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I'm not sure about the off-topic rules here, but how about this:

Why are some of the drinks so expensive, given that all of them are mostly water?

Sometimes we use the phrase "given that" to mean, "considering that". Here, we do not mean, some of the drinks are not mostly water but we are not talking about them. We mean that literally all the drinks are mostly water.

1notfnofn5mo

Jumping in here: the whole point of the paragraph right after defining "A" and "B" was to ensure we were all on the same page. I also don't understand what you mean by: and much else of what you've written. I tell you I will roll a die until I get two 6s and let you know how many odds I rolled in the process. I then do so secretly and tell you there were 0 odds. All rolls are even. You can now make a probability distribution on the number of rolls I made, and compute its expectation.

A very strange probability paradox

darrenreynolds5mo20

Yes, exactly - thank you. It depends on the interpretation of the phrase "given that all rolls were even". Most ordinary people will assume it means that all the rolls were even, but as you have succinctly explained, that is not what it means in the specialist language of mathematics. It is only when you apply the latter interpretation, that some of the rolls are odd but we throw those out afterwards, that the result becomes at first surprising.

I do find LessWrong a curious place and am not a regular here. You can post something and it will get downvoted as wrong, then someone else comes along and says exactly the same thing and it's marked as correct. Heh.

3Ben5mo

Yes, its a bit weird. I was replying because I thought (perhaps getting the wrong end of the stick) that you were confused about what the question was, not (as it seems now) pointing out that the question (in your view) is open to being confused. In probability theory the phrase "given that" is a very important, and it is (as far as I know) always used in the way used here. ["given that X happens" means "X may or may not happen, but we are thinking about the cases where it does", which is very different from meaning "X always happens"] A more common use would be "What is the probability that a person is sick, given that they are visiting a doctor right now?". This doesn't mean "everyone in the world is visiting a doctor right now", it means that the people who are not visiting a doctor right now exist, but we are not talking about them. Similarly, the original post's imagined world involves cases where odd numbers are rolled, but we are talking about the set without odds. It is weird to think about how proposing a whole set of imaginary situations (odd and even rolls) then talking only about a subset of them (only evens) is NOT the same as initially proposing the smaller set of imaginary events in the first place (your D3 labelled 2,4,6). But yes, I can definitely see how the phrase "given that", could be interpreted the other way.

A very strange probability paradox

darrenreynolds5mo-2-7

Nah.

The problem with the explanation is this line of code:

x=random.randint(1,6)

That will generate odd numbers. The introduction to the problem states, "given that all rolls were even."

That's why you're getting a surprising result. It's the differing interpretation of what 'all' means. If you think it means what it says - all rolls - then there is no surprise. But if, as the explanation implies, you think it means that some of the rolls were odd, but not those involved into the success condition, that's when you get a different outcome. It's not a paradox. It's just how the question is interpreted.

8Ben5mo

"given that all rolls were even" here means "roll a normal 6 sided dice, but throw out all of the sequences that included odd numbers." The two are not the same, because in the case where odd numbers can be rolled, but they "kill" the sequence it makes situations involving long sequences of rolls much less likely to be included in the dataset at all. As other comments explain, this is why the paradox emerges. By stealth, the question is actually "A: How long do I have to wait for two 6s in a row, vs B: getting two 6's, not necessarily in a row, given that I am post selecting in a way that very strongly favors short sequences of rolls".

The Useful Idea of Truth

darrenreynolds13y00

Thanks for engaging on this - I'm finding it educating. I'll try your suggestion but admit to finding it hard.

So, there's a Chinese rocket-maker in town and Sir Isaac Newton has been offered the ride of his life atop the rocket. This is no ordinary rocket, and it's going to go really, really fast. A little boy from down the road excitedly asks to join him, and being a jolly fellow, Newton agrees.

Now, Newton's wife is pulling that funny face that only a married man will recognise, because she's got dinner in the oven and she knows Newton is going to be lat... (read more)

1TheOtherDave13y

You've mostly lost me, here. Reading between the lines a little, you seem to be suggesting that if Newton says "It's true that we returned in time for dinner!" that's just an attempt to assert the privilege of his beliefs over the boy's, and we know that because Newton is unaware of the simulators. Yes? No? Something else? If I understood that right, then I reject it. Sure, Newton is unaware of the simulators, and may have beliefs that the existence of the simulators contradicts. Perhaps it's also true that the little boy is missing two toes on his left foot, and Newton believes the boy's left foot is whole. There's undoubtedly vast numbers of things that Newton has false beliefs about, in addition to the simulators and the boy's foot. None of that changes the fact that Newton and the boy had beliefs about the rocket and the clock, and observed events supported one of those beliefs over the other. This is not just Newton privileging his beliefs over the boy's; there really is something (in this case, the programming of the simulation) that Newton understands better and is therefore better able to predict. If "reality" means anything at all, the thing it refers to has to include whatever made it predictably the case that Newton was arriving for dinner on time. That it also includes things of which Newton is unaware, which would contradict his predictions about other things were he to ever make the right observations, doesn't change that.

The Useful Idea of Truth

darrenreynolds13y00

"Throughout this process, what I'm doing is using my observations as evidence for various propositions. "Reality" is my label for the framework that allows for those observations to occur, so what we call this process is "observing reality."

"What's confusing?"

It seems to me that given this explanation, we can never know reality. We can only ever have a transient belief in what it is, and that belief might turn out to be wrong. However many 9's one adds onto 99.999% confident, it's never 100%.

From the article: "Isn't ... (read more)

0nshepperd13y

That's progress.

0TheOtherDave13y

Yes, that's true. Mm. It sounds to me like we're not using the word "reality" at all consistently in this conversation. I would recommend trying to restate your concern without using that word. (Around here this is known as "Tabooing" the word.)

0BerryPick613y

I thought that 99.999999.... actually does equal 100, no?

The Useful Idea of Truth

darrenreynolds13y00

"It's not, we know it's not, and I bet that you yourself treat reality differently than you treat fiction, thus disproving your claim."

How do we know it's not? You might say that I know that the table in front of me is solid. I can see it, I can feel it, I can rest things on it and I can try but fail to walk through it. But nowadays, I think a physicist with the right tools would be able to show us that, in fact, it is almost completely empty space.

So, do I treat reality different from how I treat fiction? I think the post we are commenting on h... (read more)

6A1987dM13y

So f***ing what? What does solidity have to do with amount of empty space? If according to your definition of solid, ice is less solid than water because it contains more empty space, your definition of solid is broken.

5ArisKatsaris13y

Yes. I bet that if a fire happens you'll call the fire-brigade, not shout for Superman. That if you want to get something for Christmas, you'll not be writing to Santa Claus. No matter how much one plays with words, most people, even philosophers, recognize reality as fundamentally different to fiction. This is playing with words. "Solidity" has a macroscale meaning which isn't valid for nanoscales. That's how reality works in the macroscale and the nanoscale, and it's fiction in neither. If it was fiction then your ability to enjoy the table's solidity would be dependent on your suspension of disbelief. The operative word here is "less". Here's a relevant Isaac Asimov quote: "When people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together." You are effectively being "wronger than both of them put together" 1 and 0 aren't probabilities, but you're effectively treating a statement of 99.999999999% certainty as if it's equivalent to 0.000000000000001% certainty; just because neither of them is 0 or 1. That's pretty much an example of "wronger than both of them put together" that Isaac Asimov described...

The Useful Idea of Truth

darrenreynolds13y-20

Why is it accepted that experiments with reality prove or disprove beliefs?

It seems to me that they merely confirm or alter beliefs. The answer given to the first koan and the explanation of the shoelaces seem to me to lead to that conclusion.

"...only reality gets to determine my experimental results."

Does it? How does it do that? Isn't it the case that all reality can "do" is passively be believed? Surely one has to observe results, and thus, one has belief about the results. When I jump off the cliff I might go splat, but if the clif... (read more)

3TheOtherDave13y

Well, in one sense it isn't accepted... not if you want "prove" to mean something monolithic and indisputable. If a proposition starts out with a probability between 0 and 1, no experiment can reduce that probability to 0 or raise it to 1... there's always a nonzero probability that the experiment itself was flawed or illusory in some way. But we do accept that experiments with reality give us evidence on the basis of which we can legitimately increase or decrease our confidence in beliefs. In most real-world contexts, that's what we mean by "prove": provide a large amount of evidence that support confidence in a belief. So, OK, why do we accept that experiments do that? Because when we predict future experiences based on the results of those experiments, we find that our later experiences conform to our earlier predictions. Or, more precisely: the set of techniques that we classify as "reliable experiments"are just those techniques that have that predictive properties (sometimes through intermediate stages, such as model building and solving mathematical equations). Other, superficially similar, techniques which lack those properties we don't classify that way. And if we found some superficially different technique that it turned out had that property as well, we would classify that technique similarly. (We might not call it an "experiment," but we would use it the same way we use experiments). Of course, once we've come to trust our experimental techniques (and associated models and equations), because we've seen them work over and over again on verifiable predictions, we also develop a certain level of confidence in the _un_verifiable predictions made by the same techniques. That is, once I have enough experience of the sun rising in the morning that I am confident it will do tomorrow, (including related experiences, like those supporting theories about the earth orbiting the sun etc., which also serve to predict that event), I can be confident that it will

1ArisKatsaris13y

And one of the beliefs they've confirmed is "reality is really real, it isn't just a belief." :-) No. If that's all it could do then it would be indistinguishable from fiction. It's not, we know it's not, and I bet that you yourself treat reality differently than you treat fiction, thus disproving your claim.

LESSWRONG
LW

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