The Two Oracles - a statistics problem

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Re: The Two Oracles - a statistics problem

Postby SDK » Mon Dec 07, 2015 5:15 pm UTC

I don't think it matters whether or not the event has been predetermined. The oracle's answer is all you have, and that's all you need.

The oracle is rolling her four-sided dice again. She's got multiple dice and I have no idea what's on the faces, but she tells me there's a 75% chance that it will come up 1. So I bet a dollar. Turns out I'm right! Woohoo! I win a dollar! I win twice more (with the oracle stating the same) and then I lose. That's okay. I'm still up $2. Repeat this 100 times and I'll continue to win, on average 75% of the time. This is true no matter what tricks the oracle is playing with the dice. Maybe on some rolls I had no chance of winning (the event was predetermined)! That's okay though, because if that's true then on some other rolls I had a higher than 75% chance to balance out the oracle's predictions. All that matters is the average.

The same is true for the 90% oracle rolling her ten-sided dice. She can play all the tricks she wants with the dice themselves, including rolling them ahead of time, switching them out for dice with all 2's... whatever. I don't care, and I don't need to. I'm still averaging 90%.

This is why I think we take 82.5% as a legitimate number in the case where we have two oracles with different responses. The one oracle says 75%, the other says 90% for this particular roll of the dice. If we run through this 1000 times, the oracles cannot continue to give the same answers - one of them would be proven wrong which is logically inconsistent by the wording of this problem. However, to give us a full set of cases where one oracle says 75% and the other says 90%, they can switch their answers back and forth. For the first roll of the dice Oracle1 says 75% and Oracle 2 says 90%. For the second roll, Oracle 1 says 90% and Oracle2 says 75%. Continue up to 1000 rolls. To maintain their accuracy, what percentage of the time was a 1 rolled? 82.5% is the only possibility.

Or maybe that's logically inconsistent too...?

Oracle1 says 75% on all odd-rolls. Oracle2 says 90% on all odd rolls. So we have the same problem for those 500 rolls (and for the other 500 in the opposite way). That doesn't change my conclusion, but it does mean that they both need to make up the difference elsewhere. I think the logic is still sound to take the average for an individual event though. It's just that they can't be working alongside each other 100% of the time so they will have the opportunity to be wrong (or right) independant of the other.
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Re: The Two Oracles - a statistics problem

Postby PeteP » Mon Dec 07, 2015 5:44 pm UTC

Say the oracles are statisticans who don't see the future but information about the present world that they evaluate with statistics. Say oracle A knows the facts 1,2,3 and 4 based on them it says 75%. Now maybe Oracle B with 90% also know 1,2,3,4 but also fact 5 which raises the chance to 90% in that case the 75% can be ignored entirely since it is based on less data. On the other hand Oracle B might only know facts 1,2 and 3 and fact 4 lowers the probability, then Oracle B offers no additional informations. Or it knows 1,2,3 and 5 and if you knew 1,2,3,4 and 5 you might conclude the real chance is 10% that is also possible because the facts look good in isolation but bad if you combine them. Without knowing how they arrive at their probabilites you can't combine them. Taking the mean is probably a decent choice but you can't reliably state that the probability is 82.5%

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Re: The Two Oracles - a statistics problem

Postby SDK » Mon Dec 07, 2015 6:16 pm UTC

That's true, but that's not consistent with the presumption that the oracles are always correct. If they were basing it on real-world information, there's no way they'd remain accurate.
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Re: The Two Oracles - a statistics problem

Postby DrZiro » Mon Dec 07, 2015 10:14 pm UTC

Vytron wrote:Can the first oracle say the probability is 0% and the other that it's 100%?

In practice, no, but they can be arbitrarily close.

In theory - well, can an event have probability 100%, and still not happen? Technically, yes - we would say that it happens "almost surely" (there's a good Wikipedia article if you're not familiar with the concept). If the oracles answer an infinite number of questions, this could in principle happen, but I think that's making things more difficult than they need to be. Simply put, the probability of an oracle saying 100% and the event not happening, is 0%.

SDK wrote:At this point I agree that you can't know the actual probability of the event occurring (it could even be zero!), but I think you can know that the oracles giving these responses will result in the event 82.5% of the time on average.

But average over what?

Imagine if they say 50% and 100%. Will it happen 75% of the time? No, because as you say, if one says 100%, it must happen (barring weird infinity effects). If they say 50% and 99%, does it drop from 100% to 74.5%? Sounds a bit odd, doesn't it?

Cradarc wrote:If you need to determine the probability that the signal is 1 based on that single data point, clearly the best option is to use the value as your certainty.

It's a good analogy, but if you mean to use the value as the probability, I don't think it's quite that simple. If we get 0.9 for the first value, can we deduce a probability from that? Not really - it depends what the signal to noise ratio is.

Vytron wrote:If a predetermined event is equivalent to a string of all 0s or a string of all 1s in Cradarc's last post, then this is clearly different from an event that only has some probability of happening [...] if the chance that the king wins is 0.8, we can expect a string of all 0.8s with no noise

I'm not sure where you're going with this, but in Cradarc's example, we can never expect a string of all 0.8s. You might be thinking that the sent signal represents the "real" probability, but the point is, there is no such thing as a "real" probability.

SDK wrote:That's true, but that's not consistent with the presumption that the oracles are always correct.

Why not?

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Re: The Two Oracles - a statistics problem

Postby SDK » Mon Dec 07, 2015 10:21 pm UTC

SDK wrote:If they were basing it on real-world information, there's no way they'd remain accurate.
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Re: The Two Oracles - a statistics problem

Postby SDK » Mon Dec 07, 2015 10:22 pm UTC

DrZiro wrote:Imagine if they say 50% and 100%. Will it happen 75% of the time? No, because as you say, if one says 100%, it must happen (barring weird infinity effects). If they say 50% and 99%, does it drop from 100% to 74.5%? Sounds a bit odd, doesn't it?

Yes. Numbers are hard. I'd still bet with them, though. :P
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Re: The Two Oracles - a statistics problem

Postby Kingreaper » Tue Dec 08, 2015 12:44 am UTC

SDK wrote:That's true, but that's not consistent with the presumption that the oracles are always correct. If they were basing it on real-world information, there's no way they'd remain accurate.

I don't see why it's impossible.

Let's give an example of a simple scenario:

There is a box, it contains 2 dice, each with 6 sides, but with different numbers on their sides. One of them has been removed (at random) and is going to be rolled.

Oracle A knows that the box contains 1 die with sides 1, 2, 3, 4, 5, 6 and one that has 1, 6, 6, 6, 6, 6. When asked the probability of a 3 being rolled she says 1 in 12.
Oracle B also knows that the second die was the one taken. When asked the probability of a 3 she says 0.

Both Oracle A and Oracle B are perfectly accurate, and yet they're basing their conclusion on real-world information.

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Re: The Two Oracles - a statistics problem

Postby Vytron » Tue Dec 08, 2015 3:08 am UTC

SDK wrote:I win twice more (with the oracle stating the same) and then I lose.


This is impossible, as the king is always fighting the same war.

We're not talking about the probability that some war is lost or won, because it has been predetermined.

Therefore, if you play, and you win, you know you'd win ALL the time.

Therefore, if you play, and you lose, you know you'd lose ALL the time.

So a single try is enough to know the truth.

Used this way, the oracle wouldn't tell you that if you played lots of times, you'd win 75% of the time. It's telling you that there's a 75% chance that if you'd started playing you'd win all the time.

DrZiro wrote:You might be thinking that the sent signal represents the "real" probability, but the point is, there is no such thing as a "real" probability.


For a predetermined event there is a "real" probability. Of 1 if the event has been predetermined to happen. Of 0 if the event has been predetermined to not happen.

If the event isn't predetermined, say, if the king going to fight the war causes a Ghost in the Machine to flip a 80/20 biased coin and on head the king wins, then, noiseless oracless with all the information that don't lie would always report a probability of 0.8, and noisy ones would provide info that would approach 0.8 the more tries we get, instead of approaching 1 in case the war will be won.

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Re: The Two Oracles - a statistics problem

Postby Cradarc » Tue Dec 08, 2015 3:13 am UTC

DrZiro wrote:It's a good analogy, but if you mean to use the value as the probability, I don't think it's quite that simple. If we get 0.9 for the first value, can we deduce a probability from that? Not really - it depends what the signal to noise ratio is.

There is no probability to deduce. You know the signal is either 1 or 0. It's about how much certainty you can have assuming 1 or 0.
If the first data point is 0.9, you can assume 1 with 0.9 confidence that you would be correct. Likewise, if it were 0.5, you can assume 1 with 0.5 confidence (ie that of guessing a coin flip).

If you receive two 0.9s, you should be more confident when assuming the signal is 1. In fact, if you get a string of fifty 0.9s, and then you get a 0, you confidence should not go down noticeably. It is far more likely for that one data point to be an outlier than the other fifty.
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Re: The Two Oracles - a statistics problem

Postby Vytron » Tue Dec 08, 2015 5:20 am UTC

This "the king will fight a war but the event is predetermined" is problematic for many reasons. Precognition is getting in the way of discussion, and we really shouldn't be talking about whether it matters if the event is predetermined or not, because that distracts you from the main subject.

An applicable event happened in real life, and I think it can be put to the test by rewriting the OP and make it use it. DrZiro can correct me if I'm wrong on that assumption, but if I'm not then we should be able to discuss the main issue without problems.

Here goes:

My family put on the ceiling of the house, decorative lights as chirstmas decorations, one year ago.

We kept them there, so they suffered from all the weather and other bad conditions, and we're not sure if they still work.

So we went to ask the nearest oracle. The problem with this oracle is that she only answers in probabilities, but on the other hand, she is always correct (and doesn't give stupid ambiguous answers). That is, out of all the times the oracle says "50%", the thing really happens 50% of the times, and so on.

We asked her if lights still worked, and she says that the probability that they still work is 75%.

Not quite content with this, we decided to get a second opinion. We went to ask another oracle. She works the same way - always giving correct probabilities. This oracle says that the probability is 90%. We thought that sounds a lot better, but now we're confused - what probability should we expect after hearing both oracles?

--------

A ha! Please note that the lights work. Or they don't. But there's no such thing as "75% of the time, whenever we flip the lights on, they work, and the other 25% of the time, they fail".

Now, I can use this analog case (which should be identical) to argue with SDK, that taking the average of what the oracles say does not work.

We don't know how they got their numbers, but here's a possibility:

Oracle 1 took a great examination at the place where our house is, and of all the weather that happened the last year, and of all the things around that could damage the lights. And then thought that, from 4 lights undergoing those conditions, 3 would work, and 1 wouldn't. Thus the number. I.e. If we had 4 lights on there instead of 1, we'd most likely expect to see that 3 work, and the other one doesn't. This is right, so we get 75%.

Oracle 2 worked differently. She went to examine the actual lights, and see the damage they suffered. The damage was there, but she thought it wasn't major. But she wasn't sure. She concluded that if 10 lights suffered this damage, 9 would work, and one wouldn't. This is right, so we get 90%.

In this case, Oracle 2 has a much more accurate info, because it doesn't matter what kind of damage lights would get on a year on this place, but what damage our lights got. Apparently, they were lucky, and suffered... 15 percent less than average (or something.)

The real chances are 90%. Oh, but this doesn't matter, because if it's from the 9 that work, it'll work with probability 1, and if it's from the other one, it'll not work.

Problem is, the opposite could be true, Oracle 1 could know the actual chances, and Oracle 2 could be making a statement about how often lights left on these conditions are expected to break (but not a specific statement about our lights).

I built this example, specifically this way, to counter SDK's argument:

If we left 10 different lights on the ceiling, for the damages caused on this year, 9 out of 10 lights would work.

If we left a light on there every year and we had the same conditions of this year, we'd see a broken light in about 1 out of 3 cases.

Taking the average fails on both instances.

--------

Interestingly, after looking at the problem this way, I'm not sure I'd make the bet anymore, and I definitively don't agree with the 0.975 chance posted by Cradarc.

Why?

Because I can create a third oracle:

We went to visit Oracle 3 and it worked in the same way that the others did.

This one said that the probability of the lights working was 25%.

How come?

Well, it turns out this Oracle method was the following:

Take all the people that was going to ask this Oracle if the lights on the ceiling of their house worked. Turns out, from every 4 people that asks this question to this oracle, only 1 of them has their lights working.

The thing is, that we can reorder the kind of oracles this way:

Oracle A: 75% (75% would be correct if we asked the same question every year)
Oracle B: 90% (9 out of 10 people that ask this oracle have working lights)
Oracle C: 25% (If we had 4 lights on our ceiling with the damages that they have, only 1 would work)

So it's clear the real chances are from Oracle C, and that the information gotten from A and B is irrelevant.

Can we create more oracles that give irrelevant info about whether the lights work? Oh, yes, yes, we can.

So 2 oracles aren't enough to know anything, and it's possible to meet many with high percentages but that one looking at the state of the light would tell us they have 0% of working.

And so, I go back to Cradarc...

Cradarc wrote:There is no probability to deduce. You know the signal is either 1 or 0. It's about how much certainty you can have assuming 1 or 0.
If the first data point is 0.9, you can assume 1 with 0.9 confidence that you would be correct. Likewise, if it were 0.5, you can assume 1 with 0.5 confidence (ie that of guessing a coin flip).

If you receive two 0.9s, you should be more confident when assuming the signal is 1. In fact, if you get a string of fifty 0.9s, and then you get a 0, you confidence should not go down noticeably. It is far more likely for that one data point to be an outlier than the other fifty.


The problem with your analogy is that it assumes random noise. Say, an oracle that would arrive at 0.95 by some method could tell you that the probability is 1 because the noise added 0.05. but in the actual problem, the noise isn't random.

I could invent 9 different oracles that arrive at 0.9 in a row, by some methods that are irrelevant to the actual light on the ceiling (like one that speaks about the probability that someone asking the question has a working light, or the one that speaks about the probability that you'd get if you asked every year), but if you get one that tells you the chance is 0, then the chance is 0, because it must be right.

We can invent many oracles that tell you 0, like the one that went to your house and tested the lights an saw they didn't work, or ones that make a specific detection of damage (say, whether a critical part of the light is working, and it turns out they don't), but we can't invent an oracle that tells you 0 when the lights in fact, work.

And the same happens with 100 (probability 1).

So DrZiro is right, we can get arbitrarily close to 0 or to 1, but getting 0 or 1 makes us certain that you've visited an oracle that actually knows whether the lights work or not.

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Re: The Two Oracles - a statistics problem

Postby SDK » Tue Dec 08, 2015 4:37 pm UTC

You proved me wrong for an individual case when the entire point of my last big post was to make you realise that you can't take that case in isolation. The probabilities of your three oracles contradict for your one case - they can't all be right (and, in fact, they're all wrong for a predetermined case) - which is why you have to look at all the other times the oracle responds as well. When I used my dice rolling example multiple times, it wasn't because the king was going to fight multiple battles. It's because one of the die rolls was a stand-in for his success, while the rest were other equivalent predictions the oracle made at other times about other things. But you need to take them all into account to maintain consistency in the oracles' predictions.
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Re: The Two Oracles - a statistics problem

Postby Cradarc » Tue Dec 08, 2015 6:57 pm UTC

What's the point of going to the oracle if the event isn't "predetermined"? If the oracles, themselves, do not know the outcome with 100% certainty, the only way they are different than the king is their knowledge of hidden variables.

If I recall correctly, Bayesian probability always has a prior. The king's prior is whatever he knows about his own army and the oracle. What is the oracle's prior? Whatever the oracle says, there is an implicit "if my prior is true" since they also do not know the outcome with 100% certainty. If the king has no idea what the oracle's prior is, he cannot objectively establish a confidence in whatever the oracle says.
For example, I can claim I have secret knowledge of the universe and that the probability of the sun going supernova in the next 10 years is 40%. Do you believe me? Regardless if the sun goes supernova in the next decade, there is no way I can be proven wrong. The only way you can evaluate the reliability of my claim is by looking at the "secret knowledge" I am using to support it.
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Re: The Two Oracles - a statistics problem

Postby Vytron » Wed Dec 09, 2015 6:38 am UTC

SDK wrote: The probabilities of your three oracles contradict for your one case - they can't all be right


Oh, but they are right, in their secret context.

Say, the lights don't work.

I walk to the first oracle and smack them on the head. But then I found 3 working lights on my ceiling, and they work, so the chance of my light working, on this context, was 75%.

I walk to the second oracle and smack them on the head. But then 9 other people ask them the same question, and after answering 0.9 to all of them, all of their lights work, so the chance of my light working, on this context, was 90%.

I walk to the third oracle and tell them "yup, you said that my lights would only work 25% of the time, and they didn't". I visited this oracle the next 3 years after leaving my lights on the ceiling under the same weather conditions, and only 1 out of three times, the lights worked, so the chance of my light working, on this context, was 25%.

All these oracles were correct on their estimations, but they had independent reasons for their predictions. The probability of something happening can depend on the context it's based on, and you can find a dozen oracles that tell you about some context you don't really care about, and give high chances of the event happening, even though it has been predetermined that it will not happen.

Say, suppose that the lights are weather-resistant, and that it's really, really unlikely that they were damaged to the point they no longer work.

The oracles could say:

Oracle 1: The chances they work are 99% (this one answers this question to 100 people with weather-resistant lights and 1 out of 100 have lights that don't work)

Oracle 2: The chances they work are 95% (this one knows that if the experiment was repeated 100 years in a row under the same weather, 5 of those years the lights wouldn't work)

Oracle 3: The chances they work are 99.9% (this one knows that if the past year you left 1000 lights on your ceiling, 999 would work and 1 wouldn't)

Ah, but here's this:

Oracle 4: The chances they work are 0% (this one, by chance, was looking at your ceiling, and witnessed how the lights were struck by lightning, breaking them for sure.)

Note that, your lights were struck by lightning, and that's the reason they don't work, but Oracles 1, 2 and 3 don't know this.

And if you were to stay and see another 99 people ask the same question they'd get 99% and their lights would work.

And if you were to test it yearly for another 99 years, you'd see other 4 times where they didn't work.

And you could find on your ceiling another 999 lights that weren't struck by lightning, so they work.

All 4 Oracles are right, but you only really care about the one saying 0.

The problem is that, if the lights are struck by lightning but the 4th oracle isn't sure if this is fatal, he could tell you the chances they work are 10%, and if you get 9 more lights struck by lightning, you'd only find one that works, making all oracles right again.

But the point is that getting probabilities of 0.99, 0.95, 0.999 and 0.1 doesn't really tell you anything, if it turns out the one you care about is the 0.1, and no mathematical model can exist, in principle, that can get you from that set to conclude the chances are really high that the light doesn't work, because the oracle won't tell you "your lights were struck by lightning", the bastard will keep that info secret.

Cradarc wrote:What's the point of going to the oracle if the event isn't "predetermined"?


At what point does it become pointless?

Because if there's a ghost in the machine that flips a coin to decide the event, I don't think flipping a coin that falls on heads 99.999{for a Googolplexian 9s}999% of the time is much different from an event that is predetermined to happen 100% of the time. It'd certainly be pointless to go visit oracles if the event isn't predetermined and it happens 50% of the time, but if the event is "almost predetermined" (say, the sun coming up tomorrow) then this should be good enough to go see them.

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Re: The Two Oracles - a statistics problem

Postby SirBrendan » Wed Dec 09, 2015 2:53 pm UTC

This question is fundamentally flawed. In your example of two differing probabilities for the same subject, to explain how this could occur, you made use of rolling a six sided die and a of six with the option of 2 removed. The flaw lies here: the king's question about victory in war is a specific x to which both oracles are addressing. X does not equal X-2. You claimed their inconsistency evolves from a lack of information, but the 'always correct term' was defined as 'if they predict 50%, it really will occur 50% of the time'

If the oracle made a prediction of 1 in 6 of rolling a 6, but was unaware that the 2 couldn't be rolled, the 1 in 6 prediction would be false. This is the eminent point. If the oracle lacked this type of essential information, then their prediction would not occur, as you claimed, as 'always correct'. Which means there is no possible justification for the variance in their prediction as they both adressing the formula of x, not x with conditions.

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Re: The Two Oracles - a statistics problem

Postby Vytron » Fri Dec 11, 2015 4:16 am UTC

That would be true if the event wasn't predetermined.

If the king has a normal six sided die and wanted to know his chances of rolling a 2, an oracle could tell them "1/6" and be correct. But an oracle that knows the king will roll a 6 can tell them "0%", and be also correct. Both oracles are addressing x but from different perspectives and no perspective is erroneous.

As long as the outcome of the war has been predetermined "correct" can have a definition that depends on what the oracle knows.

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Re: The Two Oracles - a statistics problem

Postby ThirdParty » Fri Dec 11, 2015 6:02 am UTC

I'm late to the thread, but here's my two cents on the original problem. Consider two universes you might be in:

First universe:
There was a 9/14 chance that Oracle 1 would say "75%", Oracle 2 would say "90%", and you would win.
There was a 3/14 chance that Oracle 1 would say "75%", Oracle 2 would say "25%", and you would lose.
There was a 1/14 chance that Oracle 1 would say "50%", Oracle 2 would say "90%", and you would lose.
There was a 1/14 chance that Oracle 1 would say "50%", Oracle 2 would say "25%", and you would win.

Second universe:
There was a 1/14 chance that Oracle 1 would say "75%", Oracle 2 would say "90%", and you would lose.
There was a 3/14 chance that Oracle 1 would say "75%", Oracle 2 would say "75%", and you would win.
There was a 9/14 chance that Oracle 1 would say "90%", Oracle 2 would say "90%", and you would win.
There was a 1/14 chance that Oracle 1 would say "90%", Oracle 2 would say "75%", and you would lose.

In the first universe, the probability of winning after hearing "75%" followed by "90%" is 100%. In the second universe, the probability of winning after hearing "75%" followed by "90%" is 0%.

So the probability of winning depends on the probabilities of being in each of these two universes (not to mention the probabilities of the uncountably-many others you might be in). I don't see any obvious way to assign these probabilities from the information given.

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Re: The Two Oracles - a statistics problem

Postby DrZiro » Sat Dec 12, 2015 7:38 pm UTC

Vytron wrote:For a predetermined event there is a "real" probability. Of 1 if the event has been predetermined to happen. Of 0 if the event has been predetermined to not happen.

Yes, you can consider the actual outcome as a true probability. I'm not sure that's a useful definition.

We can think of a non-deterministic system as one where it's impossible to know the outcome before it happens. But from a mathematical standpoint, it doesn't matter what we can know, only what we do know. Whether the reality (as opposed to the model) is deterministic or not is more a philosophical question than a mathematical one. It does not affect what the king should do, nor how we should reason about it.

From that perspective, an oracle with "all the information" can't exist, since some information is impossible to know.

Cradarc wrote:There is no probability to deduce. You know the signal is either 1 or 0. It's about how much certainty you can have assuming 1 or 0.

That certainty is a probability. Probabilities are not just about things that might happen in the future - they can also apply to something that has already happened (or, equivalently, are already determined).

Cradarc wrote:If the first data point is 0.9, you can assume 1 with 0.9 confidence that you would be correct.

No, that's not true. It depends on the noise.

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Re: The Two Oracles - a statistics problem

Postby lightvector » Mon Dec 14, 2015 6:02 am UTC

ThirdParty is definitely right here (as are a few others who posted earlier pointing out the same thing). I also independently came up with the same conclusion prior to reading through the thread. The probability that represents your posterior belief/confidence can be anywhere at all from 0% to 100% based on your prior knowledge/beliefs about "what universe" you are in, or more specifically, about what correlations may exist between the two oracles' knowledge and behavior. The problem is more-or-less well-defined (from a Bayesian perspective), it's just incomplete in that the answer depends on something that's not specified.

In practice in a real-life situation, such as if the two oracles are two different weather forecasters or something like that, who had in the past proven themselves to be fairly well-calibrated (and who are simply reporting their best estimates, rather than trying to do something perverse or tricky), it's unlikely that you would conclude a value less than 75%, but anything in the range 75%-95% might be reasonable assuming you came in around 50-50 before seeing the predictions:

* A posterior belief around 95% might be reasonable, for example, if you thought that mostly the two people came up with their predictions through mostly independent sources of information (in practice, in real life they're unlikely to be entirely independent, but they could be mostly independent).
* A posterior belief around 90% might be reasonable, for example, if you observed that the 90% predictor regularly gave more confident predictions than the 75% predictor (and was usually right when he/she did so), such that it seemed like the 90% predictor's information or quality of analysis outright subsumed that of the 75% predictor so that the 75% prediction adds no value beyond the 90% prediction.
* A posterior belief around 75%-80% might be reasonable, for example, if you believed that in this particular scenario, conditioned on the outcome being true, both predictors should have had access to the same information and produced similarly confident predictions. The fact that one of them didn't and only output 75% instead of also 90% is evidence that although the outcome is still more likely than not, there's reason to be unsure.

And although it's rarer, I think there still might be real-life cases where it's reasonable to go less than 75%, or even the other way and believe something less than 50%! For example, I suspect that in certain kinds of macroeconomic or quant-finance situations, you might have metrics/predictors A, B, C, D that are each moderately reliable when they happen to make a prediction on their own. But on the day that *all of them* make strong predictions at the same time, that might be the day the crash just happened and everything actually is going the other way.

The point is, it depends heavily on your prior belief about how you think two oracles/predictors correlate among each other in this situation. If you do have such information, you *can* often say reasonable things about how confident you should be afterwards. But without this information being specified, there's not enough there to answer the question because technically anything from 0%-100% is possible depending on that prior belief.

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Re: The Two Oracles - a statistics problem

Postby Cradarc » Mon Dec 14, 2015 10:36 pm UTC

DrZiro wrote:That certainty is a probability. Probabilities are not just about things that might happen in the future - they can also apply to something that has already happened (or, equivalently, are already determined).

Yes, but the certainty is a probability that you are correct about the event. It's not a probability that you are correct about the probability that the oracle is correct.

DrZiro wrote:
Cradarc wrote:If the first data point is 0.9, you can assume 1 with 0.9 confidence that you would be correct.

No, that's not true. It depends on the noise.

Right, but we have no clue what the noise is. That is the heart of the problem.
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Re: The Two Oracles - a statistics problem

Postby SirBrendan » Tue Dec 15, 2015 1:17 am UTC

Vytron wrote:That would be true if the event wasn't predetermined.

If the king has a normal six sided die and wanted to know his chances of rolling a 2, an oracle could tell them "1/6" and be correct. But an oracle that knows the king will roll a 6 can tell them "0%", and be also correct. Both oracles are addressing x but from different perspectives and no perspective is erroneous.

As long as the outcome of the war has been predetermined "correct" can have a definition that depends on what the oracle knows.


except that it isn't perspective oriented, the correctness is determined by outcome. Remember, "if they predict it will occur 50% of the time, it really does". Their correctness is not their ability to derive appropriate probabilities given their limited information-- it is specifically defined by outcomes supporting their prediction. In other words, they're claimed to have a confidence interval of 100%. You can change the question; you can redefine "always correct", but as it stands the question is intrinsically paradoxical (and not in a clever but self-contrary way)

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Re: The Two Oracles - a statistics problem

Postby Vytron » Tue Dec 15, 2015 7:46 am UTC

SirBrendan wrote:Remember, "if they predict it will occur 50% of the time, it really does".


But what does that even mean for an event that has already happened and that will never get another chance?

If I have a brand new golden die, that has never been flipped, and throw it, and it lands on a 2, and I asks the oracles, and one tells me:

"The chance that it landed on a 2 was 1 out of six."

And the other tells me:

"The chance that it landed on a 2 was 100%."

I'd say the second oracle had more information about it, so "if they predict it will occur 1 out of six times, it really does" doesn't make sense, because for a predetermined event that has happened, since the universe started, it was predetermined that I'd eventually had the die in my hands, flip it, and get a 2. An oracle could have told me so before flipping the die.

Um... am I agreeing with you?

Basically, whether we can have oracles that give "correct" predictions depends on whether everything is predetermined or not. The first oracle can only be right in an universe where what I roll on my die isn't predetermined. So I'll go back to DrZiro:

DrZiro wrote:But from a mathematical standpoint, it doesn't matter what we can know, only what we do know. Whether the reality (as opposed to the model) is deterministic or not is more a philosophical question than a mathematical one.


I can't agree with this.

From a mathematical standpoint where outcomes aren't predetermined, all the outcomes of rolling dice will have probability 1/6. A noiseless oracle with complete information will give 1/6. A mathematical model that does the impossible (such as knowing $real_probability after getting 0.9 and 0.75 as inputs) would output 1/6 on a die.

From a mathematical standpoint where outcomes are predetermined, every time you roll a die you'll get some number with probability 1. A noiseless oracle with complete information will tell you 0 or will tell you 1 because they know what you will roll every time.

Both mathematical models are in conflict with each other, because in the latter everything that happens couldn't have happened another way, so it was going to happen with probability 1. And in this latter universe, both oracles are wrong.

Knowing the probability of an event that happened requires the possibility that it didn't happen, so it only works in a non-deterministic universe, and this isn't philosophy because it affects the numbers you get.

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Re: The Two Oracles - a statistics problem

Postby Verdigris97 » Tue Dec 15, 2015 9:10 am UTC

DrZiro wrote:That is, out of all the times the oracle says "50%", the thing really happens 50% of the times, and so on.


How did any Oracle ever give her first answer that was not 0% or 100%?

Based on the OP, this could never happen. (She can never have exactly one answer of 50%. In fact, she can never have a finite number of "50%" answers and ever answer a question with "50%" again, because it will mess up her proportions.) Which means that Oracles cannot give answers in the form of accurate, but non-certain, probabilities.

If you want to re-phrase as:
That is, out of all the times the oracle says "50%", the thing really happens at least 50% of the times, and so on.

then this is possible, but it means that to ever get started with non-certain answers an Oracle will have to low-ball some estimates. After they build up enough of a bank at any level of probability, they can give any answer they want. (And, as a corollary, the very first time they ever gave a non-certain answer the result had to happen anyway.)

Which means that the Oracles are just glorified bookies who are messing with people.

Which means that their answers (except for a neighborhood around 100%, whose size decreases based on the number of questions they've answered) essentially contain no information regarding any particular question. You don't need two contradictory Oracles to conclude this; you can get here with just one Oracle who won't give you a straight answer.

I think you should go with your prior.

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Re: The Two Oracles - a statistics problem

Postby SirGabriel » Wed Dec 16, 2015 12:36 am UTC

Verdigris97 wrote:
DrZiro wrote:That is, out of all the times the oracle says "50%", the thing really happens 50% of the times, and so on.


How did any Oracle ever give her first answer that was not 0% or 100%?

She's an oracle. She knows exactly how many times and under what circumstances she will predict 50% success in the future.

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Re: The Two Oracles - a statistics problem

Postby Verdigris97 » Wed Dec 16, 2015 12:50 am UTC

SirGabriel wrote:
Verdigris97 wrote:
DrZiro wrote:That is, out of all the times the oracle says "50%", the thing really happens 50% of the times, and so on.


How did any Oracle ever give her first answer that was not 0% or 100%?

She's an oracle. She knows exactly how many times and under what circumstances she will predict 50% success in the future.


Hmm. Okay, I can see that. I was taking the frequentist, historical view of her performance (as if someone was constantly checking up on the Oracles and making sure that they were always as accurate as advertised), instead of the omiscient view.

This point of view is not without its pitfalls, however, some even worse than my original objection. When she tells me that the probability of my event is, say "80%," that still essentially means nothing, because she has an infinite future with which she can balance any particular "didn't happen" with 4 other "yes, it happened"s. I think this has already been pointed out, though. I'd probably still stick with my prior. (I'd probably avoid asking any oracle that didn't give me a definite answer in the first place.)

Something about this still bothers me at a fundamental level. You know, at the "we're not really talking about probability or statistics, we're actually just talking about weird applications of set theory" level.

To which I guess the appropriate reply will always be, "but probability IS just weird applications of set theory."

Well played, probability. Well played.

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Re: The Two Oracles - a statistics problem

Postby Cradarc » Wed Dec 16, 2015 3:17 am UTC

Vytron wrote:rom a mathematical standpoint where outcomes aren't predetermined, all the outcomes of rolling dice will have probability 1/6. A noiseless oracle with complete information will give 1/6. A mathematical model that does the impossible (such as knowing $real_probability after getting 0.9 and 0.75 as inputs) would output 1/6 on a die.

What is the mathematical justification for the probability being 1/6? It is an argument based on symmetry. There is a lot of assumptions being made when you claim a die lands on every face with equal probability without actually rolling the die. We make these assumptions all the time in physics, and they almost always turn out to be correct. However, when dealing with things like oracles that can see into the future, we have to be more careful.
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Re: The Two Oracles - a statistics problem

Postby Vytron » Wed Dec 16, 2015 7:17 am UTC

So, the oracles of the thread work in the way Oracle 1 worked in my lights analogy?

Vytron wrote:Oracle 1: The chances they work are 99% (this one answers this question to 100 people with weather-resistant lights and 1 out of 100 have lights that don't work).


If so, I think I finally get it.

We could say that, for the first oracle, whenever 4 kings go ask if they'll win a war, it tells them "75%", then 3 out of 4 go and win it.

We could say that, for the second oracle, whenever 10 kings go ask if they'll win a war, it tells them "90%", then 9 out of 10 go and win it.

Thus, we'd go back to this:

My favorite solution then comes from Cradarc where he arrives at 0.975 probability that I brought the coin that always falls on heads and that the oracles' methods predicted high chances of this. This leaves 0.025 as the probability that the coin will fall on tails (I brought the coin that always falls on tails) and that both oracles are wrong (because we arrive from the other side: "The coin will land on heads with probability 1, but the accuracy of the oracle only allows it to predict up to 0.75" followed by "The coin will land on heads with probability 1, but the accuracy of the oracle only allows it to predict up to 0.9") even if their methods were accurate.

I agree with the second advisor. One would bet at better than 1:40 odds (i.e. the oracles would say 0.75 and 0.9 when I bring the tails always coin about 1 in 40 times.)


This is because the chance of being the king that asks the first oracle and loses AND being the king that asks the second oracle and loses is extremely low.

I had already ditched this, but I keep coming back, so I'd like to see a counter-example where the war is going to be lost and oracles of these kind give such answers more often than 1 in 40.

(Note: betting on better than 1:40 odds means that you'd bet if, say, after giving $100, you'd get a return of $102.5 (for a profit of 2.5), or better, if you win. Risking $100 to win $2.5? - that's how certain you're that you'll win the $2.5 bucks.)

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Re: The Two Oracles - a statistics problem

Postby DrZiro » Thu Dec 17, 2015 2:33 am UTC

Verdigris97 wrote:How did any Oracle ever give her first answer that was not 0% or 100%?

You are right, of course, that there is a slight oversimplification here. More accurately, the fraction of positive outcomes will approach the given probability as the number of tries approach infinity.

This is after all how probabilities usually work. If you toss a coin, you might assume that it has a 50% probability, but you only "know" that if you've tried it many times. Going by previous experience, the probability would, as you say, change a little every time you use the coin. But we simplify and say that it's really 50%.

In its simplest form, the question is just "given p(A|B) and p(B|C), what can we say about p(A|B&C)?" (where p(A|B) stands for "the probability of A given B"). That might be an easy question, at least if you have some experience with statistics, but as the various illustrations of the problem are intended to show, in many situations in real life (and, uh, mythology), the answer is not very intuitive.

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Re: The Two Oracles - a statistics problem

Postby micha » Wed Jan 20, 2016 6:24 pm UTC

I think there's a definite answer but it relies heavily on semantics/figuring out how the oracle works.

Introduction before talking about the problem :

One of the problem to solving this riddle is that events don't have a % of chance to happen.
They're either 100% to happen or 100% to not happen, only we do not know it.

Like, say you buy a lottery ticket and think you have 1 out of 1 million to win.
Well you don't. You either are 1:1 to win or 1:1 to lose.
If you win, then going back in time and replaying the situation would have you win every single time.
If it's, for example, lottery balls dropping from a machine, it'll always be the same balls that drop, the ones that gives you the win.

People say you have 1 out of 1 million to win because they can't calculate these things.
If one of the balls was eroded 1/1000th of millimeter by a hair, it might give you the win. Or 'steal' the win from you.

That's what the 1/1000000 means. It's the sum of everything you can't count.

If 1 out of 10 million people will get ran over by a ran this month, you assume that your odds are 1 out of 10 millions.
If you're very careful maybe 1 out of 100 millions. If you work at home and never go out, maybe 1 out of 10 billions ( a car would need to ram your house ).
But, in retrospect, you were or were not hit. If you were not hit... It means your odds, given everything you did, were 0%. You just didn't know it, of course, but it was still 0%, given everything you did and everything other people did.

So, back to the problem :
If the oracles CAN'T answer "You will win the war" or "You will lose the war" with a 100% certainty, it means that, like us normal people, they do not know these variables. They do not know "which balls were eroded by a hair", or how will they bounce. Or, in our current problem...
They do not know which blacksmith used cheaper metal to make swords to save money, which will result in one of your men's swords breaking in a small skirmish, and that would turn an otherwise close win, into a close loss. Which eventually will lose the war for you.

They do not know about the Persian scout shunned by his wife, who will ride a few hours to get to his mistress, which will get him tired and result in him spotting an enemy attack a minute later than he could have, and your army will win a crucial battle.

They do not know any of these details. They just know the 'basics', like the guy who can do math and buy a lottery ticket and think his odds are 1:1000000 because that's what the math says.

So they probably know army sizes, commander skills on both armies, the quality of the gear and training for the troops, the ressources available to both teams, the terrain that will help/hinder the armies, the mental state of the soldiers, and so on...
But they don't know the details, the little things that can affect the outcome of this or this battle, and snowball into something bigger and change everything.

So, going from that assumption...
How could the first one give 75% and the second one 90% if they always give the exact odds ( 'general' odds )?

Well, I only see two possibilities.

1) Something happened between the two predictions.
Maybe a river flooded a land and it'll play a drastic role in the victory. Maybe the enemy general's son died and it crushed his father's morale. Maybe the army sizes changed ( part of the enemy nation rebelled and declared independence, thus he will face a smaller army ).

2) The first oracle prediction influenced things, that ultimately made the second one higher.
Maybe the king was reckless/careless because he thought it was an easy win, but upon hearing he 'only' had 75% to win, became more serious about it and - despite him not even being aware - he will avoid making some mistakes.
Thus, hearing about the 75% made him a better commander and that alone increased his odds to 90%.

I think the easy way to understand this is...
Imagine if the King goes back to the first oracle to ask again.
What does he get?

If the oracle says 75%... I'm pretty sure this problem is flawed. I don't see how it could be possible ( well unless we get into a constant back&forth with the king getting more serious but then going careless when he thinks it's a sure thing, and so on ).
I think what would be likely is that after the second oracle says 90%, whatever the reason is... If the King was to go back to the first oracle, it would tell him 90% too. Or maybe 89% if the king got overconfident with the previous 90%, and so on.

Point is, something changed. We do not know if it's something tangible ( a death, a terrain/ressources/armies change.. ) or confidence, serious, and so on... But something changed. 2 oracles who always answer accurate predictions could not answer 2 different numbers. Has to be changing numbers, and now they'd both tell the same.

Or to put it this way :
If, after asking the first one and getting the 75%, he asked the 2 oracles at the same time, they should both answer 90%.

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Re: The Two Oracles - a statistics problem

Postby Vytron » Tue Jan 26, 2016 7:38 am UTC

micha wrote:Well, I only see two possibilities.

1) Something happened between the two predictions.
Maybe a river flooded a land and it'll play a drastic role in the victory. Maybe the enemy general's son died and it crushed his father's morale. Maybe the army sizes changed ( part of the enemy nation rebelled and declared independence, thus he will face a smaller army ).

2) The first oracle prediction influenced things, that ultimately made the second one higher.
Maybe the king was reckless/careless because he thought it was an easy win, but upon hearing he 'only' had 75% to win, became more serious about it and - despite him not even being aware - he will avoid making some mistakes.
Thus, hearing about the 75% made him a better commander and that alone increased his odds to 90%.


3) Both oracles have access to different sets of information.

You said it yourself:

So they probably know army sizes, commander skills on both armies, the quality of the gear and training for the troops the ressources available to both teams, the terrain that will help/hinder the armies, the mental state of the soldiers, and so on...


You could have it so that the oracles have pieces of information in the following manner:

First Oracle knows:
army sizes
the quality of the gear and training for the troops
the mental state of the soldiers
...

Second Oracle knows:
commander skills on both armies
the terrain that will help/hinder the armies
...

You could also have one of the Oracles knowing all the things you mentioned, while the other oracle knows the "little things", the details. It's a big leap in logic to assume the oracles can't know the details, they just might be missing some details.

It's also possible that both Oracles actually have overlapping knowledge, so both know several things, but one Oracle knows more than the other one, or they have small bits that are unique to them.

The thing is, you don't know what the oracles know. Is it that the first one is missing some info that would make the war much more likely to be won? Or is it that the first one has critical info that makes it less likely to be won, but incomplete, such that the chances to win the war are less than 75%?

The solution to the puzzle is that the information given isn't enough to make a decision, because, if it's the latter case, say, the enemy has access to time travel and has a nuclear bomb from the future, but the Oracles have no idea about it, whatever info they have could be irrelevant.

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Re: The Two Oracles - a statistics problem

Postby narrativium » Sun Jan 31, 2016 9:07 pm UTC

The first oracle says 75%. The king isn't satisfied so consults a second oracle who says 90%. The narrative link seems that the king increased his odds by asking for more advice. If the king had been satisfied and didn't consult the second oracle, would the odds of winning the war have dropped?

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Re: The Two Oracles - a statistics problem

Postby emlightened » Sun Jan 31, 2016 9:28 pm UTC

No, because the first oracle predicted the king visiting the second oracle. Assuming it works correctly.

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Re: The Two Oracles - a statistics problem

Postby notzeb » Sun Jan 31, 2016 10:44 pm UTC

Here are two boundary cases described in the framework where the oracles know the answers but sometimes screw up their answers on purpose.
Spoiler:
Let's pretend that the remainder of the Dow opening price modulo 1 is a uniformly random real number between 0 and 1. The first oracle checks tomorrow's Dow opening price, and says the truth if the remainder modulo 1 is between 0 and 0.75, and lies otherwise. The second oracle checks tomorrow's Dow opening price, and says the truth if the remainder modulo 1 is between 0 and 0.9, and lies otherwise. In this case, the chance they both tell the truth is 75%, and the chance they both lie is 10%. Given that they both gave the same answer, they are either both lying or both telling the truth, so a Bayesian with a 50% prior would conclude that the war will be won with a chance of 75%/(75%+10%) = 15/17, or about 88.24%.
Spoiler:
Again we pretend that the remainder of the Dow opening price modulo 1 is a uniformly random real number between 0 and 1. The first oracle checks tomorrow's Dow opening price, and says the truth if the remainder modulo 1 is between 0 and 0.75, and lies otherwise. The second oracle checks tomorrow's Dow opening price, and says the truth if the remainder modulo 1 is between 0.1 and 1, and lies otherwise. In this case, the chance they both tell the truth is 65%, and the chance they both lie is 0%. Given that they both gave the same answer, they are either both lying or both telling the truth, so a Bayesian with a 50% prior would conclude that the war will be won with a chance of 65%/(65%+0%) = 1, or 100%.
Mathematical analysis to show these really are the extreme cases:
Spoiler:
Let's make four (edit: prior) probabilities, a,b,c,d, and say that a is the chance that both oracles are correct, b is the chance the first is correct and the second is wrong, c is the chance that the first is wrong and the second is right, and d is the chance both are wrong (edit: since our observation is that both oracles answered the same way, the denominator in the Bayesian analysis will be a+d, the prior probability of the observed state of affairs). We know that a+b+c+d = 100%, a+b = 75%, a+c = 90%, all four of a,b,c,d are at least 0%, and our Bayesian answer in terms of a,b,c,d is a/(a+d). Solving in terms of a, we have b = 0.75 - a, c = 0.9 - a, d = a - 0.65, and our Bayesian answer is a/(a+d) = a/(2a - 0.65) = 0.5 + 0.325/(2a - 0.65), which is a monotone decreasing function of a. Since all of a,b,c,d must be at least 0, a must be between 65% and 75%, so the extreme values of our Bayesian answer are 0.5 + 0.325/(2*0.65 - 0.65) = 100% and 0.5 + 0.325/(2*0.75-0.65) = 15/17 = 88.235...%.

If the 65% that keeps popping up seems a little mysterious: it comes from the union bound.
Last edited by notzeb on Mon Feb 01, 2016 2:26 am UTC, edited 1 time in total.
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Re: The Two Oracles - a statistics problem

Postby jestingrabbit » Mon Feb 01, 2016 1:40 am UTC

notzeb wrote:Mathematical analysis to show these really are the extreme cases:
Spoiler:
Let's make four probabilities, a,b,c,d, and say that a is the chance that both oracles are correct, b is the chance the first is correct and the second is wrong, c is the chance that the first is wrong and the second is right, and d is the chance both are wrong.


Spoiler:
But aren't b and c both necessarily 0? You can't both win and lose a war.
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Re: The Two Oracles - a statistics problem

Postby notzeb » Mon Feb 01, 2016 2:11 am UTC

jestingrabbit wrote:
Spoiler:
But aren't b and c both necessarily 0? You can't both win and lose a war.
Spoiler:
Er... a+d is supposed to be the chance of what we observed so far happening, and b+c is supposed to be the chance of the opposite occurring, that is, one oracle claiming that we win the war and one oracle claiming that we lose. Computing the prior probability of what we have observed so far is sort of an important step in Bayesian analysis, and sometimes it helps to reason about the prior probabilities of events which could have happened but didn't. I really ought to have said some phrase like "prior probability" somewhere in my earlier post, but since I don't know any other way to reason about this type of problem the possibility of confusion didn't occur to me (oops).
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Re: The Two Oracles - a statistics problem

Postby Vytron » Mon Feb 01, 2016 8:53 am UTC

Spoiler:
notzeb wrote:and lies otherwise.
This doesn't work, as if the oracle lies, then his prediction is wrong, and we know whatever it says will be right.

Again, there's the example of the coin with heads on both sides, but you don't know the nature of this coin. The oracle does. Whatever method that the oracle uses that includes randomness and leads her to tell you that there's some chance that the coin lands on tails will be wrong, so the oracle can't use such methods.


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