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Topic: Gold or Silver? (Read 2649 times) |
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Sir Col
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Re: Gold or Silver?
« Reply #25 on: Jun 11th, 2003, 6:03am » |
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By my interpretation of the problem I get 100%.
If a silver coin is chosen at random from a box that is chosen at random, that would imply you have chosen a box at random and are now faced with the choice of which silver coin to select. If you chose the box with silver and gold you would not be faced with a random choice, hence you must have chosen the box with the two silver coins.
I suspect that this was the clever twist T&B intended.
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wowbagger
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Re: Gold or Silver?
« Reply #26 on: Jun 11th, 2003, 6:17am » |
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on Jun 11th, 2003, 6:03am, Sir Col wrote:| I suspect that this was the clever twist T&B intended. |
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I don't think so, but who knows? Well, okay, T&B should know.
Anyway, you can choose one coin out of one. The number of possible outcomes is C(1,1) = 1. And please don't argue about whether this choice can be considered "random". 
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| « Last Edit: Jun 11th, 2003, 6:18am by wowbagger » |
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otter
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Re: Gold or Silver?
« Reply #27 on: Jun 11th, 2003, 7:05am » |
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on Jun 11th, 2003, 2:25am, wowbagger wrote:
On the other hand, it can mean: "A coin is drawn at random, and it turns out to be silver."
In this case we have to deal with conditional probability and arrive at towr's result. |
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I believe this was the intent of the original post. One of two boxes is randomly chosen and one coin is randomly removed from the box, which turns out to be silver. As wowbagger said, we then arrive at towr's answer. 
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Icarus
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Re: Gold or Silver?
« Reply #28 on: Jun 11th, 2003, 4:01pm » |
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on Jun 11th, 2003, 6:03am, Sir Col wrote:By my interpretation of the problem I get 100%.
If a silver coin is chosen at random from a box that is chosen at random, that would imply you have chosen a box at random and are now faced with the choice of which silver coin to select. If you chose the box with silver and gold you would not be faced with a random choice, hence you must have chosen the box with the two silver coins.
I suspect that this was the clever twist T&B intended. |
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If this is what T&B intended, then he duffed the statement very badly indeed.
How about it, T&B? Any comment?
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ThudnBlunder
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Re: Gold or Silver?
« Reply #29 on: Jun 13th, 2003, 12:18pm » |
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Sorry, guys, I think I screwed up. I was trying to remember a puzzle from way back, and posed a different one.
Anyway, you seem have covered every possibility, and then some.
I think this was the one I had in mind:
A box contains two coins, either two Silver or one Silver and one Gold. A coin is chosen at random.
It is Silver. What is the probability that the other coin is also Silver?
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| « Last Edit: Jun 13th, 2003, 5:19pm by ThudnBlunder » |
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Sir Col
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Re: Gold or Silver?
« Reply #30 on: Jun 13th, 2003, 4:42pm » |
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In which case, as you said, it's been covered.
Assume box 1 contains: silver, silver and box 2 contains: silver, gold.
There are three possible ways, with equal chance, of having taken a silver coin:
Selected box 1 and took 1st silver coin; 2nd silver coin remains.
Selected box 1 and took 2nd silver coin; 1st silver coin remains.
Selected box 2 and took silver coin; gold coin remains.
Hence P(remaining coin is silver)=2/3.
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ThudnBlunder
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Re: Gold or Silver?
« Reply #31 on: Jun 13th, 2003, 5:18pm » |
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But here there is only ONE box.
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Chronos
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Re: Gold or Silver?
« Reply #32 on: Jun 13th, 2003, 11:25pm » |
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We need to know the probability on your either/or. Is it equally likely, a priori, to be a GS box or an SS box? Because if that's the case, then just say that the box in question is the box we randomly chose in the previous version of the problem. Really, would you expect the answer to change just because there's some other box floating around somewhere in the Universe?
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ThudnBlunder
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Re: Gold or Silver?
« Reply #33 on: Jun 14th, 2003, 3:19am » |
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Quote:| Is it equally likely, a priori, to be a GS box or an SS box? |
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No a priori is given. 
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| « Last Edit: Jun 14th, 2003, 3:21am by ThudnBlunder » |
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Sir Col
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Re: Gold or Silver?
« Reply #34 on: Jun 14th, 2003, 4:06am » |
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on Jun 13th, 2003, 5:18pm, THUDandBLUNDER wrote:| But here there is only ONE box. |
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 I missed that!
However, It may still hold. As Chronos said, we're dealing with the concept of a box that contains silver/silver (box 1) or silver/gold (box 2).
Having said that, and as we have no more information, I believe there are two ways to interpret the problem:
(i) Equal chance of the box being SS or SG. So P(other coin is silver)=2/3 [as outlined above].
(ii) The contents of the box can be generated from a principle of there being an equal chance of each of the two coins being silver/gold. That is, P(SS)=P(SG)=P(GS)=P(GG). Using the first given (it contains two silver or silver and gold), we reduce to three possible configurations: SS,SG or GS. We then find that the second given (the first coin taken is silver), is redundant, as all three boxes satisfy this. Hence we deduce that P(other coin is silver)=1/3.
I would suggest that we are unable to solve this problem from the information given, unless I've missed something. What is the probability of that?
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| « Last Edit: Jun 14th, 2003, 4:07am by Sir Col » |
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ThudnBlunder
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Re: Gold or Silver?
« Reply #35 on: Jun 14th, 2003, 2:10pm » |
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Quote:| I would suggest that we are unable to solve this problem from the information given... |
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I agree.
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Sir Col
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Re: Gold or Silver?
« Reply #36 on: Jun 14th, 2003, 4:13pm » |
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If we're right, isn't it strangely satisfying that an apparently simple and familiar problem cannot be solved – rather, the solution is that it has no exact solution. For all intents and purposes it appeared to be a version of the three door problem or the boy/girl problem. Thanks for sharing it, T&B.
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towr
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Re: Gold or Silver?
« Reply #37 on: Jun 15th, 2003, 8:10am » |
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on Jun 14th, 2003, 4:06am, Sir Col wrote:
I would suggest that we are unable to solve this problem from the information given, unless I've missed something. What is the probability of that? |
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Without any information the educated guess is equal chance, 50-50 in this case (since there are two options).
But there is information, and that gives the answer you gave earlier, 2/3. Because without further information the either/or gives a 50-50 chance of it being either box, which is equal in every relevant way to first choosing a box, and than randomly pulling out a coin that is revealed to be silver.
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| « Last Edit: Jun 15th, 2003, 8:11am by towr » |
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ThudnBlunder
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Re: Gold or Silver?
« Reply #38 on: Jun 15th, 2003, 9:01am » |
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Quote:| But there is information... |
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Where? '50/50' is merely a subjective preference.
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| « Last Edit: Jun 15th, 2003, 9:04am by ThudnBlunder » |
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Sir Col
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Re: Gold or Silver?
« Reply #39 on: Jun 15th, 2003, 12:07pm » |
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LOL 
When I said, "What is the probability of that?" I meant, what is the probability of me missing somehting.
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towr
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Re: Gold or Silver?
« Reply #40 on: Jun 15th, 2003, 2:31pm » |
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on Jun 15th, 2003, 9:01am, THUDandBLUNDER wrote:
Where? '50/50' is merely a subjective preference.
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The coin is silver. Which is usefull information (just apply Bayes)
50-50 is not a subjective preference. When there are two choices, the 'real' probability is either 100% or 0% because it will allways be one or the other, not (part of) both.
The educated guess without any knowledge is therefore 50% (in the middle), giving no opportunity for a 'profitable bet'.
Any information brings it closer to the 'real' probability, and perfect knowledge allows you to predict/classify perfectly (allowing you to know the future past and present).
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| « Last Edit: Jun 15th, 2003, 2:39pm by towr » |
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Sir Col
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Re: Gold or Silver?
« Reply #41 on: Jun 15th, 2003, 3:42pm » |
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Philosophically speaking, can we be certain of anything? I know that Pascal said, "It is not certain that everything is uncertain." Which is doubly ironic, as it was Pascal who invented the system of probability to settle a dispute when a gambling game was interrupted. He was asked to determine who was more likely to win, if it had been allowed to be completed.
Anyway, I digress. The reason I mention certainties is to challenge whether or not we may permit talk of certainties in probability at all. Aside from the existential philosophical issues, we can also ask, if probability is definied as a measure of the likelihood of an event happening, is it reasonable to talk of an event for which there is no measure of uncertainty?
Back to the current question. As we've established that the probability of the other coin being silver is 2/3 or 1/3, depending on the model, can we reasonably interpolate by a linear method? Is it more likely to be at one extreme or the other, or is it sensible to suppose that both are equally likely? I would dare to suggest that we have no reasonable grounds to suppose on thing or the other without further information. It would be as foolish as arguing that as I can win £1 million in a lottery or win nothing, my expected winnings is £0.5 million. Unless you are provided with the system that determines one outcome against the other, you simply cannot assume. Now what's the probability of that? 
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ThudnBlunder
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Re: Gold or Silver?
« Reply #42 on: Jun 15th, 2003, 7:01pm » |
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Quote:Without any information the educated guess is equal chance, 50-50 in this case (since there are two options).
But there is information, and that gives the answer you gave earlier, 2/3. |
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I throw a die.
If it comes up a 1, I put SS in the box;
Otherwise, I put SG in the box.
(Or I could do it the other way round. Or use another method to get whatever probabilities I choose.)
Without any a priori information on how the coins were put in the box, any 'answers' will be based on
unjustified assumptions and/or blind guesswork.
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| « Last Edit: Jun 15th, 2003, 8:42pm by ThudnBlunder » |
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ThudnBlunder
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Re: Gold or Silver?
« Reply #43 on: Jun 15th, 2003, 9:59pm » |
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Quote:Anyway, I digress. The reason I mention certainties is to challenge whether or not we may permit talk of certainties in probability at all. Aside from the existential philosophical issues, we can also ask, if probability is definied as a measure of the likelihood of an event happening, is it reasonable to talk of an event for which there is no measure of uncertainty?
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MY 2 CENTS:
In higher maths (and I'm no expert), probability is considered in terms of measure theory.
Probability is basically a special kind of measure. Considering 'impossibility', certainty that an event cannot occur:
(i) Zero probability events have a probability of occurring of zero, but are not impossible.
(ii) Impossible events also have a probabilty of occurring of zero, and are impossible.
eg, the probability of choosing a particular real in the interval [0,1] = zero, but we know that it is not impossible.
Conversely, the probability of not choosing the number is 1, but it is not certain that it will not be chosen.
Also, the Law of Large Numbers doesn't say that the ratio is guaranteed to approach the probability, p,
of the event one is observing. It merely says that the event approaches p 'with probability 1'.
Again, there's a difference. The outcome is not certain, but it is almost certain.
(To sum up, I would say that when we are dealing with infinite sets (and possibly finite sets, too),
there is no certainty. I guess Icarus will have a few things to say about your philosophical musings,
besides being well booked-up on the above.)
PS Don't think about this stuff while you are driving or you are certain to have an accident!
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| « Last Edit: Jun 16th, 2003, 10:13am by ThudnBlunder » |
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towr
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Re: Gold or Silver?
« Reply #44 on: Jun 16th, 2003, 1:49am » |
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on Jun 15th, 2003, 7:01pm, THUDandBLUNDER wrote:
I throw a die.
If it comes up a 1, I put SS in the box;
Otherwise, I put SG in the box.
(Or I could do it the other way round. Or use another method to get whatever probabilities I choose.) |
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If I know you used a die it gives extra information.
simple as that.
Quote:Without any a priori information on how the coins were put in the box, any 'answers' will be based on
unjustified assumptions and/or blind guesswork. |
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Precisely, justified blind guesswork. Because without information you are blind.
That's the whole point. Even (blind) chance, because any other assumption is unjustified.
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| « Last Edit: Jun 16th, 2003, 2:10am by towr » |
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towr
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Re: Gold or Silver?
« Reply #45 on: Jun 16th, 2003, 1:53am » |
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on Jun 15th, 2003, 3:42pm, Sir Col wrote:| Back to the current question. As we've established that the probability of the other coin being silver is 2/3 or 1/3, depending on the model, can we reasonably interpolate by a linear method? |
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Firstly the other model is wrong, since it isn't supported by any information, the other is supported by the lack of the information supporting the other model.
Lastly when there is insufficient information the uniform distribution gives the best educated guess, since it doesn't favour any outcome (and no outcome should be favoured as there is no information).
Quote:| It would be as foolish as arguing that as I can win £1 million in a lottery or win nothing, my expected winnings is £0.5 million. |
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That is _only_ the case because you _know_ the lottery isn't an even-bet game. you know they want to make a profit, you know many people enter it and only one can win.
If you didn't know anything about it, nor about human nature, nor had any intuition (which is on avarage a surprinsingly good predictor) etc, it would be foolish to assume the expected winnings was not 0.5 million.
Because without any information, you only know that either you win, or you don't win, and you do not know which is more likely, so you can only assume they are equally likely. Favouring neither outcome over the other as there is no information to justify doing so.
(But your instincts will probably tell you there is no free lunch, and an average winning of 0.5 million is impossible, so you'll never be in this situation.)
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| « Last Edit: Jun 16th, 2003, 2:04am by towr » |
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towr
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Re: Gold or Silver?
« Reply #46 on: Jun 16th, 2003, 2:46am » |
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suppose we play a game.
rules:
We can't see each other
We both flip a coin.
We write down the side that faces up on a piece of paper and hand it to an independant and fair arbitrer.
If both pieces of paper say the same thing you win, else I win.
What is the chance you win?
What is the chance you win if only I cheat and:
1. always write heads
2. write heads 1/3 of the time and tail 2/3 of the time.
3. use any other strategy
What is the chance you win if only you cheat (using whatever strategy you want)
What is the chance you win if we both cheat (and have to determine our strategy beforehand)
What is the chance you win if you know my strategy
"knowledge is power"
"when in doubt, scream and shout"
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| « Last Edit: Jun 16th, 2003, 2:46am by towr » |
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redPEPPER
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Re: Gold or Silver?
« Reply #47 on: Jun 16th, 2003, 3:52am » |
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I'm no expert in probabilities, but isn't there a difference between a probability based on complete and incomplete information?
In towr's example, the probability to win if at least one player doesn't cheat is 1/2. If you reproduce the experiment a number of times, each will win about half of the time.
But what about the probability to win if you don't know if the players cheat? You would give a probability of 1/2 for lack of information? But if you reproduce the experiment a number of times, you might not verify that probability. Say, if one player uses strategy 1 and the other strategy 2, one of them will win 2/3 of the time.
So is it better to say the probability is 1/2, or to say you can't decide for lack of information? The "real" probability is not 1/2, it's 2/3. You just have no way to say it's 2/3 but you know enough to see it might not be 1/2. So what's the value of such an uninformed "guess"?
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towr
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Re: Gold or Silver?
« Reply #48 on: Jun 16th, 2003, 4:24am » |
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For every strategy there is an exact opposite strategy.
what you win with one you loose in the other. And you don't know which strategy is being used. So it is a fair game.
The average gain would be the weighed sum over all strategies for both players. And since every strategy has an exact opposite you end up at even chance.
In the end you can always make a guess, that's why it's called a guess. If you don't have any information, in other words the educated guess is 50%, you know you can't make a profitable bet, so you might as well not bet.
(Aside from an average gain there is also something to be said for the smallest variance. The smallest variance for no gain is attained by not betting.
Betting 1 dollar against 100 on a 1/100 chance 100 times is worse than 1 against 2 dollars on a 1/2 chance 100 times )
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| « Last Edit: Jun 16th, 2003, 4:34am by towr » |
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redPEPPER
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Re: Gold or Silver?
« Reply #49 on: Jun 16th, 2003, 4:55am » |
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Okay, I understand how, if the players independently choose how to play at each iteration, the probability is 1/2. But what about a case where they use the same strategy each time, but we don't know that strategy?
I'm referring to your earlier posts, about the 1/2 probability to win the lottery if you don't have more infos, or the die to decide whether to put gold or silver in the box.
What difference do you make between a 1/2 probability because you don't know a die was thrown to decide between SG and SS, and the 1/6 probability you would have if you knew that? The situation is the same, so the "real" probability is the same, but your guess changes.
Or to look at it from the other way: what difference do you make between a 1/2 probability because you don't know how the choice between SG and SS was made, and a 1/2 probability because you know the choice was made at random, with equal chance for each? I view these as very different: the latter is an accurate probability (if you repeated the action several times you'd win 1/2 of the time) while the former is only a best guess, in which case I'd personally view "I can't know the probability" as a better answer. But you don't seem to be making that distinction, and seem to view 1/2 as a very valid answer to the first situation.
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