's behaviour through experimentation - without the experimentation, why should we still trust math?
Any chance you could post that equation? I’m kinda curious

7. ## Re: Murphy's Law

My guess is that’s it’s something like 2^-1 + 2^-2 + etc until 2^-n

8. ## Re: Murphy's Law

Or can you not add them? You can’t add them I think.
In that case it would seem the game is literally impossible to win.
Maybe our math is wrong then. Alternatively, reality is not deterministic which is a very interesting concept albeit one which is impossible to prove/disprove - I think you need to ask yourself in you believe in determinism if you want to get an answer to your question.

Personally, I think math is wrong and that reality is still deterministic (i.e. it’s possible to win that game in the long run, just that our mathematical tools are insufficient to verify that claim)

9. ## Re: Murphy's Law

Alternatively, infinity doesn’t exist or has no meaning in reality and you will always, therefore, win the game in the long run.

10. ## Re: Murphy's Law

Alternatively this statement might mean something like ‘if the game never stops, it’s 100% the case that you haven’t won’.

11. ## Re: Murphy's Law

So if you impose an additional constraint and decide to play the game until n=100 and then start over if you haven’t won, you will definitely win at some point.

12. ## Re: Murphy's Law

Originally Posted by Ganelon
Alternatively this statement might mean something like ‘if the game never stops, it’s 100% the case that you haven’t won’.
I’m actually leaning towards this interpretation right now.

13. ## Re: Murphy's Law

It's simple. You reverse the polarity of the Neutron flow.

14. ## Re: Murphy's Law

Originally Posted by SuperJack
It's simple. You reverse the polarity of the Neutron flow.
You need to cross the streams

15. ## Re: Murphy's Law

Remember tho that you’re talking to someone who sniped statistics with literally 5.0. So

16. ## Re: Murphy's Law

Originally Posted by Ganelon
Any chance you could post that equation? I’m kinda curious
Spoiler : Note that we will use the following formulae: :

[probability of winning round n]=2^-n

[probability of losing round n]=1-2^-n

[probability of independent events A and B both occuring]=[probability of A occuring][probability of B occuring]

Informally speaking, events are independent if the probability of one occuring is irrelevant to the other's probability. Suppose you roll a die. Whether you rolled an even or odd number is independent of the probability that a number less than 3 was rolled. i.e. being informed the roll is odd has no effect on whether the second condition is satisfied, and vice versa. However, whether you rolled an even or odd number is not independent of the probability that a number less than 4 was rolled. Being told your roll is even makes it rather less likely a number less than 4 was rolled. Write out all the possibilities and see why.

[probability of one of two mutually exclusive events A or B occuring]=[probability of A occuring]+[probability of B occuring]

Two events are mutually exclusive if they cannot occur together. Back to the die, rolling a 5 is mutually exclusive from rolling a 6. Hence, the probability of rolling a 5 or a 6 is simply 1/6+1/6.

Spoiler : final formula :
[probability that you win] = [probability that you win on exactly round 1] + [probability that you win on exactly round 2] + [probability that you win exactly on round 3]...

= [probability of winning round 1] + [probability of losing round 1][probability of winning round 2] + [probability of losing round 1][probability of losing round 2][probability of winning round 3]...

= [2^-1]+[(1-2^-1)(2^-2)]+[(1-2^-1)(1-2^-2)(2^-3)]+[(1-2^-1)(1-2^-2)(1-2^-3)(2^-4)]...

Spoiler : possibility of losing :

[2^-1]+[(1-2^-1)(2^-2)]+[(1-2^-1)(1-2^-2)(2^-3)]+[(1-2^-1)(1-2^-2)(1-2^-3)(2^-4)]...

< [2^-1]+[2^-2]+[2^-3]+[2^-4]...

= 1

Hence, the probability of the game being won is less than 1, and it's possible to never win the game.

I neglected to include all this to make it a little less intimidating lol. Also, idk how to make forum latex work with words so you have to suffer the horrible notation lol.

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