Let's make a deal (logic puzzle) by SergeantX

psychologically, you may feel you have 50/50 shot, but its still the same odds as the first time, because its really only the first guess that matters.

after that, why change your answer when you have no additional important information

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

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Saturday, July 22, 2006 8:18 PM

GUYWHOWANTSAFIREFLYOFHISOWN

Are you sure you don't want to change your answer?

SergeantX

"Dream a little dream or you can live a little dream. I'd rather live it, cause dreamers always chase but never get it." Aesop Rock

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Saturday, July 22, 2006 8:27 PM

yup, I'll take the 11 cans of dogfood and the puppy please

http://www.albinoblacksheep.com/flash/llama.php
-try it out, I dare you

98% of teens have smoked pot, if you are one of the 2% that haven't, copy this into your signature

I'm so into Firefly, my butt glows in the dark.

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Sunday, July 23, 2006 4:54 AM

Nobody got it exactly right, but PhoenixRose was closest.

You actually double your chances if you switch.

SergeantX

"Dream a little dream or you can live a little dream. I'd rather live it, cause dreamers always chase but never get it." Aesop Rock

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Sunday, July 23, 2006 5:44 AM

i don't see how. you choose one out od 3 doors, either picking a dud or a prize. you already know one out of the remaining two doors is a dud, which the host shows you. what information have you gained on your door? It seems like your odds of picking the right door are the same as before, wo why change your answer. If you can explain why being shown the one dud door givers you more information to change your answer, do tell.

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

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Sunday, July 23, 2006 6:22 AM

The difference is that by changing your choice, you're effectively getting to open two doors instead of one. You get the benefit of having the one with the clunker already ruled out. So the odds of your original choice being correct are 1/3 and the odds of the remaining door having the prize are 2/3.

It took me several days to wrap my head around this one when I first read it. In fact, the puzzle column in the newspaper where I first saw it, was receiving mail from math professors telling them they were wrong, but the column writer stuck to her guns and proved her point in the next weeks column. Fascinating stuff.

SergeantX

"Dream a little dream or you can live a little dream. I'd rather live it, cause dreamers always chase but never get it." Aesop Rock

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Sunday, July 23, 2006 6:24 AM

STINKINGROSE

Because you now know which one the dud's behind, therefore you know there's something nifty behind one of the two doors left two doors not 3. Your choice of picking the "nifty" is 1 in 2 instead of 1 in 3 at this point so long as nobody turns it into the shell game on you. It's now like guessing if the coin came down heads or tails before you peek, but you get to call the other side if you want to at the last minute.
Statistics and Probability, and I thought that college class would never do me any good. Just don't ask me to calculate the regression of the mean, OK?

So yes, some of us are familiar with the gameshow, why do you ask?

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Sunday, July 23, 2006 6:31 AM

but you get to have 2 doors opened regardless of whether you change your choice. Essentially, if they do this every time, its awlays a 50/50 chance, because its only the last decision that matters.

i'd like a link to where the author proved himself correct, because i'm inclined to agree with the dissenting mathmeticians.

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

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Sunday, July 23, 2006 6:37 AM

Stinkingrose: nope, as SergeantX pointed out, the odds are actually 2 in 3 that you'll get the prize if you switch.

The key is the fact that when you pick the first door, what are the odds that you'll pick the door with the prize? 1 in 3, very good. As has been pointed out, the opening of one "bad" door doesn't alter the knowledge you have but if you don't think of it as "Do you want to swap the door?" but instead "Do you want the combined odds of opening the other two doors?"

Okay, think of it this way. Imagine it wasn't three doors but 300,000. Pick one. Door number 231,496; a very good choice. Now I'm going to open up every door other than 116,766 and your choice, 231,496. So the prize is either behind your choice of door or 116,766. Which one is it behind?

That's right, unless you got the right door first time (at odds of 1 in 300,000), it's door 116,766.

No? Well, there's an even fuller explanation in The Curious Incident of the Dog in the Night Time by Mark Haddon, which is a cracking read too.

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Sunday, July 23, 2006 6:38 AM

Look at it this way. The odds that you chose correctly on the first guess are 1/3. That won't change. If you switch your guess to whichever of the doors are remaining, you get the complement of those odds, 2/3. If you switch you are, in a sense, being given two doors, the door already opened and the remaining closed door. It's like having two lottery tickets instead of one, so you're chances double.

EDIT: (like SimonWho said.) :)

SergeantX

"Dream a little dream or you can live a little dream. I'd rather live it, cause dreamers always chase but never get it." Aesop Rock

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Sunday, July 23, 2006 6:41 AM

You can study the maths here (along with a simplified explanation):

http://en.wikipedia.org/wiki/Monty_Hall_problem

Or why not just try it yourself with three cards?

(This problem of course assumes that the gameshow is fair and they a) always show a dud door and b) they don't shuffle what is behind the doors according to your choice)

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Sunday, July 23, 2006 6:47 AM

i'm sorry, but i still don't buy it.

You pick a door, your odds are 1/3
a door is opened revealing a dud. a dud which you knew had to be in one of the two remaining doors.
you now have 2 doors to choosed from. you have a 1/2 possibility of picking the right one. you stick with your door, which means you've got a 1/2 possibility. you picked the other door, which is 1/2 possiblity. The first choice is null and void therefore you always had a 1/2 possibility.

This 2/3 doesn't make any sense because it is always going to be the dud door, never the right one. simply is always 1/2 and they just make it another choice because they want that segment of the show to go on longer.

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

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Sunday, July 23, 2006 6:50 AM

STINKINGROSE

But they're treated as discrete events, two separate choices. First pick is 1/3, 2nd pick (same or change) is now 1/2 because you've eliminated the third door as an option. Unless you really want the old shoe behind the ick door.
I *did* pass my college math classes, honest. I just never went into higher level courses than I had to.
If you were to include all the possible choices not as discrete events and add a cumulative variable and come up with an equation to make it all work out you could. You can do anyting with mathematics, but my brain will explode if I try it without a net.
Mathematicians are always having fun arguing with each other and stretching reality by describing it with numbers and such. I know a couple. They are scary people, but fun to listen to. I do not speak the language.

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Sunday, July 23, 2006 6:54 AM

Quote:
You can study the maths here (along with a simplified explanation):

http://en.wikipedia.org/wiki/Monty_Hall_problem

Or why not just try it yourself with three cards?

(This problem of course assumes that the gameshow is fair and they a) always show a dud door and b) they don't shuffle what is behind the doors according to your choice)

ah, i see, that makes sense, i wouldn't have thought of it that way

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

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Sunday, July 23, 2006 6:55 AM

You don't have to do anything with mathematics.

Try it!

http://math.ucsd.edu/~crypto/cgi-bin/monty2?2+11627

(This one has ongoing results)

Your odds will be 2/3 if you switch.

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Sunday, July 23, 2006 6:57 AM

Check out the wiki page SimonWho linked to.

One way to look at it is to assume there are a 100 doors. You pick one, after which the host opens all the remaining 99 doors except for one. You're left with a choice. Assume that your original choice was right, or switch to the other unopened door. Does that make it any clearer?

SergeantX

"Dream a little dream or you can live a little dream. I'd rather live it, cause dreamers always chase but never get it." Aesop Rock

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Sunday, July 23, 2006 7:08 AM

LISSA37

Quote:
Originally posted by JubelLate:
i'm sorry, but i still don't buy it.

You pick a door, your odds are 1/3
a door is opened revealing a dud. a dud which you knew had to be in one of the two remaining doors.
you now have 2 doors to choosed from. you have a 1/2 possibility of picking the right one. you stick with your door, which means you've got a 1/2 possibility. you picked the other door, which is 1/2 possiblity. The first choice is null and void therefore you always had a 1/2 possibility.

This 2/3 doesn't make any sense because it is always going to be the dud door, never the right one. simply is always 1/2 and they just make it another choice because they want that segment of the show to go on longer.

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

I agreed completely with this until I read the wikipedia page (and I see you changed your mind, too, if I'm reading your other post right).

I think this is what's throwing people off... the 2/3 odds are 2/3 CHOICES (to switch or not to switch), not 2/3 doors... does that even make sense? Maybe not, but that's how it clicks in my head.

*****
"I'm a leaf on the wind..." - Wash

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Sunday, July 23, 2006 7:14 AM

But that's the beauty of the puzzle - common sense takes a bath on this one.

Even if you can't make sense of it, the 2/3rds odds is right, that's for certain.

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Sunday, July 23, 2006 7:16 AM

Quote:
Originally posted by Lissa37:
Quote:
Originally posted by JubelLate:
i'm sorry, but i still don't buy it.

You pick a door, your odds are 1/3
a door is opened revealing a dud. a dud which you knew had to be in one of the two remaining doors.
you now have 2 doors to choosed from. you have a 1/2 possibility of picking the right one. you stick with your door, which means you've got a 1/2 possibility. you picked the other door, which is 1/2 possiblity. The first choice is null and void therefore you always had a 1/2 possibility.

This 2/3 doesn't make any sense because it is always going to be the dud door, never the right one. simply is always 1/2 and they just make it another choice because they want that segment of the show to go on longer.

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

I agree with this. Your argument makes perfect sense to me. I don't understand the 2/3 concept, either.

*****
"I'm a leaf on the wind..." - Wash

I understand it now. The wikipedia link explains it.

Basically, your odds of getting it wrong on the first guess are 2/3. One choice is removed, leaving only a right and a wrong answer. Because it is 2/3 likely you are on the wrong answer, picking the other one switches the odds.

O.1 pick a goat
O.2 pick a goat
O.3 pick a car

you're likely to pick a goat, then the other goat is shown

now:
O.1 your option
O.2 goat or car

well your option is 2/3 likely to NOT BE car, so by picking the second one, its 2/3 likely to BE car.

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

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Sunday, July 23, 2006 7:18 AM

LISSA37

Yeah, I see it now. That's why I edited my post. It always seems that way... after I ask a question, then the answer sinks in on its own. But, of course, it's too late to take it back. lol

*****
"I'm a leaf on the wind..." - Wash

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Sunday, July 23, 2006 7:20 AM

STINKINGROSE

I followed along with the link's explanation. I tried the game, I got the goat.
I like goats. I used to have goats. They use less gas than sportscars anyway, plus there's cheese and curry. Try eating your Ferrari.
Intuitively it feels like you should have a 1/2 once you've eliminated the definitely not something you want door from the pool of choices. If you picked the one that had the thing nobody wants, you'd definitely want to change your choice..but you'd still only be making a choice from two doors once the third one had been revealed.
This is where the arguing starts in this exercise I think. You can either take the stand that the third door is still included in your choices because you can decide to pick the one that was just opened, or you can say it is no longer an option.
But who in a practical application is going to pick the bag of cow poop? (Okay maybe a gardener..) (;
All possiblilities exist until you choose one. The cat is in a state of quantum uncertainty until you open the box.
I'll stick to chickens and leave the maths to those of you with the brains that can turn corkscrews.
Mine works better with the bottle opener approach. (On or off, not how many turns round until we can get the cork out.. though I will cheerfully drink the wine with you, and I'll share my beer.)
Have fun continuing with the discussion, I'm dropping out at this point unless directly addressed because I can not possibly contribute anything useful to it.

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Sunday, July 23, 2006 7:26 AM

Now this one will make you cry:

I offer you two envelopes, both of which have money in them, one of which has twice as much as the other.

You pick an envelope and find $100 inside. Nice, eh? But... I then ask if you'd like to switch to the other envelope (don't worry, this is a different mind bender).

You know the other envelope contains either $200 or $50. So you could win $100 or lose $50. Now assuming that you're not incredibly desperate for cash, it seems to make sense that you swap (i.e. you could win twice as much as you would lose with 50/50 odds).

But... if you had opened the other envelope and it had contained $200, you would work out the other envelope would contain either $400 or $100. Therefore when I offer the chance to swap, you could gain $200 or lose $100. Ergo, you should swap. Similarly if the other envelope contained $50, you could win $50 or lose $25.

Therefore it seems that it always makes sense to swap which can't be right. Or can it?

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Sunday, July 23, 2006 7:30 AM

how many envelopes are we dealing with, do any two envelopes contain the same amount of money? what happens to the envelopes once they're given back?

The urge to save humanity is almost always a false front for the urge to rule. – H.L. Mencken

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Sunday, July 23, 2006 7:31 AM

Okay, my explanation is obviously rubbish: here's a better one.

You are taking part in a game show. The host introduces you to two envelopes. He explains carefully that you will get to choose one of the envelopes, and keep the money that it contains. He makes sure you understand that each envelope contains a cheque for a different sum of money, and that in fact, one contains twice as much as the other. The only problem is that you don't know which is which.

The host offers both envelopes to you, and you may choose which one you want. There is no way of knowing which has the larger sum in, and so you pick an envelope at random (equiprobably). The host asks you to open the envelope. Nervously you reveal the contents to contain a cheque for 40,000 pounds.

The host then says you have a chance to change your mind. You may choose the other envelope if you would rather. You are an astute person, and so do a quick sum. There are two envelopes, and either could contain the larger amount. As you chose the envelope entirely at random, there is a probability of 0.5 that the larger check is the one you opened. Hence there is a probability 0.5 that the other is larger. Aha, you say. You need to calculate the expected gain due to swapping. Well the other envelope contains either 20,000 pounds or 80,000 pounds equiprobably. Hence the expected gain is 0.5x20000+0.5x80000-40000, ie the expected amount in the other envelope minus what you already have. The expected gain is therefore 10,000 pounds. So you swap.

Does that seem reasonable? Well maybe it does. If so consider this. It doesn't matter what the money is, the outcome is the same if you follow the same line of reasoning. Suppose you opened the envelope and found N pounds in the envelope, then you would calculate your expected gain from swapping to be 0.5(N/2)+0.5(2N)-N = N/4, and as this is greater than zero, you would swap.

But if it doesn't matter what N actually is, then you don't actually need to open the envelope at all. Whatever is in the envelope you would choose to swap. But if you don't open the envelope then it is no different from choosing the other envelope in the first place. Having swapped envelopes you can do the same calculation again and again, swapping envelopes back and forward ad-infinitum. And that is absurd.

That is the paradox. A simple mathematical puzzle. The question is: What is wrong? Where does the fallacy lie, and what is the problem?

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Sunday, July 23, 2006 8:18 AM

SHINY

This straight-forward analysis only works if the host *always* does this. However, if the host only *sometimes* shows the player an empty door and offers the chance to change, you then have to consider whether or not the host is making the offer because he knows your first choice was correct.

---

I don't need a gorram back-spaceship driver!!!

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Sunday, July 23, 2006 10:46 AM

LISSA37

Quote:
Originally posted by SimonWho:
The host then says you have a chance to change your mind. You may choose the other envelope if you would rather. You are an astute person, and so do a quick sum. There are two envelopes, and either could contain the larger amount. As you chose the envelope entirely at random, there is a probability of 0.5 that the larger check is the one you opened. Hence there is a probability 0.5 that the other is larger. Aha, you say. You need to calculate the expected gain due to swapping. Well the other envelope contains either 20,000 pounds or 80,000 pounds equiprobably. Hence the expected gain is 0.5x20000+0.5x80000-40000, ie the expected amount in the other envelope minus what you already have. The expected gain is therefore 10,000 pounds. So you swap.

Does that seem reasonable? Well maybe it does. If so consider this. It doesn't matter what the money is, the outcome is the same if you follow the same line of reasoning. Suppose you opened the envelope and found N pounds in the envelope, then you would calculate your expected gain from swapping to be 0.5(N/2)+0.5(2N)-N = N/4, and as this is greater than zero, you would swap.

I have one word for you: WOW.

Okay, with the help of Wikipedia, I managed to follow the other puzzle. But, this one has my mind blown.

I've quoted the part that I don't understand.

I see the calculation you did and what its pieces are, but I don't understand why you did it. What does that tell you? How does it create an expected gain of 10,000 pounds? You can't gain 10,000 pounds if the envelope you're holding has 40,000 in it. You will either gain 40,000 or lose 20,000... right? Or am I just completely lost here?

*****
"I'm a leaf on the wind..." - Wash

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Sunday, July 23, 2006 11:50 AM

KANEMAN

When you first pick a door your odds are 1 in 3. After they show you that one of the remaining doors was a dud. Even if you don't change your chances are 1 in 2. If you change it's still 1 in 2. Why would changing your door increase your odds? Two choices original door or new door. 50 50.

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Sunday, July 23, 2006 11:51 AM

KANEMAN

New puzzle...When you hit enter once and you get two posts for one push.................

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Sunday, July 23, 2006 11:58 AM

KANEMAN

Aha I get it..chances just doubled

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Sunday, July 23, 2006 12:07 PM