Interesting. I've never watched that show.
Jn
Deceptive math?
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Primula Baggins
- Living in hope
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Let's you and me go have a cup of tea.
“There, peeping among the cloud-wrack above a dark tor high up in the mountains, Sam saw a white star twinkle for a while. The beauty of it smote his heart, as he looked up out of the forsaken land, and hope returned to him. For like a shaft, clear and cold, the thought pierced him that in the end the Shadow was only a small and passing thing: there was light and high beauty for ever beyond its reach.”
― J.R.R. Tolkien, The Return of the King
― J.R.R. Tolkien, The Return of the King
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Sassafras
- still raining, still dreaming
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Most of them stay put.
What I've observed is that when the large amounts on the board are reduced to three or less .... the majority of contestents seem driven by a belief in their luck and they almost always walk away with a very small amount of money.
I always think that I'd wait for a substantial offer from the banker and then make the deal.
A bird in the hand, you know
Btw, the wiki link was fascinating and lead to an even more fascinating link which finally made me understand why switching was the better option assuming that the host KNOWS what's behind each door.
Counter-intuitive indeed.
But then I'm a dolt at these probability things.
What I've observed is that when the large amounts on the board are reduced to three or less .... the majority of contestents seem driven by a belief in their luck and they almost always walk away with a very small amount of money.
I always think that I'd wait for a substantial offer from the banker and then make the deal.
A bird in the hand, you know
Btw, the wiki link was fascinating and lead to an even more fascinating link which finally made me understand why switching was the better option assuming that the host KNOWS what's behind each door.
Counter-intuitive indeed.
But then I'm a dolt at these probability things.
Ever mindful of the maxim that brevity is the soul of wit, axordil sums up the Sil:
"Too many Fingolfins, not enough Sams."
Yes.
I think I can explain where the confusion lies. Most people view the game as two choices you have to make. The first choice is for a random door out of three. Then, the Host removes a wrong choice, and you have a second choice, between two doors.
If the two choices were independant. Then it is a 50/50 chance you're right.
However, with the rules set up, so that the host always chooses a door with a rat to show you, and then offers the deal to switch... there are NOT two choices.
The first thing you do is not to choose a door. The first thing you do is divide the doors into two groups. The first group is "your door (one)" The second group is "The host's doors (two)." The only CHOICE you make, is to take what's behind your door, or the host's doors. your door has a 1/3 chance of being the right one, the hosts doors have a 2/3 chance of having the right one.
The fact one of the host's doors is wrong doesn't affect your choice.
If the two choices were independant. Then it is a 50/50 chance you're right.
However, with the rules set up, so that the host always chooses a door with a rat to show you, and then offers the deal to switch... there are NOT two choices.
The first thing you do is not to choose a door. The first thing you do is divide the doors into two groups. The first group is "your door (one)" The second group is "The host's doors (two)." The only CHOICE you make, is to take what's behind your door, or the host's doors. your door has a 1/3 chance of being the right one, the hosts doors have a 2/3 chance of having the right one.
The fact one of the host's doors is wrong doesn't affect your choice.
Interesting thread though, just scanning through them since I am busy with "take home" work and slacking around for awhile.
But I do love Quantitative Methods back when I was in Business School. I got an A+ in Statistics classes and just love probability questions esp. TREES and the two tail test.
I never seen Monty Hall.
Argh! Now I'm going to read the whole thread again out of curiosity, but my work awaits me.
But I do love Quantitative Methods back when I was in Business School. I got an A+ in Statistics classes and just love probability questions esp. TREES and the two tail test.
I never seen Monty Hall.
Argh! Now I'm going to read the whole thread again out of curiosity, but my work awaits me.
“Lawyers are the only persons in whom ignorance of the law is not punished.” - Jeremy Bentham (1748 - 1832)
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Primula Baggins
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Eww. Bergamot. No, no Earl Grey. (Did you know you can get it with extra bergamot? Eww.)
China oolong? Or the smoky warmth of lapsang souchong?
. . . A thread like this truly reminds me why I ought not to go around thinking of myself as smart, and therefore it is probably a public service.
China oolong? Or the smoky warmth of lapsang souchong?
. . . A thread like this truly reminds me why I ought not to go around thinking of myself as smart, and therefore it is probably a public service.
“There, peeping among the cloud-wrack above a dark tor high up in the mountains, Sam saw a white star twinkle for a while. The beauty of it smote his heart, as he looked up out of the forsaken land, and hope returned to him. For like a shaft, clear and cold, the thought pierced him that in the end the Shadow was only a small and passing thing: there was light and high beauty for ever beyond its reach.”
― J.R.R. Tolkien, The Return of the King
― J.R.R. Tolkien, The Return of the King
I got a headache now after reading this thread all over again.
Last edited by Lurker on Wed Dec 13, 2006 1:37 am, edited 1 time in total.
“Lawyers are the only persons in whom ignorance of the law is not punished.” - Jeremy Bentham (1748 - 1832)
I think I know exactly what you mean!Primula Baggins wrote:Eww. Bergamot. No, no Earl Grey. (Did you know you can get it with extra bergamot? Eww.)
China oolong? Or the smoky warmth of lapsang souchong?
. . . A thread like this truly reminds me why I ought not to go around thinking of myself as smart, and therefore it is probably a public service.
I'm so glad you don't like Earl Grey tea!
I like ordinary every day tea, myself. Lapsang Souchong is interesting, but what my tea-loving soul requires about this time of day ( 4 to 4:30 here) is a cup of good strong Tetley's or Lipton's Yellow Label or Murchie's English Breakfast. I put the milk in the teacup first and pour in the hot strong tea and breathe in the lovely aroma and take a sip.
And then I observe to myself, "There is NOTHING like a cup of tea."
It is extra nice if you have a slice of some good cake or a Peek Frean's Digestive to go with.
Yes, I am a lowly soul, grubbing away down here, far from the Realm of Higher Mathematics. I see those Statistics nodding sagely at the Probabilities up there --- see? Up there on the mountain, just at the tree line?
"5 to 1, you say? Oh, no, dear fellow. The chances are, in fact, 8 to 4.57 that Annabel will pull a yellow stocking from her stocking drawer!"
So intones Statistics.
Then, Probablity shakes his heavy bald head and smiles a mean smile. "Oh, what prodigious nonsense you utter, mon ami. I must say I find your intransigence most troublesome!"
*sigh*
There IS nothing like a cup of tea, is there?
Dig deeper.
Lol, Prim, but you are smart!
Actually...it's not so much that the host knows what's behind the doors. Even if he didn't know....if what he reveals is a rat/goat, you're best switching. The problem (as stated in the letter to the newspaper) is clear, but not general. Meaning, for that particular case, you can solve the problem, but insights you gain cannot be applied to any other situations.
If you double your bet every time (at blackjack, say), you will eventually win (or run out of money, whichever comes first). If you pick a game in which the odds are that you will win long before you run out of money....then you are practically guaranteed to beat the house. The house doesn't mind, because they know you will be tempted to try it again, and if you keep playing, they'll win. That, and people get nervous when they have to go so far in the hole (doubling your bet ten times in a row gets very scary very quickly - especially if you lose all ten times). Meaning - its not all just odds and math, there is a lot of psychology to this sort of thing.
But as for the original complaint - that puzzles like this aren't fair cause they make people feel stupid....well, they're puzzles! Cryptic crosswords make me feel stupid, but I don't assume the worst of the people who make them. Putting such a problem on a test is unfair, but challenging people with it or using it as an example is fine.
I mean, what is the difference between that problem and this one?
You are travelling to see the king, and you are taking with you a wolf, a chicken, and a sack of grain. You come to a river, where you can cross in a boat. But there is a problem! Only two of the items can fit in the boat with you at any one time. If you leave the wolf with the chicken, it will eat it, and if you leave the chicken with the grain, it will eat it. How do you get you (and all three items) safely across the river?
That's from memory, so I may have flubbed it up, but it should work. It is silly and pointless, and there is a "trick" to it - but that's the fun of the puzzle....
Actually...it's not so much that the host knows what's behind the doors. Even if he didn't know....if what he reveals is a rat/goat, you're best switching. The problem (as stated in the letter to the newspaper) is clear, but not general. Meaning, for that particular case, you can solve the problem, but insights you gain cannot be applied to any other situations.
If you double your bet every time (at blackjack, say), you will eventually win (or run out of money, whichever comes first). If you pick a game in which the odds are that you will win long before you run out of money....then you are practically guaranteed to beat the house. The house doesn't mind, because they know you will be tempted to try it again, and if you keep playing, they'll win. That, and people get nervous when they have to go so far in the hole (doubling your bet ten times in a row gets very scary very quickly - especially if you lose all ten times). Meaning - its not all just odds and math, there is a lot of psychology to this sort of thing.
But as for the original complaint - that puzzles like this aren't fair cause they make people feel stupid....well, they're puzzles! Cryptic crosswords make me feel stupid, but I don't assume the worst of the people who make them. Putting such a problem on a test is unfair, but challenging people with it or using it as an example is fine.
I mean, what is the difference between that problem and this one?
You are travelling to see the king, and you are taking with you a wolf, a chicken, and a sack of grain. You come to a river, where you can cross in a boat. But there is a problem! Only two of the items can fit in the boat with you at any one time. If you leave the wolf with the chicken, it will eat it, and if you leave the chicken with the grain, it will eat it. How do you get you (and all three items) safely across the river?
That's from memory, so I may have flubbed it up, but it should work. It is silly and pointless, and there is a "trick" to it - but that's the fun of the puzzle....
Last edited by MithLuin on Wed Dec 13, 2006 1:49 am, edited 1 time in total.
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Primula Baggins
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There must be another constraint on the wolf-grain-chicken problem, Mith, because as it is you take the wolf and the grain over and then go back for the chicken.
“There, peeping among the cloud-wrack above a dark tor high up in the mountains, Sam saw a white star twinkle for a while. The beauty of it smote his heart, as he looked up out of the forsaken land, and hope returned to him. For like a shaft, clear and cold, the thought pierced him that in the end the Shadow was only a small and passing thing: there was light and high beauty for ever beyond its reach.”
― J.R.R. Tolkien, The Return of the King
― J.R.R. Tolkien, The Return of the King
No, this is wrong too. There are very specific conditions that have to hold. The host has to know what is behind the doors, and always open a rat door among the two you didn't choose. The problem as stated to Vos Savant was not actually clear enough, because it wasn't specifically stated that the host always opens a rat door.Actually...it's not so much that the host knows what's behind the doors. Even if he didn't know....if what he reveals is a rat/goat, you're best switching. The problem (as stated in the letter to the newspaper) is clear, but not general. Meaning, for that particular case, you can solve the problem, but insights you gain cannot be applied to any other situations.
The difference is that a key assumption is often left out of the Monty Python problem.But as for the original complaint - that puzzles like this aren't fair cause they make people feel stupid....well, they're puzzles! Cryptic crosswords make me feel stupid, but I don't assume the worst of the people who make them. Putting such a problem on a test is unfair, but challenging people with it or using it as an example is fine.
I mean, what is the difference between that problem and this one?
Only one item fits in the boat, rather than two.You are travelling to see the king, and you are taking with you a wolf, a chicken, and a sack of grain. You come to a river, where you can cross in a boat. But there is a problem! Only two of the items can fit in the boat with you at any one time. If you leave the wolf with the chicken, it will eat it, and if you leave the chicken with the grain, it will eat it. How do you get you (and all three items) safely across the river?
What is the probability of the wolf eating me?MithLuin wrote: You are travelling to see the king, and you are taking with you a wolf, a chicken, and a sack of grain. You come to a river, where you can cross in a boat. But there is a problem! Only two of the items can fit in the boat with you at any one time. If you leave the wolf with the chicken, it will eat it, and if you leave the chicken with the grain, it will eat it. How do you get you (and all three items) safely across the river?
I'll take the wolf first cause the probability of the chicken eating all the grain until I get back is slim, my sack can still pass off as full depending on how you look at it, but the probability of the wolf eating the chicken is 100%.
“Lawyers are the only persons in whom ignorance of the law is not punished.” - Jeremy Bentham (1748 - 1832)
Oh, yeah, right - two total (you plus one other critter).
Apparently, Lurker, the wolf does not eat you, and your mere presence is enough to cow both the wolf and the chicken into submission.
So, there is a way to do it without loosing any grain, honest. But it takes several trips... (It can also be a wolf, a goat and a cabbage, if you prefer that combination)
But anyway, Faramond, I agree with you that for the solution to always make sense, there need to be constraints. But for the solution to work that one time, there needn't be. That's why it's fine as written, so long as you don't take it as universal. You're saying the host has to always open the rat-door; I'm saying you only play the game once, and he opens a rat-door that time.
I guess my question is, is the problem with puzzles that are not what they appear, or is the problem with asking questions that aren't straightforward? Because, I have no problem asking convoluted questions (so long as you label them a puzzle or cipher).
Puzzles that annoy me are ones that are not thoroughly thought out. The potions puzzle in the first Harry Potter book is a logic puzzle, but it has two possible solutions (if you can't see the bottles), and the poison isn't placed anywhere interesting. This bothers me. If you are going to design a trap, it should have the pitfalls in the right place! (That being said, it is sufficiently puzzling, and cannot be figured out just by reading it - you'd have to jot some stuff down). Also, and most importantly, it does work (it just requires a diagram that isn't provided).
Solution to the potion puzzle
Apparently, Lurker, the wolf does not eat you, and your mere presence is enough to cow both the wolf and the chicken into submission.
So, there is a way to do it without loosing any grain, honest. But it takes several trips... (It can also be a wolf, a goat and a cabbage, if you prefer that combination)
But anyway, Faramond, I agree with you that for the solution to always make sense, there need to be constraints. But for the solution to work that one time, there needn't be. That's why it's fine as written, so long as you don't take it as universal. You're saying the host has to always open the rat-door; I'm saying you only play the game once, and he opens a rat-door that time.
I guess my question is, is the problem with puzzles that are not what they appear, or is the problem with asking questions that aren't straightforward? Because, I have no problem asking convoluted questions (so long as you label them a puzzle or cipher).
Puzzles that annoy me are ones that are not thoroughly thought out. The potions puzzle in the first Harry Potter book is a logic puzzle, but it has two possible solutions (if you can't see the bottles), and the poison isn't placed anywhere interesting. This bothers me
. If you are going to design a trap, it should have the pitfalls in the right place! (That being said, it is sufficiently puzzling, and cannot be figured out just by reading it - you'd have to jot some stuff down). Also, and most importantly, it does work (it just requires a diagram that isn't provided).Solution to the potion puzzle
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Angbasdil
- The man, the myth, the monkey.
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This thread has moved on since I started typing this, so I apologize if it's redundant or irrelevant. But I've got to go to bed and can't read that last page right now. :|
How's this for an intuitively understandable answer?
If the host's actions are random, then the outcome is random (i.e. 1/2).
If the host's actions are not random, then the outcome is not random (i.e. 2/3).
I'm a big fan of Marilyn vos Savant, and I own the book wherein she spends an entire chapter on this conundrum. (or I used to own it - I think I may have lent it to my in-laws.) So I thought I understood this thing. But after reading through this thread, I wasn't so sure any more. The good thing about these small scale probability problems is that you can work them out "longhand" fairly easily by examining every possible scenario. So let's do that.
First, let's assume that you pick door #1. It doesn't really matter which one you pick, so whichever one you pick we'll just rename it door #1 and you'll just have to live with that. Now there are three equally likely possibilities - the car is behind door #1, door #2 or door#3. (Duh!) And the host can open either door #2 or door #3.
Scenario 1 - the car is behind door #1
It doesn't matter at this point which door the host opens, or why. If you stay with #1 you win. If you switch, you lose. But for the sake of completeness, let's break this down into both possibilities.
1A - the host opens door #2
1B - the host opens door #3
the odds at the beginning of the game of each of these happening is 1/6.
Scenario 2 - the car is behind door #2
This gets a little trickier. Let's break it down again.
2A - the host opens door #2. If this happens you lose right then and there.
2B - the host opens door #3. If you switch, you win.
Note that if the host's actions are random, then each of these has a 1/6 chance of happening. But... if the host always shows you a goat, then 2A will never happen. In that case, if the car is behind door #2, the host will open door #3. And since the odds of the car being behind door #2 is 1/3, the odds of 2B occuring are 1/3. And the odds of 2A are 0.
Scenario 3 - the car is behind door #3
See scenario 2 above for the explanation, just swap the numbers around.
3A - the host opens door #2
3B - the host opens door #3
Intuitively, one would think that your odds of winning this thing would be 1/3. Three doors, one chance, it sounds like a no-brainer. And if it's all random, those are your odds at the beginning of the game. Above you see six equally likely scenarios, two of which cause you to lose as soon as the host opens the door (2A and 3B). But in the question as posed here, you've already dodged those two bullets. At that point your odds improve from 2/6 to 2/4. Which two options are winners depends on whether or not you switch doors, but either way it's 50-50.
But if the host's choice is not random, all that goes out the window. Now 2A and 3B are impossibilities, and 2B and 3A are more likely (1/3 each). Let's look at the possibilities.
Scenario 1 - if you switch, you lose.
Scenario 2 - if you switch, you win.
Scenario 3 - if you switch, you win.
There you have it. Three equally likely scenarios. Two of them give you a win if you switch.
If the host's choice is random, then it doesn't matter whether or not you switch. But if the host's choice is informed, then it matters very much. So assuming that the host made an informed choice has a significant potential payoff and no potential drawback. Therefore, the best tactical move is to assume that the host is making informed choices and behave accordingly. Switch doors.
How's this for an intuitively understandable answer?
If the host's actions are random, then the outcome is random (i.e. 1/2).
If the host's actions are not random, then the outcome is not random (i.e. 2/3).
I'm a big fan of Marilyn vos Savant, and I own the book wherein she spends an entire chapter on this conundrum. (or I used to own it - I think I may have lent it to my in-laws.) So I thought I understood this thing. But after reading through this thread, I wasn't so sure any more. The good thing about these small scale probability problems is that you can work them out "longhand" fairly easily by examining every possible scenario. So let's do that.
First, let's assume that you pick door #1. It doesn't really matter which one you pick, so whichever one you pick we'll just rename it door #1 and you'll just have to live with that.
Now there are three equally likely possibilities - the car is behind door #1, door #2 or door#3. (Duh!) And the host can open either door #2 or door #3.Scenario 1 - the car is behind door #1
It doesn't matter at this point which door the host opens, or why. If you stay with #1 you win. If you switch, you lose. But for the sake of completeness, let's break this down into both possibilities.
1A - the host opens door #2
1B - the host opens door #3
the odds at the beginning of the game of each of these happening is 1/6.
Scenario 2 - the car is behind door #2
This gets a little trickier. Let's break it down again.
2A - the host opens door #2. If this happens you lose right then and there.
2B - the host opens door #3. If you switch, you win.
Note that if the host's actions are random, then each of these has a 1/6 chance of happening. But... if the host always shows you a goat, then 2A will never happen. In that case, if the car is behind door #2, the host will open door #3. And since the odds of the car being behind door #2 is 1/3, the odds of 2B occuring are 1/3. And the odds of 2A are 0.
Scenario 3 - the car is behind door #3
See scenario 2 above for the explanation, just swap the numbers around.
3A - the host opens door #2
3B - the host opens door #3
Intuitively, one would think that your odds of winning this thing would be 1/3. Three doors, one chance, it sounds like a no-brainer. And if it's all random, those are your odds at the beginning of the game. Above you see six equally likely scenarios, two of which cause you to lose as soon as the host opens the door (2A and 3B). But in the question as posed here, you've already dodged those two bullets. At that point your odds improve from 2/6 to 2/4. Which two options are winners depends on whether or not you switch doors, but either way it's 50-50.
But if the host's choice is not random, all that goes out the window. Now 2A and 3B are impossibilities, and 2B and 3A are more likely (1/3 each). Let's look at the possibilities.
Scenario 1 - if you switch, you lose.
Scenario 2 - if you switch, you win.
Scenario 3 - if you switch, you win.
There you have it. Three equally likely scenarios. Two of them give you a win if you switch.
Your problem here is that you're thinking logically. You should be thinking tactically. Look beyond the likelihood of the rules and consider how your choice could potentially affect the game.Jny wrote: I don't remember now what rule resulted in 2:3 for never switching. I'd have to rethink the puzzle. The point remains, of course, that if different rules result in different outcomes, and you don't know what rule the host is using, then there's no reason to proclaim one outcome 'right' and the other one 'wrong.' It seems to me that I also calculated what the odds would be if the host switched rules at random! So if you don't know the rule, then what should you choose? But again, I'd have to recreate the tables to remember what answer I got.
If the host's choice is random, then it doesn't matter whether or not you switch. But if the host's choice is informed, then it matters very much. So assuming that the host made an informed choice has a significant potential payoff and no potential drawback. Therefore, the best tactical move is to assume that the host is making informed choices and behave accordingly. Switch doors.
Could it be a rock, a scissor, and a paper?So, there is a way to do it without loosing any grain, honest. But it takes several trips... (It can also be a wolf, a goat and a cabbage, if you prefer that combination)
If you don't know the circumstances under which the host opened that door, then you really can't calculate a probability. For any probability to make sense, you need to be able to reliably repeat the experiment. Either repeat it in the real world, or be able to at least theoretically repeat it as a thought experiment. A probability is a limit of a ratio, but that's probably beyond the scope of this thread.But anyway, Faramond, I agree with you that for the solution to always make sense, there need to be constraints. But for the solution to work that one time, there needn't be. That's why it's fine as written, so long as you don't take it as universal. You're saying the host has to always open the rat-door; I'm saying you only play the game once, and he opens a rat-door that time.
The point is you can't have a "one-off" probablity. You need to know enough to reasonably model the important things leading up to the outcome. In the real world you don't know everything, but if you know enough you can make assumptions or simplifications to be able to reasonably calculate a probability. But here we have a host making a decision and opening a door. We have no idea what he knows or why he opened that door, so we can't repeat the experiment. We know he opened a rat door, but that's not enough. If we know how he makes his choice each time, then we can repeat the experiment, at least in the abstract, and calculate a probability. But the way the question is usually written ( though not the way I wrote it ) we can't model his behavior. We just don't know.
I guess my question is, is the problem with puzzles that are not what they appear, or is the problem with asking questions that aren't straightforward? Because, I have no problem asking convoluted questions (so long as you label them a puzzle or cipher).
I enjoy convoluted questions, as long as you are given enough information, or you aren't asked to make inobvious assumptions.