The Logic of Probability
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The Logic of Probability
This one reminds me of Monty for some reason. I think I get it, but I was hoping you guys could help me out and get some enjoyment from it. I apologize for the theme. My teacher really likes potter.
6 wands on a table. Each of the four wizards pick a wand one after each other. 4 wands will produce a dragon. 2 will produce a toad. When your wand conjures each animal you can't see it, but everyone else can. Wands are removed after being used.
1st wand Dragon
2nd wand (you) unkonwn.
3rd wand Dragon
4th wand Dragon
What is the probability you conjured a toad.
What is the probability you conjured a dragon.
Have fun.
edited for pedantry of respondents. jr
6 wands on a table. Each of the four wizards pick a wand one after each other. 4 wands will produce a dragon. 2 will produce a toad. When your wand conjures each animal you can't see it, but everyone else can. Wands are removed after being used.
1st wand Dragon
2nd wand (you) unkonwn.
3rd wand Dragon
4th wand Dragon
What is the probability you conjured a toad.
What is the probability you conjured a dragon.
Have fun.
edited for pedantry of respondents. jr

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Re: The Logic of Probability
Spoiler:
Mighty Jalapeno wrote:Tyndmyr wrote:Роберт wrote:Sure, but at least they hit the intended target that time.
Well, if you shoot enough people, you're bound to get the right one eventually.
Thats the best description of the USA ever.

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Re: The Logic of Probability
Spoiler:
Or are there factors I'm not considering? I'd love an explanation of why this train of logic is false.
Re: The Logic of Probability
Spoiler:
Pseudomammal wrote:Biology is funny. Not "haha" funny, "lowest bidder engineering" funny.
 jestingrabbit
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Re: The Logic of Probability
EricH wrote:Spoiler:
*groan*
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
 Yakk
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Re: The Logic of Probability
The usual problem with this kind of logic puzzle is that it doesn't include information required to solve it.
Describing the sequence of events that happen is not sufficient to determine the distribution of probability.
As an example, suppose you know (are 100 certain) that the last wand is a frog wand. And when you picked a wand, you picked the last wand. This fits the described events, yet in this case the probability that you summoned a frog is 100%. Similarly, if you knew the first wand was a dragon wand, and you picked it, then the probability you summoned a dragon is 100%  and this also fits the described events.
So to solve this problem, we need a description of the algorithm that each of the wizards used to select their wand. Now, we could invent an algorithm  but that makes this a mindreading puzzle rather than a logic puzzle. The Monte Hall puzzle has a similar problem.
It is true that entire classes of algorithms that pick wands (or doors) lead to the same result. But there are classes of algorithms that are not restricted by the description that lead to results that don't agree. So short of assigning a probability to each algorithm (either explicitly, or implicitly by picking one and saying that is the one that is used), the question lacks sufficient information to answer.
This is also the reason why you don't place bets with magicians.
Describing the sequence of events that happen is not sufficient to determine the distribution of probability.
As an example, suppose you know (are 100 certain) that the last wand is a frog wand. And when you picked a wand, you picked the last wand. This fits the described events, yet in this case the probability that you summoned a frog is 100%. Similarly, if you knew the first wand was a dragon wand, and you picked it, then the probability you summoned a dragon is 100%  and this also fits the described events.
So to solve this problem, we need a description of the algorithm that each of the wizards used to select their wand. Now, we could invent an algorithm  but that makes this a mindreading puzzle rather than a logic puzzle. The Monte Hall puzzle has a similar problem.
It is true that entire classes of algorithms that pick wands (or doors) lead to the same result. But there are classes of algorithms that are not restricted by the description that lead to results that don't agree. So short of assigning a probability to each algorithm (either explicitly, or implicitly by picking one and saying that is the one that is used), the question lacks sufficient information to answer.
This is also the reason why you don't place bets with magicians.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: The Logic of Probability
Yakk wrote:The usual problem with this kind of logic puzzle is that it doesn't include information required to solve it.
Describing the sequence of events that happen is not sufficient to determine the distribution of probability.
As an example, suppose you know (are 100 certain) that the last wand is a frog wand. And when you picked a wand, you picked the last wand. This fits the described events, yet in this case the probability that you summoned a frog is 100%. Similarly, if you knew the first wand was a dragon wand, and you picked it, then the probability you summoned a dragon is 100%  and this also fits the described events.
So to solve this problem, we need a description of the algorithm that each of the wizards used to select their wand. Now, we could invent an algorithm  but that makes this a mindreading puzzle rather than a logic puzzle. The Monte Hall puzzle has a similar problem.
It is true that entire classes of algorithms that pick wands (or doors) lead to the same result. But there are classes of algorithms that are not restricted by the description that lead to results that don't agree. So short of assigning a probability to each algorithm (either explicitly, or implicitly by picking one and saying that is the one that is used), the question lacks sufficient information to answer.
This is also the reason why you don't place bets with magicians.
It's generally assumed that these things are chosen uniformly at random, unless stated otherwise. Also... your example with picking the last wand knowing it will conjure a frog is basically saying "If your wand will conjure a frog, then your wand will conjure a frog".
Blue, blue, blue
 Yakk
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Re: The Logic of Probability
Yes, I agree, things about these kind of probability problems are often left to be assumed by the reader. As noted by other posters, different things are assumed by different people, leading to different results in this case. And then people yell at each other implicitly over whose assumptions are the right ones (be it with Monte Hall or other kind of problem".
If you qualify the assumptions  ie, insert "each wizard picks a wand uniformly and randomly from the ones present", which is pretty easy, some major assumptions are removed. Or you could mention it in the answer.
The point is that you cannot, from the observations described, exactly determine the probability you summoned a dragon. Ie, if you where in an equivalent situation, making a bet with real cash at 1:3 or 1:5 odds would be foolish, as the described events aren't sufficient to generate enough knowledge about the odds of the situation. If you disagree with this and think you can build a strong probability model from a description like the above, would you like to pick a pea from under a shell?
If you qualify the assumptions  ie, insert "each wizard picks a wand uniformly and randomly from the ones present", which is pretty easy, some major assumptions are removed. Or you could mention it in the answer.
The point is that you cannot, from the observations described, exactly determine the probability you summoned a dragon. Ie, if you where in an equivalent situation, making a bet with real cash at 1:3 or 1:5 odds would be foolish, as the described events aren't sufficient to generate enough knowledge about the odds of the situation. If you disagree with this and think you can build a strong probability model from a description like the above, would you like to pick a pea from under a shell?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: The Logic of Probability
Making a bet with real cash would not be foolish at all. It would be foolish if you made the bet with the wizards who were picking up the wands, or anyone who could control the outcome.Yakk wrote:The point is that you cannot, from the observations described, exactly determine the probability you summoned a dragon. Ie, if you where in an equivalent situation, making a bet with real cash at 1:3 or 1:5 odds would be foolish, as the described events aren't sufficient to generate enough knowledge about the odds of the situation. If you disagree with this and think you can build a strong probability model from a description like the above, would you like to pick a pea from under a shell?
If I flipped a coin what would you say the probability is that it was heads? Keep in mind that it MIGHT be a double sided coin. I suppose you could take into account that doubleheaded coins are more common than doubletailed coins (or something), but 50/50 is PROBABLY the right answer, especially if I didn't have a motive for tricking you.
You don't have to treat logic puzzles as if someone is trying to "get you".
Adam
Re: The Logic of Probability
Yakk wrote:would you like to pick a pea from under a shell?
Sounds good, as long as I get to be the guy with the suspiciously similar accent who has the first go.
 Yakk
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Re: The Logic of Probability
Adam H: No, I treat claims of fact and proof as if I'm trying to "get it". This helps find errors in the chain of reasoning.
Which is why things like "by convention, we will assume", which admits an arbitrary choice, are less problematic than simply assuming it and then thinking that the results are actually facts. Conditional facts that admit it are more true than those that don't.
I think this position is wise, because I see people arguing over logic puzzles like this one and Monte Hall over basically this very problem: they make different assumptions, don't state them clearly, and then get different results. Both positions often end up with answers which "fit" the description of the problem, yet disagree on the answer. If you add in the assumptions, it becomes clear why they disagree  because the question isn't clear. (Not to say that there aren't also people who answer differently because they are just wrong!)
Which is why things like "by convention, we will assume", which admits an arbitrary choice, are less problematic than simply assuming it and then thinking that the results are actually facts. Conditional facts that admit it are more true than those that don't.
I think this position is wise, because I see people arguing over logic puzzles like this one and Monte Hall over basically this very problem: they make different assumptions, don't state them clearly, and then get different results. Both positions often end up with answers which "fit" the description of the problem, yet disagree on the answer. If you add in the assumptions, it becomes clear why they disagree  because the question isn't clear. (Not to say that there aren't also people who answer differently because they are just wrong!)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: The Logic of Probability
The problem has a lot of extraneous information that seems to be throwing people off.
Spoiler:
Re: The Logic of Probability
That depends on your interpretation as frequentist or bayesian.
An interpretation which cannot give you a probability that you picked a dragon wand, given the information you have, has bad limits to its applicability.
An interpretation which cannot give you a probability that you picked a dragon wand, given the information you have, has bad limits to its applicability.
 Yakk
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Re: The Logic of Probability
More importantly, there are uses of the mathematics of probability that describe levels of certainty that something is true given limited information.
It doesn't matter if you are a frequentist or bayesian, the bayesian approach can be used to analyze certain situations, and generates mathematically valid results. You might be uncomfortable using the world "probability" to describe what it gives you, but that is a problem with semantics, not the math.
It doesn't matter if you are a frequentist or bayesian, the bayesian approach can be used to analyze certain situations, and generates mathematically valid results. You might be uncomfortable using the world "probability" to describe what it gives you, but that is a problem with semantics, not the math.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: The Logic of Probability
mfb wrote:An interpretation which cannot give you a probability that you picked a dragon wand, given the information you have, has bad limits to its applicability.
We are not constrained by such limitations in this case. We can easily determine the probability that the experiment will (note future tense) result in a toad or dragon for a particular wizard in any position in the choosing queue, given that the results produced by previous wands in the same experiment.
If a wizard has a betting partner who can also see the results of the other wizards' wands, but not the results from the wand of his betting partner, fair bets can be made between them, even after the experiment is complete and the results observed by the other wizards. That does not imply that the probability of the wand producing a dragon is different from what I previously stated.
Consider:
Spoiler:
Re: The Logic of Probability
As I said, it depends on the interpretation.
The 345,876,912th digit has a ten percent probability of being a 3 (for me, with my current knowledge).
I can.
We cannot say that the 345,876,912th digit has a ten percent probability of being a 3.
The 345,876,912th digit has a ten percent probability of being a 3 (for me, with my current knowledge).
I can.
Re: The Logic of Probability
Halleck wrote:4 wands will produce a dragon. 2 will produce a frog.
What is the probability you conjured a toad.
Zero!
V V V I'm sorry, was just amazed it went this far without anyone noticing!
Last edited by Tunga on Wed Apr 11, 2012 9:51 am UTC, edited 2 times in total.
 jestingrabbit
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Re: The Logic of Probability
You have moved me to edit the OP.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
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