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 Cognitive Illusion 
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Chibi-Czar
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Post Cognitive Illusion
A researcher puts three boxes in front of you. In one of the three boxes is a ten dollar bill. You do not know which box the money is in but the researcher does. You select one box. The researcher then opens one of the remaining boxes, always an empty one. The researcher then offers you the chance to exchange boxes. Do you switch boxes or keep your original choice, and why?

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Wed Apr 25, 2007 3:15 pm
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Post Re: Cognitive Illusion
Short Answer: No.

I'll break this down.

Anony-mouse wrote:
A researcher puts three boxes in front of you.


"This better be good..."



Quote:
In one of the three boxes is a ten dollar bill. You do not know which box the money is in but the researcher does.


So what's he need me for?


Quote:
You select one box.


Right-click, save as.

Quote:
The researcher then opens one of the remaining boxes, always an empty one.


'ALWAYS' an empty one? Isn't this the first time we did this?

Quote:
The researcher then offers you the chance to exchange boxes. Do you switch boxes or keep your original choice, and why?


DEAL... NO DEAL! WAIT! ERRR... ARGH!

I just don't want to go home with only one dollar...


Anyway... The wording is odd. If I sit down with a researcher, and he 'always' opens an empty box, this probably means we're doing this several times, and I know that some time during my choice, he probably slipped the ten dollar bill into his wallet. Seriously though. All this really is, is an extended Fifty Fifty choice. All he does is show me which of the boxes I didn't pick was empty, leaving two already and a choice to 'repick' for a fifty fifty chance.

No matter how many times I sit down with him, I get a fifty fifty chance the moment I pick my first box. Might as well just keep the one I picked first.

Besides... even if I lose, I get paid $500 or so for participating in this research project. I win anyway. :twisted:

EDIT: There is one other possibility. The wording leaves a gap that also allows him to close the empty box, and open the other one. If He's opening both boxes, and they're both empty. I have the ten dollar box in my hand anyway. :roll: I'd expect something like that from a brain twister.

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Wed Apr 25, 2007 4:46 pm
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Wrong.

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Wed Apr 25, 2007 8:34 pm
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This is no illusion, it's just playing on words to get the reaction you want across. This is such an open-ended puzzle that it's basically only accurate if you already have the answer. But hey, you're taking the time to share these things from wherever you're gleaming them from, so I'll give you Ye Ol College Try.

So, within the stated rules, I would do one of four things:

-Before choosing, ask him which box has the money. Offer to split it with him if he looks particularly under-fed by his grant-money. :)

-If he doesn't give me the obvious "here is where the money is" answer or a similar poker-face failure while I'm choosing, I'll just look in my box, see if there is money, then politely agree or disagree to trade depending on if there is money in it. After all, he peeked (or we both know he picked an empty one), and there is no rule against me doing the same before making the trade. If I have no money, then that means both our boxes = empty and I grab the last one as a trade, assuming I don't need to exchange my empty box with his, and can instead substitute it with the one we now know has the bucks.

-If there is a rule about looking inside my box: Say Yes, and take the remaining third box. Why: Because as the researcher, he has no percevable desire for the money since he's taking the time to set up the situation anyway and clearly wants something else out of it. If greed is not motivating him, then his suggestion to take the last box may be simply a test of trust to see if I will allow him to direct me to my prize.

-If I cannot ask about, look in or otherwise negate the idea that I only have one chance, and/or pick a second box after, then I would simply tell the researcher how my mom got scared and said: "You're moving with your antie and uncle in Bel-Air!". I whistled for a cab and when it came near the license plate said fresh and it had dice in the mirror. If anything I can say this cab is rare but I thought 'Naw forget it' - 'Yo homes to Bel Air'. I pulled up to the house about 7 or 8 And I yelled to the cabbie 'Yo homes smell ya later.' I looked at my kingdom I was finally there, to sit on my throne as the Prince of Bel Air

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Wed Apr 25, 2007 11:35 pm
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A cognitive illusion is to thought what an optical illusion is to sight. If you put a certain pattern of black and white shapes on a spinning disk, you'll start seeing colors. Whether or not you know that there are only black and white shapes on the disk does nothing to alter your perception. This is due simply to the way our eyes work. Similarly, because of the wa our brains work, we simply suck, and suck hard, at certain types of thinking. No matter what you know to be true, your brain will tell you something else.

One of the things our brains suck at most is statistical analysis. This problem is inherently a statistical one. As such, people consistently get it wrong. There are three boxes and only one bill. That means there is a 1-in-3 chance that any given box will have the ten dollars in it. This means that the remaining two boxes have a combined probability of 2/3. When the researcher opens one of the two remaining boxes, this probability colapses into the remaining box. Thus, while your box still has only a 1/3 chance of being the correct box, the other box has a 2/3 chance of being right. Thus you always choose the other box. Always.

Right now you're probably thinking 'that's wrong.' And that's the point. Your brain will tell you that this solution is wrong, even though it's right. Your brain is playing a trick on you. Nothing you could say or think would change this feeling. It will always feel wrong, just like an optical illusion will always look different than it actually is. If you don't believe me, get three boxes and a friend, and do this test around a hundred times for a good sample size, then post your findings. It feels wrong but it's right.

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Wed Apr 25, 2007 11:50 pm
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Yup.

It's a fun one to prove to people too, it's an easy enough experiment to replicate - record your own observations, it'll match up pretty well to the 1/3 -> 2/3 shift.

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Wed Apr 25, 2007 11:58 pm
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Sorry, I'm going to have to call: WTF??

If you want to go statistics, it's actually a 1/2 chance. Where are you getting 1/3 > 2/3 from? The "out of three" statement is actually the only misleading portion of this so-called illusion, as the rules and the actions put forth by the researcher reduce your choices from 3 to 2. How do you justify keeping a completely known and negated portion of the equation in when deciding the final outcome? You're basically just padding the mathematics with junk data to produce a desirable "chance".

Another example to illustrate:

A poll of 3 people:
-1 says "I like the World"
-1 says "I don't like the World"
-1 says "THERE ARE BEES IN MAH HEAD, SALLY!! I LOVE CAKE! MOOSE!!"

You do NOT count person 3 to create a poll result that says: "Only 1 out of 3 people said they Like the World, therefore public opinion on life is very low right now."

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Thu Apr 26, 2007 1:29 am
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Quote:
There are three boxes and only one bill. That means there is a 1-in-3 chance that any given box will have the ten dollars in it. This means that the remaining two boxes have a combined probability of 2/3. When the researcher opens one of the two remaining boxes, this probability colapses into the remaining box. Thus, while your box still has only a 1/3 chance of being the correct box, the other box has a 2/3 chance of being right. Thus you always choose the other box. Always.


WTF? When the researcher opens one of the two remaining boxes, there is no 'collapse' of probability.

If the box is empty. The equation changes to a 1/2 chance of the bill being in the box I have as opposed to the other box.

And I chose to keep the one I already had, not the 'other' box. Explain THAT.

Your illusions do not work on me Evangeline.


EDIT: And I know why.

This illusion is something that has to be done on the spot with real boxes and such. If it's done on the spot, we've got no time to actually think about it and break the illusion like we've already done by typing our response. Not only that, but you told us this would be an illusion, and when we think illusions, we instantly hunt for the trick. We were anylizing this from the start, which ruins the effect.

With it ON THE SPOT, the illusion is that the remaining box has a higher probability.

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Thu Apr 26, 2007 4:54 pm
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If the researcher has some way of knowing which box holds the bill without opening it, which he does, then what happens is:

1. I pick a box. right now, there is a one in three chance my box is right.
2. the researcher selects one of the two boxes I did not pick, one of which MUST be empty, making this selection a one in two. however, he knows ahead of time which of the boxes is empty, so he is always right.
3. there are now two boxes in play, one of which has money and one of which does not.
4. I pick a box. there is a one in two chance that I am right.

It doesn't matter that I already picked a box. After the researcher opens one of the empty boxes, there are two boxes and I now have a one in two chance of making the right guess and picking the correct box. The fact that I previously picked a box has no bearing on the second choice, because the two choices are independant events.

The "illusion" is the jump from 1/3 chance to 2/3 chance, and putting this question to people on the street will show that most of them will choose to switch boxes.

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Fri Apr 27, 2007 9:04 pm
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>_<

Okay, let me go through this again. Your instincts on probability are wrong. Completely wrong. That's the whole point of this, to show how bad the human brain is at probability. The math is right, there is a 66.666% chance that the other box has the bill, but the box you picked only has 33.333% chance of being the right one. I didn't come up with this. This problem was designed by cognitive neurologists to underscore their findings. I can't say it enough, the math is right.

Perhaps it would be helpful to rephrase the answer? The box you choose has a two-in-three chance of being wrong. This is because there are three boxes. The other two boxes have a combined one-in-three chance of being wrong. When the researcher opens an empty box, your box still has a two-in-three chance of being wrong. The other two boxes still have a combined one-in-three chance of being wrong. This is because, and this is where the brain fools you, THERE ARE STILL THREE FUCKING BOXES!!! The only thing that has changed is that now you have a single, absolute value for one of the boxes, 0% chance of being right. Your box still has a two-in-three chance of being wrong. The other unopened box, when considered alone, also has a two-in-three chance of being wrong. But see, you're not considering it alone. You're considering it along with the other box. Those two boxes together have only a one-in-three chance of being wrong, because there are two of them; remember, an opened box IS STILL A FUCKING BOX and thus still part of the equation. The only thing that has changed is that you know for sure that the one-in-three chance of being wrong, that is to say a two-in-three chance of being right, isn't evenly distributed. All the chance of being right shifts to the unopened box. Once one box is opened, once the probability of one of the boxes is locked down you know the distribution of the combined probability of one of the two boxes being right. That means the unopened box has a 66.666% chance of being right, and you should always choose the second box.

The way the illusion works is based on the way your brain groups objects. Your brain tells you that the three unopened boxes are all the same and thus in a group. When you open one box, it makes that box different from the other boxes. Your natural inclination is naturally to regroup the boxes mentally into two groups, boxes that are open and boxes that are unopened. Your brain focuses on the boxes that it thinks are still grouped together, namely the two unopened boxes. In reality, all three boxes are still in play. The grouping is actually like this: the box you picked is in a group by itself; the boxes you didn't pick, even though one is open and the other isn't, are in another group.

I detailed in my response post that your brain was fooling you. You insisted your brain was just fine, even though this thread is title Cognitive Illusion. I showed that my math was right. It just isn't intuitive. That's rather the point of this whole thing. I showed where your brain fools you. It has to do with how your brain handles abstract groupings. It groups objects by their similar characteristics, not by their abstract mathematical values. Do you now understand why the choice to switch boxes is not 50/50?

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Sat Apr 28, 2007 2:58 am
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No. In REALITY.

Once the box has been opened and proven to have nothing in it, it is negated from the equation. That's three boxes that might have a prize, minus one box we KNOW doesn't have it, equals two boxes remaining that might have it.

3 - 1 = 2.

Probability of two boxes with one item in them is one out of two. 1/2 or 50-50.

We are not having an 'instinctive' reaction to your illusion. Your presentation of it has failed. We had time to THINK. This illusion DOES NOT WORK typed up on a message board. Your math doesn't matter. You can throw calculus equations at us for all I care, but it won't change the fact that we're not operating on instinctive gut feelings, AND you TOLD US it was an illusion from the start, which pretty much set us all in gear to expect tricks so we anylized the problem.

Image

I know what the illusion is SUPPOSED to be, but none of us here fell for it. Trying to tell us we're 'wrong' is the incorrect way to go about what you're explaining, and it will only piss us off more.

Image

Now, for the sake of your sanity...


What he's saying... Since his communicaiton skills suck worse than anything...

When presented to an 'off the street person', the three box event tricks the brain. Despite being opened, the unprepared brain still treats the open box as part of the BOX catagory, so it still thinks in terms of three items subconciously, rather than dropping to two.
In the two remaining boxes, the probability that should be negated from the reveal of the third box is incorrectly sent to the second box. That foulup makes the brain believe the second box has the highest probability of having the money... inciting a person to want to switch boxes.

Is this what you are trying to say?

How many here wanted, even for a moment to type that 'yes' you wanted to switch boxes when presented with the choice, before you had thought it over?

It's THAT gut instinct I believe the illusion plays on. A-M's just been spending too much time sniffing that rat-poison again to be able to communicate correctly. *Slaps A-M with a fish.*

Enguard!

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Sat Apr 28, 2007 6:11 am
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Stupid graphics aside, you got it exactly backwards. The illusion is you think that opening the box magically removes it from the probability equation. The fact that you think it becomes a 50/50 choice MEANS THAT YOU HAVE SUCCUMB TO THE ILLUSION! The human brain is inherently bad at probability, and you're proving that by insisting that it both boxes you have the option of choosing are of equal likelihood of being correct. They don't. The chance of your box being the right is one-in-three, no matter what. The chance of a box that isn't yours being right is two-in-three, no matter what.

There is no trick. It's just a simple math problem. Your brain tricks itself. Your brain, by virtue of its functioning won't let you see the right answer. It's like an optical illusion. You know the circles are the same size, but that in noway affects the way you see them.

Image

You can know the right answer, because I'm giving it to you and I got my answer from the neurologists and psychologists who came up with this test and have checked the math already. The problem is you mind won't let you just think the problem through and see the math. It's the same thing. Your mind is fooling you like your eyes did above.

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Sat Apr 28, 2007 9:06 am
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Now you're just full of shit.

Quote:
Stupid graphics aside, you got it exactly backwards. The illusion is you think that opening the box magically removes it from the probability equation.


It DOES remove the third box from the equation. The box is no longer 'might' be empty, or 'might' be full. the box IS EMPTY. It has CEASED TO BE It is an EMPTY BOX. This is an EX-PARROT!

Sorry, wrong skit.

It is no longer part of the realm of probability, but the realm of proven fact.


Quote:
The fact that you think it becomes a 50/50 choice MEANS THAT YOU HAVE SUCCUMB TO THE ILLUSION!


I think you are an illusion put on this board to taunt us.


Quote:
The human brain is inherently bad at probability, and you're proving that by insisting that it both boxes you have the option of choosing are of equal likelihood of being correct. They don't.


You're beating the same dead horse.


Quote:
The chance of your box being the right is one-in-three, no matter what. The chance of a box that isn't yours being right is two-in-three, no matter what.


No, by opening an empty box, that empty box is proven not to have the money. It no longer exists. It is no longer a box to deal with. It is null and void. If I take that empty box, and BURN that empty box to dust so that is no longer a BOX, I have TWO boxes. One of which will still have money, the other will not. That third box that was opened is no longer part of the equation.

Or perhaps you want to say next that there was actually a One in FIVE chance, seeing as you start with three, and end with two? Or perhaps we're falling upwards instead of downwards.

Image

Now link us. I want to see how its supposed to be worded and how much you screwed up their 'illusion'.

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Sat Apr 28, 2007 10:29 pm
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Agreed. Source of this so called illusion please.

As I stated with my "keeping in unnecessary data" example, I understand what you're trying to say, but you're doing exactly what I pointed out, either unintentionally by not posting it correctly, or by the problem itself being undefined in certain key areas. Saying "1/2" doesn't mean our brains magically cause the 3rd box to stop existing, it just means that we can adapt to the problem changing and shifting. It's no different than suddenly claiming 1/3 turns into 2/3...except for maybe 1/3 > 1/2 makes actual logical and practical sense.

ATC, please limit "making a point" images to ONE PER THREAD. At least I wrote out the song for mine. :P

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I've taken courses in both cognative science and statistics (okay, statistics was in high school, so it's rusty), and this is still sounding wrong to me.

Here's my problem:

Anony-mouse wrote:

Perhaps it would be helpful to rephrase the answer? The box you choose has a two-in-three chance of being wrong. This is because there are three boxes. The other two boxes have a combined one-in-three chance of being wrong. When the researcher opens an empty box, your box still has a two-in-three chance of being wrong. The other two boxes still have a combined one-in-three chance of being wrong. This is because, and this is where the brain fools you, THERE ARE STILL THREE FUCKING BOXES!!! The only thing that has changed is that now you have a single, absolute value for one of the boxes, 0% chance of being right. Your box still has a two-in-three chance of being wrong. The other unopened box, when considered alone, also has a two-in-three chance of being wrong. But see, you're not considering it alone. You're considering it along with the other box. Those two boxes together have only a one-in-three chance of being wrong, because there are two of them; remember, an opened box IS STILL A FUCKING BOX and thus still part of the equation. The only thing that has changed is that you know for sure that the one-in-three chance of being wrong, that is to say a two-in-three chance of being right, isn't evenly distributed. All the chance of being right shifts to the unopened box. Once one box is opened, once the probability of one of the boxes is locked down you know the distribution of the combined probability of one of the two boxes being right. That means the unopened box has a 66.666% chance of being right, and you should always choose the second box.



Okay. So there are three boxes. At the outset, we have a 1/3 chance of being correct. The researcher then shows us that one of the other two boxes is not the correct answer. Since we still have three boxes, and we are counting three boxes, math tells us that the probability that the other box is correct is...1/3.

Why? Simple. We have three boxes. You say we must always count all the boxes. Therefore, even though it is completely unintuative to do so, we count all three boxes in our calculation. Even though we know that one cannot possibly be the correct answer.

So what do we do? We change the equation. Instead of having 3 boxes, we now have two. Which takes us to a probability of 1/2. Now, before you start yelling about the math being wrong and we can't discount a box...we do it all the time. It's called an "outlier", and scientists and engineers discard these all the time. There are for pieces of data that don't fit in the nature of the model.

Maybe that's a bad example. But the reason why (I think) it's a neuroscience problem is not because our brains suck at probability, but because what is actually happening is not entirely obvious while it's happening. Our brains don't actually catch it. To me, it's very similar to a demonstration my cognative science professor showed us:

He brought out a dollar bill and asked the class to bid on it, starting at $0.05. The price started going higher and higher, until it hit the $.99 mark.
And then it went higher. Two people got into a bidding war over a $1.00 bill...and the guy who won ended up paying $1.05.

Now, these are college students, and reasonably intelligent ones at that. Why on earth would they pay $1.05 for something that has a known value of $1.00? Because there's more going on in the human brain than simply processing mathematics. There are deeper levels of motivation, and that's what the problem is about.

Of course, this is just my interpretation based on your description of the experiment. If I had the journal article or paper it's from, I can confirm or refute it better. Hell, if I could read the source, I'd even cheerfully admit I'm wrong.

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Last edited by Ultranos on Sun Apr 29, 2007 2:00 am, edited 1 time in total.

Sun Apr 29, 2007 1:57 am
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Kahneman, D., & Tversky, A. (1973). On the Psychology of Prediction. Psychological Review, 80, 237-251.

Baron, J. (2000). Thinking and Deciding (3d ed.). Cambridge University Press.

Tversky, A., & Kahneman, D. (1982). Evidential Impact of Base Rates. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), Judgment under Uncertainty: Heuristics and Biases. Cambridge: Cambridge University Press.

Bibliographies are such wonderful things, aren't they?


ATC: I'd like to point out that there are two empty boxes, not one. And because that box is open and you know it is empty doesn't mean it no longer has a probability of being right. It just means it has a 0% probability of being right. IT STiLL COUNTS IN THE EQUATiON.


Here is a link detailing the many ways human thought goes wrong. You're suffering from like four or five cognitive biases right now.

Here's another link from Wikipedia that uses a similar problem.

Ultranos: You can only disregard data when it is less than a statistically significant value. This is usually 2-4%. Since most experiments use massive data sets, they can disregard a lot of spurious results. You can't just ignore 1/3 of the problem like you're trying to do.

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Sun Apr 29, 2007 1:57 am
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Anony-mouse wrote:


ATC: I'd like to point out that there are two empty boxes, not one. And because that box is open and you know it is empty doesn't mean it no longer has a probability of being right. It just means it has a 0% probability of being right. IT STiLL COUNTS IN THE EQUATiON.



No, I burned the box to dust when I found it was empty. It has been eliminated from the equation by brute force of superior firepower. We have two boxes.


Quote:
Ultranos: You can only disregard data when it is less than a statistically significant value. This is usually 2-4%. Since most experiments use massive data sets, they can disregard a lot of spurious results. You can't just ignore 1/3 of the problem like you're trying to do.


The pile of dust that used to be the box I burned has a 0% probability in your words. This is under the 2-4%. Since the box has a statistical value of zero... it can be disreguarded.


And just for the record, I think you're trying to test out something else entirely. The 'If I repeat it enough times LOUD ENOUGH, people will think I'm right.' effect. Which a lot of people tend to fall for.

Incidently, your links are wackypedia articles. I want to see the actual report you got your illusion from, not a list of General biases of the human brain.

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SC's right. It took me a while to see it, and I had to flash back to the (not-too-expensive, fortunately) lesson from my stepmother's bartender about three-card monte, but he's right.

Think of it as a casino game: the box you selected has a 1-in-3 chance of holding the money. Inverse of that, the dealer has a 2-in-3 chance of retaining the box with the money. Here's the sleight-of-hand part: because the dealer knows which box has the money and which one doesn't, he always shows you the empty one. That's the decoy. He's still got the 33% advantage going in, and he's counting on you not seeing it.

Working the advantages is the way casinos stay in business. F'r instance, in a crap game, the aggregate odds are 1.4% against the shooter (Chances of craps on the coming out roll, 1-in-9, chances of a natural 1-in-4, all else roughly 2-in-3, BUT, after that, the most likely outcome is 7-and-out at a little more than 1-in 5; been a while since I've read the matrices in Hoyle's Games, but that's fairly close). Therefore, the smart bet over the long term is against the shooter. But you can walk into any casino in the world, and you will almost never see any money on the "Don't Pass" bar. Because people won't "queer the luck" for the shooter...[/i]

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Sun Apr 29, 2007 8:31 am
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Michael J Doyle wrote:
SC's right. It took me a while to see it, and I had to flash back to the (not-too-expensive, fortunately) lesson from my stepmother's bartender about three-card monte, but he's right.

Think of it as a casino game: the box you selected has a 1-in-3 chance of holding the money. Inverse of that, the dealer has a 2-in-3 chance of retaining the box with the money. Here's the sleight-of-hand part: because the dealer knows which box has the money and which one doesn't, he always shows you the empty one. That's the decoy. He's still got the 33% advantage going in, and he's counting on you not seeing it.

Working the advantages is the way casinos stay in business. F'r instance, in a crap game, the aggregate odds are 1.4% against the shooter (Chances of craps on the coming out roll, 1-in-9, chances of a natural 1-in-4, all else roughly 2-in-3, BUT, after that, the most likely outcome is 7-and-out at a little more than 1-in 5; been a while since I've read the matrices in Hoyle's Games, but that's fairly close). Therefore, the smart bet over the long term is against the shooter. But you can walk into any casino in the world, and you will almost never see any money on the "Don't Pass" bar. Because people won't "queer the luck" for the shooter...[/i]



All that still doesn't explain where the Mouse is getting the idea that the remaining box suddenly has a two-third chance of being right rather than a one half chance of being right.

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Sun Apr 29, 2007 8:44 am
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The Dealer knows which box has the money. The 50-50 chance would only apply if he was guessing at which one had the money, same as you. But, he knows - he's showing you the empty as a distractor.

The odds shook out when you first picked. Everything after that is sleight-of-hand and window dressing.

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Sun Apr 29, 2007 8:59 am
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MJD: Oh thank god! I nearly burst into tears when someone finally got it! It felt like I was ramming my skull into a wall!

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Sun Apr 29, 2007 2:19 pm
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Anony-mouse wrote:
MJD: Oh thank god! I nearly burst into tears when someone finally got it! It felt like I was ramming my skull into a wall!
*bows*

Tell ya what - let's you and me open a casino - we'll clean some pigeons out. :twisted:

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Sun Apr 29, 2007 9:06 pm
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*scratches head* ah whut?

See? I horribly sucked at this sort of thing in high school... and my mom teaching me the tricks of gambling never worked either (way to go ma at teaching me to be a card counter)

I read this when the post first happened, and it gave me a headache... so I have left it alone... but all I have to say is...

Mouse, you made my head hurt, NO CUPCAKE FOR YOU! *passes them out to everyone else* :P

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Mon Apr 30, 2007 12:53 am
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See... the way I see this. The only choice that matters is the 1/3 choice at the start. That's why I said "No, I won't switch boxes." After the 1/3 choice is out of the way, the 1/2 choice is redundant, because I've already made my choice, and if I pick AGAIN, the new value is just fifty fifty and it stays that way whether I change up, or stay the same. What's the illusion supposed to be in plain terms?

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Mon Apr 30, 2007 3:06 pm
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Ok, so I did what mouse said (about testing the example).

I tested 5 people 100 times.

68% of the time people will pick either the center or the right side box (or left if their left handed)

50% of the time they picked the other box after removing one false box
25% of those times they picked the correct box

50% of the time the correct box was picked the first time

So the experiment shows that you have a 50% chance no matter which box you chose at first. A third box is irrelevant and is the true illusion, making you think you have a 1 in 3 chance. However, if you decide to switch boxes, your chances of picking the correct box is reduced to 25%.

Thus, never switch boxes.

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Thu May 03, 2007 3:26 pm
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