The Two Oracles - a statistics problem

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DrZiro
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The Two Oracles - a statistics problem

Postby DrZiro » Thu Nov 26, 2015 2:52 pm UTC

The king of Lydia is planning to attack Persia, and wants to know if the war will be successful. So he goes 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. He asks if the war will be successful, and she says that the probability of winning the war is 75%.

Not quite content with the risk, the king decides to get a second opinion. He goes to ask another oracle. She works the same way - always giving correct probabilities. This oracle says that the probability is 90%. The king thinks that sounds a lot better, but now he's confused - what probability should he expect after hearing both oracles?

He asks his advisors. The first one says, "surely now the probability must be somewhere in between".

The second one says: "By Bayesian logic, no answer at all is equivalent to 50%. If we get a 90% answer and a 50% answer, that means the 50% oracle just didn't know, so the resulting probability is still 90%. If we get 90% and 75%, the resulting probability must be higher than 90%."

The third one says: "We must use the naive Bayes assumption, and postulate that the probabilities are independent. Then we can calculate a resulting probability."

The fourth one says: "We will also need to know the a priori probability, but by the nature of war, that must be 50%."

So what is the probability? Would you be willing to bet on the king's success?

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Re: The Two Oracles - a statistics problem

Postby Vytron » Thu Nov 26, 2015 4:00 pm UTC

DrZiro wrote: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. He asks if the war will be successful, and she says that the probability of winning the war is 75%.

Not quite content with the risk, the king decides to get a second opinion. He goes to ask another oracle. She works the same way - always giving correct probabilities. This oracle says that the probability is 90%.


This is a contradiction.

Suppose that I have a magic coin that is biased and either falls on Heads 75% of the time and on Tails 25% of the time, or it falls on Heads 90% of the time and on Tails 10% of the time.

The coin is magical because, after I flip it, if it's Heads the war is won and if it's Tails the war is lost.

Please note that the coin is already one way, or the other. That is, it's not 50% chance of being one or 50% chance of being the other, but it's already one or the other, but you don't know which one.

The oracles, as posted, have access to this coin. They can flip it infinite times and reach conclusions.

The first oracle then would flip the coin a large number of times, and be able to figure out the split of the coin with great accuracy.

The second oracle would do the same.

Both oracles would approach the same number as they continue to flip the coin. This thread shows how 8000 flips of such a coin are enough to gain mind-boggling accuracy, but it also shows how, if you limit the flips to 20, both oracles reach numbers very close to each other almost surely.

If there's a chance that the war is won and it is knowable, then both oracles know it and give the same number.

Otherwise:

--------

The chance that the king wins the war is unknowable, because any event that allows you to know it at a given time changes it.

The contradiction is solved if the chances to win the war were 75% when the king asked the first question. However, the status quo has changed after the king has this new information, because the actions he was going to take if he didn't know his chances were going to be different.

So, knowing that he has 75% chance of winning apparently would have made him change his strategy, probably for a better one (maybe one that was more risky, because he knew the chances of him to succeed were high) which changes his chances to 90%.

But he doesn't know it. After asking the second oracle he knows that his chances are 90%. But again, this new information changes things. Mainly, the king is now confused, so his chances will change depending on his actions. If he still does the same he was going to do after knowing his chances were 75%, then his chances will be 90%, but if he goes back to the original plan, his chances will be only 75% again.

We don't know what the king will do, and if we were to ask the oracles again we could know. However, the chances given by the oracle are only right at the moment they're spoken. After they are heard, they change.

Note that you don't need two oracles for this, the same oracle would have given first a 75% chance and then a 90% chance.

(and the oracle would know this, so it could also say something like "If I told you that your chances were 75% they'd raise to 90%". Presumably there's some "self-fulfilling prophecy" strategy in which a number given by the oracle causes that number to be correct after the king receives the new information, even though that number was wrong when the question was asked.)

--------

In any case, this should be wrong:

DrZiro wrote:The second one says: "By Bayesian logic, no answer at all is equivalent to 50%. If we get a 90% answer and a 50% answer, that means the 50% oracle just didn't know, so the resulting probability is still 90%. If we get 90% and 75%, the resulting probability must be higher than 90%."


"The oracle just didn't know" would violate the statement of the puzzle where we're told the oracle is always right.

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Re: The Two Oracles - a statistics problem

Postby SDK » Thu Nov 26, 2015 5:17 pm UTC

I agree with Vytron that the only way this is logically consistent is if the answer of the first oracle affected the outcome (and therefore also affected the answer of the second - which would have been the same had he just asked the first oracle a second time). Therefore, I conclude that the probabily of winning the war is 90%, the second answer taken alone.

Of course, the second answer may have affected the outcome as well, so it's probably best if the King asks repeatedly until the answers come to some equilibrium. Funny though that these oracles seem to have an incomplete knowledge of the future (if the problem statement is to be believed). Certainly the first oracle could not have foreseen the king asking the second or else their answers would have been the same...
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Re: The Two Oracles - a statistics problem

Postby Cradarc » Thu Nov 26, 2015 7:27 pm UTC

This sounds like the Anti-Gambler Fallacy thread....
In Bayesian statistics there's also an uncertainty associated with the knowledge about the oracles themselves. So it ultimately comes down to the King's trust for each oracle rather than the oracles themselves.

DrZiro wrote: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.

There's no way to know this for sure. If you ask the oracle if the sun will rise tomorrow, she could answer 90%. Yet the sun has never failed to rise each day for the past decade. This could be "luck" because technically it doesn't refute the 0.9 probability, but it would cause a Bayesianist to seriously doubt the authenticity of the oracle's claim.
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Re: The Two Oracles - a statistics problem

Postby Vytron » Thu Nov 26, 2015 11:18 pm UTC

Cradarc wrote:
DrZiro wrote: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.

There's no way to know this for sure. If you ask the oracle if the sun will rise tomorrow, she could answer 90%. Yet the sun has never failed to rise each day for the past decade. This could be "luck" because technically it doesn't refute the 0.9 probability, but it would cause a Bayesianist to seriously doubt the authenticity of the oracle's claim.


That's why I mentioned the "magical coin analog". There's no way to know for sure, but you can know it with arbitrary accuracy.

In this case, I wouldn't be surprised if the oracle answered 100% to such a question, because it turns out no matter what they answer and no matter what anyone does, it's impossible to stop the sun from rising tomorrow. We couldn't know it, but the oracle can see the future, so it can make such an statement.

In fact, for the original problem, suppose that the king actually is unstoppable, so he'd win any war it'd get into. The king doesn't know it but the oracles do. An oracle could answer "100%" and be right.

Ah, but the problem is, if the oracle said "100%", there's some non-zero chance that the king doesn't believe it, and that he's more interested in testing the accuracy of his oracles than of winning this war. So the king, with the mission to attempt to test the oracle, sends a lone ill soldier to fight the war for himself, and he dies, so the war is lost, proving the oracle wrong.

If this is the case, then when the answer is "90%", it may not be because if the king goes to war there's some 10% chance of losing, but that 90% is as close as 100% the oracle can tell the king without the king going self-defeating.

We should not assume that if the king goes to war he'd do his best to win (and so, the chances of winning only change because of the strategy he uses), because it's possible the king doesn't actually want to win, he just wants to know his winning chances.

Because of the words by the advisors I highly suspect we're not providing the discussion the OP expected, though, I wonder if the puzzle could be worded differently so that the advisors's opinions would make sense (mainly, one where the second advisor could be right.)
Last edited by Vytron on Thu Nov 26, 2015 11:39 pm UTC, edited 1 time in total.

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Re: The Two Oracles - a statistics problem

Postby Vytron » Thu Nov 26, 2015 11:35 pm UTC

SDK wrote:Of course, the second answer may have affected the outcome as well, so it's probably best if the King asks repeatedly until the answers come to some equilibrium.


This comment makes me question what are we talking about in the first place...

What does "60% chance" means at all in this context? This is just like trying to predict the weather.

Sure, if I tell you there's a 60% chance of raining tomorrow, and it doesn't rain, I can tell you that it fell into the other 40%... However, that holds if I tell you 75%, 90% or 99%, so what is the value of such a statement? Do you flip a coin to decide if you bring out your umbrella and then you manage to carry it out 95% of the time when it rains and not bring it 95% of the time it doesn't rain, or something?

Curiously, it does make sense with a biased coin that nobody knows how it will fall, but when winning a war is at the stake, what does a 75% chance of winning even mean?

Does it mean that there's an infinite number of parallel universes, and in those where the war happens, you win on 3 out of each 4 universes? But what is happening differently on those universes? If things are carried the same over and over, what is causing the loss of the war in those universes? And why are there an infinite number of universes where it is lost?

Getting an equilibrium of 90% might mean nothing.

This reminds me of the bird in hand paradox... I have a bird in my hands, and I ask the oracle "what is the chance that I'll kill the bird?" Now, what I'll do is bring a magic biased coin with me, that I can control how biased is it, and I'll set it to the opposite of what the oracle says.

If the oracle says there's a 0% chance that I kill it, I kill it. With 100%, I'll not kill it. With 25%, I'll flip the 75% biased coin... Heck, if the oracle says 50% I may just go ahead and kill it, surely, without flipping any coin, so the oracle can't say "50%" because it'll always be wrong.

I can have a list of such actions depending on what the oracle says and always prove it wrong, so the oracle can't make a prediction at all.

However, this isn't much different from the problem in the OP (other that the bird fighting back and attempting to defend itself when I try to kill it... or a suicidal bird that will attempt to get killed by me), so against a self-defeating king an oracle can't answer at all, because the king can always prove the oracle wrong.

This comes down to "free will" vs "predicting the future". If instead of an oracle the king had some glasses that allowed him to look at the most likely future, the king could do his best to prove the glasses wrong, and would turn that outcome unto the least likely, so the glasses don't have anything to show.

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Re: The Two Oracles - a statistics problem

Postby DrZiro » Fri Nov 27, 2015 12:53 am UTC

As with many logic problems, there are some practical issues that get in the way. First, can the king change the outcome of the war after he knows the probability? Realistically, yes - he could lose on purpose, for example. This is not meant to be part of the problem; we should assume that the war is already determined, so to speak. Perhaps the problem needs to be expressed differently. He might instead ask "do the Persians have iron chariots", or something.

Second, what do the oracles know? Do they already know the outcome of the war, and are just messing with the king? This is also not part of the problem; all we know is that out of all the times an oracle says "x will happen with 75% probability", x really does happen 75% of the time. I think this should be a clear enough definition. It might be necessary to add that the oracles don't affect each other - one does not change her answer based on the other's.

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Re: The Two Oracles - a statistics problem

Postby SDK » Fri Nov 27, 2015 6:14 pm UTC

Yeah, but you're asking us to take their rightness at face value - it's in the problem statement: they're both always right (presumably based on thousands of trials in the past to ensure their statistical accuracy). Now you tell us that they gave different answers for this particular question. They can't both be right.

If over the years a thousand trials were conducted where similar questions were asked and given a thousand conflicting answers, what would happen? You gather the results and (statistically) prove one of three things, 1) The oracle who said 75% is actually always right, 2) The oracle who said 90% is actually always right, or 3) Neither oracle is always right. Those are the only options, so asking what the "real" probability here is impossible to answer. First we need to revise our assumptions about both of these oracles always being right, since that is now impossible.

If you're thinking that we should take this problem as just a one-of instead of as part of a larger statistical set, then the answer is meaningless. Any individual outcome, whether right or wrong, would say nothing about the correctness of the oracles or about the actual predetermined probability (with the exception of 0% and 100% answers, of course).
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Re: The Two Oracles - a statistics problem

Postby DrZiro » Fri Nov 27, 2015 6:59 pm UTC

It depends what you mean by "always correct". I have stated quite clearly what I mean - you may not agree with that definition.

It is easy to fall into the trap of thinking that there is one true probability, and that if the oracle is really correct, she should give that probability. But there is no such thing as one true probability, at least not on any mathematically meaningful level.

According to the given definition of "correct", the two oracles can in fact give different probabilities and still both be correct.

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Re: The Two Oracles - a statistics problem

Postby SDK » Fri Nov 27, 2015 7:34 pm UTC

That can be true once or twice or three times (or, at least, we can't prove they're incorrect). But not a thousand times. That's what I'm getting at, and is what's implied by your own definition of "correct" in the OP - that it can be tested against multiple events. The only way to test that these oracles give correct probabilities is by doing a thousand trials. If you did a thousand trials where the two oracles gave different percentages, you'd determine that one (or both) of them are wrong.

Probabilities are mathmatically meaningful. Whoever told you otherwise is wrong too, probably because they were falling into the same trap you are now, looking at only the outcome of an individual case. If you want an example, take half lives of radioactive isotopes. That is 100% probability, and a very meaningful number. Talking about the probability that an individual atom will decay over a certain time period has only one correct answer, even if it's impossible to know exactly when that atom will decay. Likewise, talking about the probability that the King will win this war has only one correct answer, even if it's impossible to know whether or not he'll win.
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Re: The Two Oracles - a statistics problem

Postby SirGabriel » Fri Nov 27, 2015 8:44 pm UTC

DrZiro wrote:out of all the times the oracle says "50%", the thing really happens 50% of the times,

One consistent interpretation: the oracles actually know whether the events will happen. They then arbitrarily decide on a probability to tell you, making sure that 90% of the times they say 90% it is for events that will actually happen, etc. This means both oracles can be right even if nothing has changed.

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Re: The Two Oracles - a statistics problem

Postby SDK » Fri Nov 27, 2015 9:03 pm UTC

That's true. Good point. Is that what you meant, DrZiro?

If that's the case, the 75% and the 90% are completely independant of each other since they were both picked arbitrarily. Seems to me our best guess here is the average between the two: 82.5%.
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Re: The Two Oracles - a statistics problem

Postby DrZiro » Sat Nov 28, 2015 3:16 am UTC

SDK: Yes, you could do a thousand trials, and they could give different probabilities, and they could still both be correct. Here is an example with 20 trials:
Spoiler:

Code: Select all

n     Or1     Or2     fact
1     75% T   90% T   T
2     75% T   90% F   F
3     75% F   90% T   F
4     75% F   90% F   F
5     75% T   90% T   T
6     75% T   90% F   F
7     75% T   90% T   T
8     75% F   90% F   F
9     75% F   90% F   F
10    75% T   90% F   T
11    75% T   90% F   F
12    75% T   90% T   T
13    75% F   90% F   F
14    75% F   90% F   F
15    75% F   90% T   T
16    75% F   90% F   F
17    75% F   90% F   F
18    75% T   90% T   T
19    75% T   90% F   F
20    75% T   90% T   T

Probabilities are certainly meaningful, but "what is the probability of x happening" is not strictly speaking a meaningful question in itself - you also need to know what the given information is. If different people have different information, they might also have different probabilities. For example, I just rolled a (normal) die - what's the probability that it shows a six? To you, it's 1/6, but not to me, because I can see what it actually shows. If I tell you that it's not 1, suddenly you have a different probability.

Radioactive decay seems like something that has one true probability, but that's assuming a) you know the half-life of the atom, and how many atoms you have, and b) the event has not happened, and you can't magically predict the future.

SirGabriel: That is indeed one thing that the oracles could be doing. One possible interpretation, more specifically, is that the oracle chooses a probability at random, with equal distribution. Let's say she rolls a 100-sided die, to pick the probability. If she gets 90, she will tell the king 90%. Then she rolls the die again to determine whether to say win or lose; if that roll is 90 or lower, she tells the truth, otherwise she lies.

If that is what they're doing, and we assume that the a priori probability of winning a war is 50%, then we can calculate a resulting probability after hearing the two oracles. If my calculations are correct, it's not 82.5%.

But it's also important to note that
Spoiler:
this is not the only consistent interpretation, and we want to know what we can say about the probability without making that kind of assumptions.

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Re: The Two Oracles - a statistics problem

Postby Vytron » Sat Nov 28, 2015 4:36 am UTC

I honestly don't get all this discussion.

DrZiro wrote:This is not meant to be part of the problem; we should assume that the war is already determined, so to speak.


If this is the case, then either there's a 100% chance that the war will be won, or 0% chance that the war will be won.

The oracles know it.

Since it's an outcome that has already happened in the future (note: this is equivalent to the outcome of the war being already determined), both oracles tell either 100% or 0%.

Proof:

Suppose the king wants to know about the outcome of a war that has already happened somewhere else. It happened yesterday (or a year ago, it doesn't matter.) He brings his oracles. One tells him that the war was won. The other tells him that the war was won.

It's... such a boring history but equivalent to this one:

One tells him that the war had 100% chance of being won. The other tells him that the war had 100% chance of being won.

Now, suppose the king learns that the war was won. In this case, the king has become an oracle himself, if someone asks him what happened on the war, he can tell them the war was won or that it had a 100% chance of being won.

It's the same on the OP, because the war of the future has already happened. It has been determined that the war will happen, and it has been determined that it will be won.

So asking about the status of the war being won or lost is a matter of dialectics and time (the war will have had been won or something), but doesn't change the fact that it has been determined. Nothing can change its outcome.

And so, the oracles saying 75% and 90% are wrong (because there's no such 25% chance of it being lost, or a 10% chance of it being lost. It's always won.)

DrZiro wrote:If that is what they're doing, and we assume that the a priori probability of winning a war is 50%, then we can calculate a resulting probability after hearing the two oracles. If my calculations are correct, it's not 82.5%.


Assuming an a priori probability of 50% of the war being lost or won makes no sense. We have oracles that already know how the outcome of the war has been determined, any method that they use to obfuscate this information isn't part of the puzzle as stated on the OP.

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Re: The Two Oracles - a statistics problem

Postby Cradarc » Sat Nov 28, 2015 4:44 am UTC

SDK wrote:If that's the case, the 75% and the 90% are completely independant of each other since they were both picked arbitrarily. Seems to me our best guess here is the average between the two: 82.5%.

I don't think that's how it works. The event will happen if either Oracle is correct.
There's a 0.75 chance the first oracle is correct and a 0.25*0.9 chance the second will be correct despite the first being wrong. The yields a 0.975 probability of success.
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Re: The Two Oracles - a statistics problem

Postby Vytron » Sat Nov 28, 2015 6:22 am UTC

But the oracle says "there's a 75% chance that you'll win the war", not "there's a 75% chance you'll win the war and 75% chance that I'm right about it."

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Re: The Two Oracles - a statistics problem

Postby PeteP » Sat Nov 28, 2015 10:35 am UTC

DrZiro wrote:SDK: Yes, you could do a thousand trials, and they could give different probabilities, and they could still both be correct. Here is an example with 20 trials:
Spoiler:

Code: Select all

n     Or1     Or2     fact
1     75% T   90% T   T
2     75% T   90% F   F
3     75% F   90% T   F
4     75% F   90% F   F
5     75% T   90% T   T
6     75% T   90% F   F
7     75% T   90% T   T
8     75% F   90% F   F
9     75% F   90% F   F
10    75% T   90% F   T
11    75% T   90% F   F
12    75% T   90% T   T
13    75% F   90% F   F
14    75% F   90% F   F
15    75% F   90% T   T
16    75% F   90% F   F
17    75% F   90% F   F
18    75% T   90% T   T
19    75% T   90% F   F
20    75% T   90% T   T


If you define it like that a prophet could say the probability of a fair coin resulting in head is 1% if they arrange it so that 99% of the time I say it before tail appears because they can see the future. Giving a 90% probability means in 10% of the cases they will say that when it did not happen. That isn't a problem if they are impartial and don't manipulate people for fun, but if they do their motivations begin to matter and they could give good information for small things so that they can mislead for big things. So are they impartial and is it random whether they give a high or low probability for something that will happen? (Random as in they will give high probabilities more often if something will happen but there is no other pattern.)

Spoiler:
And I find it weird to call something the probability of an event if someone who knows a result gives a probability for it. If something happened yesterday that I know about but you don't and I tell you "there is a 75% probabiliy that X happened" I wouldn't consider that the probability of the event (since it already has happened) even if in 75% of all cases where I say something like that it did happen. I would consider 75% the probability that I wouldn't make that statement if it did not happen.

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Re: The Two Oracles - a statistics problem

Postby DrZiro » Sat Nov 28, 2015 3:24 pm UTC

Vytron wrote:Since it's an outcome that has already happened in the future (note: this is equivalent to the outcome of the war being already determined),

Yes.

Vytron wrote:both oracles tell either 100% or 0%.

No, as originally stated, they do not. Whether they know the outcome or not is not known to us. The fact that something has already happened does not mean that we can't talk about probabilities (other that 0 and 100%). See the example above about the die; to you, the probability is not 0 or 100%, because you don't have all the information.

Vytron wrote:And so, the oracles saying 75% and 90% are wrong

As we already mentioned, as far as the king knows, the oracles might know more than they're letting on, in which case you could argue that they are not correct. But that's a semantic issue, since it's clearly explained what "correct" is taken to mean here.

Vytron wrote:Assuming an a priori probability of 50% of the war being lost or won makes no sense. We have oracles that already know how the outcome of the war has been determined, any method that they use to obfuscate this information isn't part of the puzzle as stated on the OP.

It's not the oracles who assume an a priori probability, it's the problem-solver. Sorry if that was unclear.

Cradarc wrote:The event will happen if either Oracle is correct.
There's a 0.75 chance the first oracle is correct and a 0.25*0.9 chance the second will be correct despite the first being wrong. The yields a 0.975 probability of success.

I'm not sure what you mean by "if either oracle is correct", but this isn't quite right either. For one thing, the order of the oracles should be independent (as I forgot to mention in the first post, but added in the second).

PeteP wrote:So are they impartial and is it random whether they give a high or low probability for something that will happen? (Random as in they will give high probabilities more often if something will happen but there is no other pattern.)

This is a good point. We can't assume (in the original problem) that they know the outcome. For example, an oracle might know the size of the armies, and give a probability based on experience. But we should assume that they are impartial with regards to who is asking the question.

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Re: The Two Oracles - a statistics problem

Postby Elvish Pillager » Sat Nov 28, 2015 4:57 pm UTC

SirGabriel wrote:
DrZiro wrote:out of all the times the oracle says "50%", the thing really happens 50% of the times,

One consistent interpretation: the oracles actually know whether the events will happen. They then arbitrarily decide on a probability to tell you, making sure that 90% of the times they say 90% it is for events that will actually happen, etc. This means both oracles can be right even if nothing has changed.

SirGabriel is correct. Mathematically, the puzzle gives us no information whatsoever about the probability of winning.

Imagine, for instance, that both oracles know the war will surely be lost, but they both want to troll the King about it. So, yesterday, the first oracle said, "there is a 50% chance that the sun will rise tomorrow", and each day for the last 9 days, the second oracle said, "there is a 90% chance that the sun will rise tomorrow". After preparing this way, when the King arrives, they are both free to give their misleading predictions about the war.

Without knowing whether the oracles are trolls (or, indeed, any information about their behavior at all), we have no information about the actual probability.
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Re: The Two Oracles - a statistics problem

Postby Cradarc » Sat Nov 28, 2015 8:12 pm UTC

DrZiro wrote:
Cradarc wrote:The event will happen if either Oracle is correct.
There's a 0.75 chance the first oracle is correct and a 0.25*0.9 chance the second will be correct despite the first being wrong. The yields a 0.975 probability of success.

I'm not sure what you mean by "if either oracle is correct", but this isn't quite right either. For one thing, the order of the oracles should be independent (as I forgot to mention in the first post, but added in the second).

The probability I calculated is independent of order. 0.75 + 0.25*0.9 = 0.9 + 0.1*0.75. However, you are correct in that I made an erroneous assumption.

Based on the new interpretation of the problem it is equivalent to:
The king asks about an event.
The oracle randomly selects from an infinite number of bags. Each bag contains a set of balls corresponding to the counting numbers {0,1,2,3...}. Each bag is also assigned a unique value X = a/b, where a and b are integers, a <= b. A ball that is labeled with k such that |k mod b| < a is black, and all other balls are white.
The oracle offers the king the selected bag and tells him the value of X corresponding to that bag.
The king blindly draws from the bag. If the ball is black, the event will happen with 100% certainty, otherwise the event will not happen with 100% certainty.

In the problem, the king receives two bags from two oracles. The oracles chose their bags independently, however the king has not selected a ball from either bag! This brings up the conundrum: What if the king selects a black ball from one bag and a white ball from the other? Note that this can occur even if the oracles pick the same bag X.
How I resolved this issue was that if a white ball is chosen, no information can be given about the event. So only the choice of a black ball matters. This is not consistent with how you defined the interpretation of the problem.
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Re: The Two Oracles - a statistics problem

Postby DrZiro » Sat Nov 28, 2015 9:10 pm UTC

Elvish Pillager wrote:Without knowing whether the oracles are trolls (or, indeed, any information about their behavior at all), we have no information about the actual probability.

Indeed, as PeteP was also getting at. I should add this to the problem description as well - the oracles are not biased against the king, or anything like that.

Cradarc wrote:The probability I calculated is independent of order.

Oh! Sorry, my mistake.

So, if one oracle says "the chance of winning war A is 75%", and the other one says "the chance of winning war B is 90%", then your calculation gives the chance of winning at least one war. But that doesn't work if they're talking about the same war.

I'm trying to understand the balls example. So the bag is just a randomiser with probability X, right? But I don't quite follow, "If the ball is black, the event will happen with 100% certainty" - does the ball change the outcome of the war?

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Re: The Two Oracles - a statistics problem

Postby Elvish Pillager » Sun Nov 29, 2015 12:14 am UTC

"they are not biased against the king" is not sufficient information to draw any further conclusions.

Do you have a specific answer in mind, which you're hoping for us to find?
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Re: The Two Oracles - a statistics problem

Postby DrZiro » Sun Nov 29, 2015 1:23 am UTC

I think I have an answer to the problem, although it might not be very satisfying.

I think the semantics of the problem has been a bit confusing (which, admittedly, is kind of what makes it fun). Here is another formulation of what should be the same problem, which might be less confusing:

Spoiler:
Suppose we know that 75% of the people in Kiruna are men, and 90% of miners are men. Given that Gren is a miner from Kiruna, what are the chances that Gren is male? (It is unclear whether we should assume that half of all people are men; we can think of it as two versions of the problem.)

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Re: The Two Oracles - a statistics problem

Postby Elvish Pillager » Sun Nov 29, 2015 2:32 am UTC

That still doesn't give enough information to solve it. For instance, imagine that Kiruna is a region where tradition forbids men from being miners. Unless we make additional assumptions about the situation, it's unsolvable.

We could solve it with this assumption: "50% of all people are men, and 'is this person in Kiruna', 'is this man a miner', and 'is this non-man a miner' are statistically independent." In that case, the answer is
Spoiler:
27-to-1, I believe. Normally, there are 9 man-miners for every one non-man-miner, but here, there are 3 times as many men as non-men.

If we retain the statistical independence, but abandon the assumption that 50% of all people are men, then the answer can still be anything between arbitrarily-close-to-100% (e.g. all men are miners, and they are a tiny fraction of the population) and arbitrarily-close-to-0% (e.g. 25 miners in Kiruna are the only non-men in existence, and the 225 male miners are diluted among a very large population).

There's even a further tricky answer: suppose we said that being in Kiruna was statistically independent from being a miner in general. In that case, the region in question has the same percentage of miners as everywhere else. (Note that in the explicit answer I gave above, Kiruna has a higher percentage of miners than other regions do.) If 50% of all people are men, then men in Kiruna are less likely to become miners than other men are. Then, if we said, let's say, that the chances of a non-man being a miner are independent of whether they live in Kiruna, then we'd have 19-to-1 instead of my earlier answer. Or if we said that the chance of a person being a non-male miner was independent of whether they live in Kiruna, then we'd have a 9-to-1 split.

You can't just say, oh, the probabilities are independent. You have to actually say which things are independent of which other things, and that can actually rule out independence between the remaining things.
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Re: The Two Oracles - a statistics problem

Postby DrZiro » Sun Nov 29, 2015 3:20 am UTC

Well, sounds like it's time for answers.

First, let's look at the special case we talked about, where the oracles know the truth but might decide to lie. This is a simple calculation:
Spoiler:
The chance of both oracles being honest is 75% * 90% = 67.5%.
The chance of both lying is 25% * 10% = 2.5%.
We know that it's one of those cases, so we get a resulting chance that they are telling the truth of
67.5% / (67.5% + 2.5%) ≈ 96.4%.


As for the original problem, we can first consider the advisors', uh, advice. I'm a little confused by it, but this is what I get:
Spoiler:
Let us call the event of winning the war A, the event that oracle 1 says "75% win" B, and the event that oracle 2 says "90% win" C. We assume, as advisor 4 says, that p(A) is 50%. We know that p(A|B) = 75%, that is, the probability of A given B is 75% (this is what I mean by the oracles being "correct"), and p(A|C) is 90%.

Bayes' theorem gives, for any X and Y,
p(X|Y) = p(Y|X) * p(X) / p(Y)
so we have
p(A|B&C) = p(A) * p(B&C|A) / p(B&C)

Now, as advisor 3 suggests, we can try to apply the Naive Bayes assumption, saying that B and C are independent. In that case,

p(A|B&C) = p(A) * p(B|A) * p(C|A) / (p(B) * p(C))

Then we can use Bayes’ theorem again, and get p(B|A) = p(A|B) * p(B) / p(A)
p(C|A) = p(A|C) * p(C) / p(A)

Put together,
p(A|B&C) = p(A) * p(A|B) * p(B) * p(A|C) * p(C) / (p(B) * p(C) * p(A)^2) =
= p(A|B) * p(A|C) / p(A) = 90% * 75% / 50% = 135%

Well, that didn't work. Unless I've made some mistake, I can only conclude that the assumption was wrong - that is, it's impossible for B and C to be independent. This is not necessarily a surprise - the oracles can freely choose the probabilities, but they can't choose the probability + win/lose combination, because that's dependent on the actual outcome.

(This would suggest that if the oracles kept their probabilities under 1/√2, they could be independent. I have no idea what significance this might have.)


Finally, the answer to the actual question, what is the probability - I believe the answer is,
Spoiler:
we have absolutely no idea.

EP: I guess you meant to question the problem statement, and therefore didn't put it in a spoiler tag, but, well, that's actually the intended answer. It's one of those questions where the answer is not in the set of what seems like possible answers. A mudskipper, as we say in Swedish. What's that in English, a "trick question"? I like "mudskipper" better, really.

Anyway, I found the problem interesting, because it seems that we have information, and then we get more information, and suddenly somehow we have no information. In reality, surely we should be able to make some kind of educated guess? If we're faced with that kind of situation, and allowed to make a bet, would we really have no idea what to bet on? Is there some other reasonable assumption I'm missing?

This is not a hypothetical question about oracles and miners, of course, but a real issue I'm struggling with in my research. If we have two ways of determining a probability, how can we combine them?

Also, EP, I'm trying to understand your calculations there, but I might have to try again when I'm less tired.

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Re: The Two Oracles - a statistics problem

Postby Elvish Pillager » Sun Nov 29, 2015 4:00 am UTC

Spoiler:
Yes, "trick question" is the right term.

A trick question, you say? Well, you sure tricked me into thinking it was a statistics problem, so I replied as such :P

The question about how you would respond in reality is an interesting one. Unfortunately, I think the answer is that, in reality, before making a bet, you would try to get answers to all the annoying questions that I and others have asked in this thread. :P

What's your research, by the way? If I knew more about how you got the two probabilities, I might be able to help.
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Re: The Two Oracles - a statistics problem

Postby Elvish Pillager » Sun Nov 29, 2015 4:42 am UTC

Spoiler:
Also, I've been looking over your Bayesian logic a bit. I think I found a flaw: you cannot convert p(B&C|A) into p(B|A) * p(C|A) just because B and C are independent. To make that conversion, they must be independent *given A*. You only end up with a problem when you assume that they are both independent given A and independent in general.

Incidentally, B and C can be independent from each other while having p(A|B&C) be anything between 0% and 100%. And B and C can be independent from each other *given A* while having p(A|B&C) be anything between 100% and, um, some weird fraction. In practice, it's unlikely that you would know either of those conditions anyway. In the case of the oracles, assuming any sort of independence is almost certainly wrong, since they may have had access to some information in common.
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Re: The Two Oracles - a statistics problem

Postby Cradarc » Sun Nov 29, 2015 8:51 am UTC

DrZiro wrote:So, if one oracle says "the chance of winning war A is 75%", and the other one says "the chance of winning war B is 90%", then your calculation gives the chance of winning at least one war. But that doesn't work if they're talking about the same war.

Maybe I'm still misunderstanding how the Oracles work...
This is how I'm understanding it:

Let the set A75 be of all the times Oracle A have ever said (and will ever say) an event happens with 75% likelihood. Out of all the elements in A75, 75% addressed events that occurred (or will occur), and the rest didn't (or will not) occur.
Similarly, let the set B90 correspond to all the times Oracle B have ever (and will ever) say an event happens with 90% likelihood.

Since the oracles are independent, knowing the response of Oracle A is in A75 should not give you information about the response of Oracle B. That's the definition of independence!
Suppose we are given a particular event in A75. It is part of the subset that truly will happen. What can we say about the same event in B90? If independence is to hold, we can't say anything more than "it has a 90% chance of truly happening".

If you assume something that "truly happens" in A75 implies the same thing "truly happens" in B90, then you are breaking the independence of Oracle A and B.

DrZiro wrote:I'm trying to understand the balls example. So the bag is just a randomiser with probability X, right? But I don't quite follow, "If the ball is black, the event will happen with 100% certainty" - does the ball change the outcome of the war?

Yes, the color of the chosen ball is the outcome of the war. If the bags are independent, there is no reason why you can't pick balls of different colors from each. However, this is inconsistent when it comes to describing a single event. One of the Oracles will have to be bogus, which leads to to a contradiction with the fact that they are both truthful.

In your solution, you assumed that it is impossible to draw balls of different colors from the two bags received from each Oracle. This means the two bags somehow share information, which is absurd.
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Re: The Two Oracles - a statistics problem

Postby DrZiro » Sun Nov 29, 2015 2:08 pm UTC

Cradarc wrote:Since the oracles are independent, knowing the response of Oracle A is in A75 should not give you information about the response of Oracle B. That's the definition of independence!

I don't think I've said that they're independent, although I may have been more or less intentionally misleading. The oracles don't affect each other, but as I'm sure you know, correlation doesn't imply causation, and conversely, non-causation doesn't imply non-correlation. Even though they don't affect each other, they can be affected by the same thing, and thus not statistically independent.

Elvish Pillager wrote:Well, you sure tricked me into thinking it was a statistics problem, so I replied as such :P

Well, it's still a statistics problem, isn't it? You need an understanding of statistics to get the answer. :D

Elvish Pillager wrote:What's your research, by the way? If I knew more about how you got the two probabilities, I might be able to help.

My research is about text classification. By analysing some properties of a text - vocabulary, syntax, punctuation, length of paragraphs, or what have you - we can try to answer various questions - who wrote it, was it a man or woman, is the text positive or negative, fact or fiction, etc. For example, we see that women use more pronouns than men, so we could potentially figure out a probability of the writer being a woman based on the frequency of pronouns. We can also see that men write longer sentences, so we can find a probability of the writer being a woman based on average sentence length. But if we combine the two there's no obvious way of combining the probabilities.

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Re: The Two Oracles - a statistics problem

Postby Elvish Pillager » Sun Nov 29, 2015 2:33 pm UTC

Okay, thinking about it. It's obviously a hard problem, because we don't know if the variables are independent. Doing it properly would probably use a more sophisticated algorithm that analyzes a whole corpus of text and interrelates the variables. But I think there ARE some naïve assumptions we could make that are more reasonable than the naïve assumptions we could make in the oracle problem or the miner problem.

So, you have p( man | long sentences) and p(man | many pronouns), and you want to combine them? I'm thinking it would make sense to start with p(long sentences | man), p(long sentences | ~man), p(many pronouns | man), and p(many pronouns | ~man). You have that information, right? If you assume those are independent, then from those 4 variables (and p(man)), it should be relatively straightforward to compute p(man | long sentences & many pronouns) and the others.

That's the simple version, anyway. Obviously, there are a lot of complications to consider after that (sentences aren't just classified into "long" or "not long", writers aren't always exactly one of "man" and "woman", etc.)
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Re: The Two Oracles - a statistics problem

Postby Vytron » Mon Nov 30, 2015 6:30 am UTC

DrZiro wrote:No, as originally stated, they do not. Whether they know the outcome or not is not known to us. The fact that something has already happened does not mean that we can't talk about probabilities (other that 0 and 100%). See the example above about the die.


Ah, but here's the problem:

If the oracles talk about the probabilities of a deterministic event, then the event can only happen one way, and the oracles know it. Or the oracles don't know it, and they can't give an useful answer because they don't have more information than we do.

That is, suppose I want to roll a die.

If the oracle knows I'll roll a 6, then, if I ask them what's the chance that I'll roll a 2:

The oracle that knows I'll roll a 6 will answer "you will roll a 2 with probability 0."

The oracle that doesn't know what I will roll will answer "you will roll a 2 with probability 0.16"

But note we already knew the probabilities that the second oracle was going to give, so it's pretty useless.

Where it gets interesting is where the dice is biased. Suppose that actually there's a 50% chance that I'll roll a 2, and 50% chance that I'll roll something else. But I'll roll a 6.

Now, we get these answers:

The oracle that knows I'll roll a 6 will answer "you will roll a 2 with probability 0."

The oracle that doesn't know what I will roll will answer "you will roll a 2 with probability 0.5"

That's about it. The second oracle is able to tell us the real bias of the dice, so we know that if we flipped it many times we'd get a 2 about half the time. That's useful but we still don't know what will we roll next. Also, this information is useless if we're going to roll the die once in our lifetime and it'll land on a 6.

The mistake is that:

On this second example, it's erroneous to assume an a priori probability of 1/6 for the die to fall on a 2. Just because there's 6 different options doesn't mean we should assume they're equally likely.

That's the problem with the puzzle on the OP: You can't assume there's a 50% chance that the war is won or lost and use it anywhere in your calculations, because that's what you're trying to calculate ("if there's a 50% chance that a card is red, what is the chance that the card is red given this new information.")

Curiously I have described two different oracles that could exists, so we can actually combine what they say:

The king has a die and wants to know if it is biased. It asks two oracles:

The first tells him that the die is unbiased and that each face has 1/6 chance of appearing.

The second tells him that the die is biased and that 2 has a 50% chance of appearing with the rest appearing equally the rest of the time.

So you don't know what oracle is right, but indeed, the 2 has a higher chance of appearing than that the second oracle says.

However, one of the oracles is clearly wrong.

Conclusions:

If winning the war is predetermined then its probability is 0 or 1. If the oracles tell us some other probability then they're useless.

If winning the war is not predetermined then 1 oracle is at least wrong, and it's possible both are wrong.

-If the probability of winning the war is 0.75 then the first oracle is right and the second one is wrong.
-If the probability of winning the war is 0.90 then the second oracle is right and the first one is wrong.
-If the probability of winning the war is {something else} (what we are trying to figure out, and to which we can't assume anything, including 50% of winning/losing) then both oracles are wrong.

DrZiro wrote:
Spoiler:
Suppose we know that 75% of the people in Kiruna are men, and 90% of miners are men. Given that Gren is a miner from Kiruna, what are the chances that Gren is male? (It is unclear whether we should assume that half of all people are men; we can think of it as two versions of the problem.)


Spoiler:
Do you claim that this problem and the OP's problem are equivalent? If so, after Elvish Pillager's analysis, I'm very impressed.

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Re: The Two Oracles - a statistics problem

Postby DrZiro » Mon Nov 30, 2015 10:09 pm UTC

Vytron wrote:If the oracles talk about the probabilities of a deterministic event, then the event can only happen one way, and the oracles know it. Or the oracles don't know it, and they can't give an useful answer because they don't have more information than we do.

They may have more information without having all the information. It is also possible that they have more information than they are letting on.

As we said before, an oracle might know that the war will be won, but still answer with a (non-100%) probability. You could argue that this would be a lie; in that case, imagine instead that the oracle gets its information from a "super-oracle", and it's the super-oracle that lies. In that case we get the exact some thing, but the oracle is not lying.

Vytron wrote:You can't assume there's a 50% chance that the war is won or lost and use it anywhere in your calculations, because that's what you're trying to calculate

You can (which is not to say that you should), because those are not the same probabilities. We can use an a priori probability to calculate a resulting probability.

Vytron wrote:Do you claim that this problem and the OP's problem are equivalent?

That's the idea.
Spoiler:
Maybe it's known that the king can't be defeated by any man (wasn't that a plot point in "Lord of the Rings"?), and one of the oracles knows that the ruler of Persia is from Kiruna, and the other knows that the ruler of Persia is a miner? Okay, the problem versions don't mix very well.

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Re: The Two Oracles - a statistics problem

Postby Vytron » Mon Nov 30, 2015 11:57 pm UTC

DrZiro wrote:As we said before, an oracle might know that the war will be won, but still answer with a (non-100%) probability.


I don't like this, because it requires a custom definition of "correct" that isn't in the dictionary. If the king will win the war with 90% certainty, then any oracle saying anything else will be wrong.

Say, if the war is won 100% of the time and the oracle says the chance it'll be won is 1%, how can the oracle be right? And what is the minimum percentage that the oracle can say anyway, for a war that will be won for certain?

We can use an a priori probability to calculate a resulting probability.


But it will be wrong, and probably useless. We know that if the outcome is predetermined, the a priori probability is either 0 (the war will be lost) or 1 (the war will be won), but we know that using any value in-between will be definitively wrong because the probability is certainly not 0.5.

--------

Anyway, I like the idea of oracles with incomplete information. And the idea that we aren't trying to figure out the chance of a predetermined event (which is like trying to figure out the chances of an even that has already happened) but about what we'd expect to see if the event was repeated many times (so, 75% chances of winning the war means you'd expect to win 3 wars and lose one.)

Back to the die, suppose the die has 75% chance of landing on a 2, and 5% chance of landing on a 1, 3, 4, 5, and 6, respectively.

Oracle 1 has some information, like: 4, 5, and 6 have each 5% of happening, for a total of 15%, but doesn't know anything else.

Oracle 2 has some other information, like: 1, 3, 4, 5 have each 5% of happening, for a total of 20%, but doesn't know anything else.

If the die lands on a 2 the king wins the war.

The king asks the first oracle what's the chance of landing on a 2. The oracle doesn't know, but knows there's 85% chance that it lands on 1, 2 and 3, so it tells the king the chance of rolling a 2 is 28.3%.

The king, unhappy, goes and asks the second oracle.

The oracle doesn't know, but knows there's 80% chance that it lands on 2 and 6, so it tells the king the chance of rolling a 2 is 40%.

Is this equivalent to the puzzle on the OP with different numbers?

Because if so, the task would be to get 28.3, 40%, and arrive to the real 75%.

Taking into account that each oracle could have answered anything from 16.6% to 75% (given by an oracle that actually knows the actual value.)

Interestingly, can't we play with the numbers and have the situation match the OP?

Say, a die has 95% chance of landing on a 2, and 1% chance of landing on another number, respectively.

Then, an oracle says 2 has % chance of landing - knows:

16.6% - Nothing
19.8% - The chance of one number that isn't 2
24.5% - The chance of two numbers that aren't 2
32.5% - The chance of three numbers that aren't 2
48.0% - The chance of four numbers that aren't 2
95.0% - The chance of 2

Hmm, so it can't me made to fit the model, which means I have an answer to the OP's question:

DrZiro wrote:Would you be willing to bet on the king's success?


Yes, no matter what is the probability, the chances are really high that the war has been predetermined in the King's favor, and if it hasn't, the chances that the king wins with the number given are at least 75% and possibly much higher than 90%.

I'd basically bet all my possessions for the king if I was told the war was predetermined (or I knew the war has already happened) if I was assured the oracles were right and not trying to con me (as the OP claims.) Because building a model where the king has been predetermined to lose and the oracles give such high numbers seem implausible. The king will win with 100% certainty.

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Re: The Two Oracles - a statistics problem

Postby DrZiro » Tue Dec 01, 2015 8:19 am UTC

Generally, the problem involves two different methods of finding a probability. In this case, one of them says 75% and one says 90%. Both methods are, on their own, accurate - the probabilities are both correct. When we express it as being about oracles, the word "correct" gets a bit awkward, but the point is to let the reader know that there is not a contradiction between the two probabilities. I agree with you that in the case where the oracles know the outcome and refuse to tell, it's dubious to say that they're correct. More accurately, the first oracle could say "after hearing only this information, you would be correct in assuming a probability of 75%".

Vytron wrote:
DrZiro wrote:We can use an a priori probability to calculate a resulting probability.

But it will be wrong, and probably useless.

The a priori probability is the probability we have at the beginning, before hearing any added information ("the beginning" is of course relative, but it should be obvious what it means in this problem, namely "before asking the oracles"). There is no reason to think that it would be 0 or 100%.

Vytron wrote:the idea that we aren't trying to figure out the chance of a predetermined event (which is like trying to figure out the chances of an even that has already happened) but about what we'd expect to see if the event was repeated many times (so, 75% chances of winning the war means you'd expect to win 3 wars and lose one.)

There may be a philosophical difference, but mathematically they are the same thing. The king doesn't know the outcome; whether it's predetermined or not is not a meaningful distinction. What we would see if the event was repeated many times is basically the definition of probability.

Vytron wrote:Is this equivalent to the puzzle on the OP with different numbers?

I think so, more or less. But note that this is just one of many ways that the oracles could arrive at those numbers.

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Re: The Two Oracles - a statistics problem

Postby Vytron » Thu Dec 03, 2015 5:41 am UTC

DrZiro wrote:Generally, the problem involves two different methods of finding a probability. In this case, one of them says 75% and one says 90%. Both methods are, on their own, accurate - the probabilities are both correct.


This is the part I don't understand.

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

If so, how can both be accurate statements?

If not, what's the limit of the divergence both statements can have while remaining accurate?

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Re: The Two Oracles - a statistics problem

Postby SDK » Thu Dec 03, 2015 6:28 am UTC

0% and 100% doesn't work. That would necessitate one of them being incorrect. 0% and 99% could work. 1% and 100% could work. This problem goes away once you realise that the probability the oracles are giving is not the probability that the event happens, exactly. This probability is a measure of the oracles themselves, not of the future. The set they're looking at isn't a wide range of possible futures where 75% of the futures result in the king winning. No, the set they're looking at is the hundred other similar responses that particular oracle has given. Within the set of "oracle responses = 75%", the event happens 75% of the time, but that doesn't really have much to do with the event itself exactly. The oracle could say 75% no matter what the actual probability of the event happening is (even if that's 0% or 100%) - she then just needs to make up the difference in some future 75% response to compensate.

The dice example is a good one. We've got a four sided die with three sides showing a 1 and one side showing a 2. The oracle rolls it and looks at the die (it's a 1). She then tells you the probability that a 1 is showing is 75%. But that's clearly not true - the probability that a 1 is showing is 100%! Repeat this twice more (with two more 1 rolls), then she rolls a 2 and says the same thing; the probability that a 1 is showing is 75%. Still strictly not true - the probability is 0%. But now that we've run this trial four times, her accuracy is back to par. The set "oracle responses = 75%" does equal 75% despite being incorrect all four times.

To take this one step further, the oracle takes a four-sided die with sides that are unknown to you (all the faces are 1, but you have no idea). Now the oracle rolls and says the probability that a 1 is showing is 75%. Turns out a 1 is showing, as she fully knew and expected. Repeat this twice more (with two more 1 rolls), then she switches dice. The new die is four-sided with all the faces 2 (but again, you don't know this). The oracle rolls and says the same thing; the probability that a 1 is showing is 75%. Tada! She was wrong one out of four times, just as expected! This example shows that even with a predetermined future that the oracle fully knows she can still be consistent in her probabilities simply because she doesn't care what probability the event actually has (the event is known and therefore meaningless to talk about probability), she only cares about how often she herself is wrong.

Once you realise all that, then it really doesn't matter what numbers the oracles say. They can give different answers and it still says nothing concrete about possible futures. We're basically not even talking about probability at this point; at least, not the probability that any particular event happens. Only the probability that the oracles maintain their accuracy.

Because of that I'd still bet in line with the oracles "probabilities". Maybe I don't fully understand DrZiro's conclusion that we learn nothing, but if you know the oracles are correct 100% of the time, you'd be a fool not to bet in line with their prediction. 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.
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Re: The Two Oracles - a statistics problem

Postby Vytron » Thu Dec 03, 2015 9:52 pm UTC

Thanks SDK. So my mistake was thinking that the oracles were referring to the probability of the event occurring.

In any case, I disagree with your conclusion because I think the average is meaningless.

This is because we're talking about a predetermined event.

Such an event is equivalent for me to bring a biased coin, that always falls on heads (the war is won) or that always falls of tails (the war is lost) - this is not unlike your dice will all sides the same.

So you don't know this (you may assume I brought a coin that would fall on either side), but both oracles do.

Now, both oracles do their unknown thing, and one says there's a 0.75 probability that the coin will fall on heads (in reality, they're talking about the chance the method they used accurately predicts that I brought the coin that always falls on heads). The other oracle uses some other method, and arrives at 0.90.

What the oracles say doesn't matter, because I already have the coin with me, so the chance I have another coin or that this coin falls on a different way, are 0.

What are the chances that I bring the coin that always lands on tails and that the accurate statements of both oracles lead the wrong way? The chances of this should be very small.

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.)

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Re: The Two Oracles - a statistics problem

Postby Cradarc » Fri Dec 04, 2015 6:04 am UTC

After thinking about this some more, I noticed it is actually kind of related to communication systems.
You are sent a signal across a very noisy channel. The sent signal is either all 1s or all 0s. If the sent signal is 1, count it as a success, otherwise it's a failure. Due to the extreme noise, you cannot expect to see 1s. In fact, 1s will occur really rarely. Instead you will see a distribution of values in the range [0,1].
Asking the first oracle is like taking the first data point from the signal stream. 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. Asking the second oracle is akin to taking the second data point. Now things get interesting. You can process multiple data points in different ways.

SDK suggests averaging them (which is like applying a low pass filter) to get a sense of the general signal. As more bits come in, the average will hover above 0.5 if the sent signal is 1 and below 0.5 if the sent signal is 0.
I considered it by thinking about each data point as an independent variable. There is some likelihood that the first data point could be a false positive, but the likelihood of the first and second data point both being a false positive is much lower.

SDK's method is more robust when it comes to deducing an arbitrary signal, however we already know that the sent signal can be only one of two options (all 1s or all 0s). This extra knowledge should allow our certainty to converge toward 1 despite always sampling from noisy data. For example, we know 0.8 could not have been sent, so receiving a chain of ten 0.8s in a row should make you more confident than receiving a single 0.8, because each 0.8 received actually increases your confidence in 1 rather than your confidence in 0.8.
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Re: The Two Oracles - a statistics problem

Postby Vytron » Sat Dec 05, 2015 2:23 am UTC

I agree, and I think that's the crux of my disagreement with DrZiro.

DrZiro wrote:There may be a philosophical difference, but mathematically they are the same thing. The king doesn't know the outcome; whether it's predetermined or not is not a meaningful distinction. What we would see if the event was repeated many times is basically the definition of probability.


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 (say, it hasn't been predetermined whether the king will win or not. "The dice hasn't been rolled", so to speak. So if the chance that the king wins is 0.8, we can expect a string of all 0.8s with no noise - with noise we can take the average as SDK says. And so, no matter what, there's at least some 0.2 chance that the king loses the war, I'd not bet if I knew the event hasn't been predetermined, or at least would only bet with high odds in my favor. With a signal of all 1s expecting to give 0.75 and 0.90 I'd bet most of the time.)

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Re: The Two Oracles - a statistics problem

Postby SirBrendan » Mon Dec 07, 2015 12:50 am UTC

Vytron wrote:But the oracle says "there's a 75% chance that you'll win the war", not "there's a 75% chance you'll win the war and 75% chance that I'm right about it."


This is the core of the problem and why the problem isn't just a trick question, but flawed. We are informed that the oracles are always correct about their predictions. Always correct is stated as an accuracy within their prediction of X's probability. X is the query, so if the statement is made that 75% chance that x is to occur then there is a 75% that the war will be won. It doesn't matter if the oracles sometimes lie or obscure their own vision of the determined event-- this is irrelevant-- provided that their prediction of x matches its probability. If x isn't the prediction of victory, then the question is wrong as it doesn't imply that's what x is, it states it. The query was unambiguous so neither can the answer to it be (answering 50% (of you wearing a blue dress(y)) when I asked you if i will go bald in the next year (x) doesn't mean x was undefined, as this would preclude the previous statement that their predictions of x are perfectly accurate. If such bias existed in their truthfulness, then their accuracy would be x amount of times x is right, as vytron stated.

From the rest of the thread, you seem to be speaking as though there is a nuanced obscurity in the language that allows for uncertainty, but there isn't. A singular event does not act as isolated from its dataset no matter its outcome; probability is a natural law, not an aspiration. For them to have absolute accuracy in their predictions, as is claimed, it is required that their range have a 100% confidence interval.

I hope this effectively explains the problem with the problem


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