I like to throw the same teaser puzzle at people I know:

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You are on a game show, playing a game that consists of three doors and one prize. You have to guess which door contains the prize. After you pick a door, the game show host opens up a door (of the two that you did not choose) that he knows does not contain the prize. He then asks you, "Do you want to keep your original choice or switch to the other closed door?"

Most people (math "nerds" excluded) stubbornly believe that the odds are 50-50 for each of the two doors that are closed. But really, you have 2/3 chance if you switch and only 1/3 if you keep the door. The game show host is essentially saying, "Would you rather pick one door or two doors?" But most people stick with their first choice, instead of picking the other two - because they only see the puzzle as 50-50.

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Yes, it's old, and has even been mentioned here before. I bring it up because a friend of mine (whom I stumped with the puzzle) had the "Price is Right" on today, when a certain game came on...

There were four "doors," and three of them had prizes. The contestant would get all three prizes if she correctly picked all three spots at random. If she missed any, she got none of them. So after picking three doors at random, Bob Barker opened up two correct doors that the contest had picked, leaving two more closed.

Now, as I'm sure you've guessed, the contestant had the choice: would you like to switch your initial pick to the other closed door or keep it the same? Bob even offered the contestant $500 to stay with her original choice. Thinking the odds were 50-50 while keeping $500 no matter what, she stayed.

And she lost. Now I wonder if Bob had asked her: "Do you want $500 plus a 25% chance of $6000 or $0 with a 75% chance of $6000?", what she would have done?

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Someone here, I think perhaps Evert, has said several times that people are bad at understanding or relating with natural probabilities. This is just a classic example...

If any of you have some puzzles that stump people (and cause them to stubbornly argue their case), I'd like to hear them.

We had that brought up in class a while ago, and one of our TA's explained it.

I've completely forgotten the logic we were told then.

[edit]
Okay I remember now....
I'll be honest though, it's something you need to think about

err... i don't understand why it's not 50/50

Which is also like betting $500 dollars that you are going to get $6000 with a 75% chance. What I mean, is if you were playing cards and you knew you had a 75% chance of winning a pot of $6000, would you bet $500? Believe it or not, that's still kind of iffy to me. If you win, you increase your profit by 600%, but that 25% that you might lose is still a big gap in my opinion. Now, if the pot was like $10,000 or higher, I'd probably go for it. And if I were rich, I'd go for the $6000 or even $501.

I believe most people don't switch because, if you stay and the prize is in the other door, you have bad luck, but if you switch and the prize was in the door you originally chose, you are an idiot. Psychology, I would call it.

I'd take those odds all day long. I play twenty times. I lose $500 five times. I gain $6000 fifteen times. $87,500 in my pocket.

Of course you only play once, not getting the benefit of winning over time. But also, you aren't losing $500 because you never had it to begin with. By that logic, if you keep the $500 (and miss) and don't get the $6000, you went home a $5500 loser.

Heh, that's why I love this puzzle. Put it this way:

Scenario 1
You pick door 1.
Host says, "Would you rather open doors 2 AND 3 instead?"

What would you do? You'd pick doors 2 and 3 if you weren't stupid. Obvious, eh? Change it slightly:

Scenario 2
You pick door 1.
Host says, would you rather open doors 2 AND 3 instead? For suspense, I will open an incorrect door first.

What would you do? You'd still pick doors 2 and 3 if you weren't stupid. Change it slightly:

Scenario 3
You pick door 1.
Host opens door 2, revealing that it is empty. He asks, would you like to open door 3 instead?

Now people scratch their heads and think it's 50-50, but it's the same thing as the first two options.

If that doesn't help, think of it this way: you have 2/3 chance of getting wrong initially. If you swap, you'll get it right. Thus 2/3 chance of winning if you swap. On the other hand, you have a 1/3 chance of getting it right initially. If you don't swap, you'll be right 1/3 of the time.

Yes, but most people think it's 50-50. If they knew they had better odds, they would switch. However, in that context I agree that they psyche themselves out by having this "gut feeling" that they were correct the first time, because lo and behold, there are two doors left standing - including the one they initially picked!

Anyway, submit your puzzle ideas! .999...=1 is good, but hard to get the common person to have interest. I like to confuse people.

I think most people don't know the host has set parameters for what door he opens, they think it's all random.

In the puzzle or on The Price is Right?

The way I see it:

Pr(A) = Pr(winning by picking correct and sticking to it) = 1/3
Pr(B) = Pr(winning by picking wrong and swapping it to correct) = (1/2*1/3)+(1/2*1/3) = 1/3
Pr(C) = Pr(winning at all) = Pr(A)+Pr(B) = 2/3

Pr(pick wrong #1) = 1/3
Pr(pick wrong #2) = 1/3
Pr(swap to correct after picking wrong #n) = 1/2

Pr(win by swapping from wrong #1) = (1/3)*(1/2) = 1/6
Pr(win by swapping from wrong #2) = (1/3)*(1/2) = 1/6

Pr(win by swapping from wrong #n) = (1/6)+(1/6) = 2/6 = 1/3

In short, swapping all the time would not lead to a 2/3 probability of winning, but the probability of winning is 2/3 if all contingencies are taken into account.

Or did I misunderstand the initial problem/what you were trying to say? ..wording is so much fun.

First game, stayed with my original option, won. I retire.

My wife and I went to a party last weekend. There were eight other couples there. At the beginning everyone circulated, shaking hands with each other. No-one shook hands with their partner, of course.

Later on, I asked everyone how many people they had shaken hands with, and, amazingly, everyone had shaken hands with a different number of people.

How many people did my wife shake hands with?

(This was in the Guardian this weekend)

Pete

Actually there would be 18 people

(8x2(other people) + 2(you +wife))

Friends don't shake hands between each other, pat him in the back, kiss her in the cheek, etc. Since your wife is pretty friendly, I guess she shook nobody's hands