Lets see who gets this right.

[w733commando]

You are on a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and there are goats behind the other two doors. He asks you to pick a door. You pick a door but the door is not opened. Then the game show host opens one of the doors you didn't pick to show a goat (because he knows what is behind the doors). Then he says you have one final chance to change your mind before the doors are opened and you get a car or a goat. So he asks you if you want to change your mind and pick the other unopened door instead.

 

What should you do and why?

[joeking27]

You should always switch because that gives a 2/3 chance of winning.

 

Just out of interest, do you know who Mark Haddon is?

[w733commando]

Yes, but I had already heard about this way before reading that book.

[ruchik]

Switch. If you pick a door that has the prize behind it, then it's all good to begin with. But if you pick a door and he doesn't open it, then the prize obviously isn't behind it. Out of the other two doors, he has to open the door with the goat behind it, so by switching, your odds of winning are almost guaranteed.

[w733commando]

Switch. If you pick a door that has the prize behind it, then it's all good to begin with. But if you pick a door and he doesn't open it, then the prize obviously isn't behind it. Out of the other two doors, he has to open the door with the goat behind it, so by switching, your odds of winning are almost guaranteed.

<{POST_SNAPBACK}>

 

He opens a door that you haven't picked.

[ruchik]

Are we assuming that if you open the door with the prize behind it right off the bat, you win, or regardless of which door you pick, prize behind it or not, he opens a door you haven't picked?

[w733commando]

You are on a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and there are goats behind the other two doors. He asks you to pick a door. You pick a door but the door is not opened. Then the game show host opens one of the doors you didn't pick to show a goat (because he knows what is behind the doors). Then he says you have one final chance to change your mind before the doors are opened and you get a car or a goat. So he asks you if you want to change your mind and pick the other unopened door instead.

 

What should you do and why?

<{POST_SNAPBACK}>

[ruchik]

Well, in that case, I still say switch. Let's say you pick door A with the car behind it, and the host opens door B with a goat behind it, then switching to door C will mean that you lose. But if you pick Door B, and the host opens door A and you switch, you win. If the host opens door C, and the host opens door B and you switch, you win again. Your chances of winning double by switching, from 1/3 to 2/3. Then again, it'll only work if the host knows what's behind each door to begin with.

[ruchik]

Double post

[Sale]

If there are three doors and one car. One door opened and it didn't have the car in there, it doesn't matter which of the remaining doors you pick. (I.E. Do you switch or not.) At this point it's 50/50 whether you get the car or not.

 

Unless the host is using a certain pattern when it comes to opening the doors. But then there's no definitive answer. He could be tricking you into switching, or into thinking that you should stick to your choice.

 

-Sale

[fanta101]

If there are three doors and one car. One door opened and it didn't have the car in there, it doesn't matter which of the remaining doors you pick. (I.E. Do you switch or not.) At this point it's 50/50 whether you get the car or not.

 

Unless the host is using a certain pattern when it comes to opening the doors. But then there's no definitive answer. He could be tricking you into switching, or into thinking that you should stick to your choice.

 

-Sale

<{POST_SNAPBACK}>

 

 

/agree

[otherrandomhero]

Yep, sale's got it right. I don't know what kind of crazy logic you other guys are going by, but it's 50/50. There's no 2/3 chance anything. It's like "is that your final answer?" on Who wants to be a millionare. You could be right just as easily as you could be wrong.

[Sale]

I somehow feel that I got it wrong. I'm really bad with trick questions, and this one feels so easy it could be one. If this isn't a trick question, then it's definitely 50/50.

 

-Sale

[transist]

If you pick switch, you'd have to assume that if you pick the right one, they wouldn't ask you again anyways.

 

I'd say don't switch because having you lose would be advantageous to them, so if you guessed right they'd naturally ask if you're sure in an attempt to throw you off.

 

Without adding the psychology to the equation, however, it doesn't really matter because it just changes your odds from 1/3 to 1/2.

[doopydoo]

It's a trick. In the riddle, you're an Amish person, and you really want the goat, so your odds went from 2/3 to 1/2.

 

Of course the answer is still the same, now... doesn't matter which you pick.

[ruchik]

Assuming the host knows exactly what is behind each door, and will only open doors that have goats behind them, then your chances are 2/3, not 50/50. Again, let's say you pick door A, which has the car behind it, host opens door B, you switch to C you lose. That's 1/3. But if you pick door B or C to begin with, then the host has to open the door with the other goat behind it, so if you switch, then you get the car 2/3 times. As I said, this would only work if the host knows what is behind each door, and will only open doors that have goats behind them until you pick your final answer. If the host has no idea what is behind each door and opens them at random, then yes your chances are 50/50.

[otherrandomhero]

That still makes no sense. Since one door has already been opened and has the goat, only 2 remain: one with a car, and one with the other goat. If the host automatically opens a door with a goat before offering you the possibility of switching, it will always end up like this. There are no thirds involved. After the initial opening of one goat door, it's 50/50. You either switch or you don't. you have no idea whether or not switching will get you the car or not. It's 50/50.

[ruchik]

But there's three different scenarios here-one where you pick the right door, switch, then lose: and two where you pick the wrong door, switch, and win. The possibility of you picking the winning scenario is still 2/3, not 50/50. A 50/50 odds would be true if the doors were randomized and each scenario was independent of each other. I.e., if you flip a coin ten times, you are just as likely to flip 10 heads in a row as you are at getting 5 heads and 5 tails, because each result is independent of each other. In this case it is not. Odds are, 1/3 times you will pick the losing scenario, while 2/3 times you will end up with the winning scenario.

[Chimpy]

The probability is 2/3.

 

It's a famous puzzle.

 

Basically the probability of the first door the contestant picks being correct doesn't alter by the host opening a door and showing a goat. That means with only one remaining open door the probability that the car is behind that must be 2/3. So you are always better off switching.

 

As ruchik says this is due to the events happening one after the other. A similar analogy would be the probability of rolling a one on a dice and then removing one of the sides for each subsequent throw.

[w733commando]

Clever boy chimpy.

2/3 chance of getting a car if you switch, everyone who said 50/50. You are wrong

[Chimpy]

It's not really clever because I was taught it. It's a famous puzzle that I imagine most people do get taught in probability lessons in maths. It's useful because it demonstrates that the probability of an event is not always intuitive. Whether they remember is is another thing altogether.

 

[Sale]

I still don't get it. Someone please tell me where I go wrong.

 

Initially there are two goats and one car. You have 1/3 chance of getting a car, and 2/3 for a goat.

 

When one door is opened with the goat behind it, it is ruled out. One goat is out of the game, and you can't pick that door anymore. Just one question at this point: The game host would open one door with a goat no matter what you choose? Or if you picked one with the goat behind it, would it be opened right away and you'd lose at this point already?

 

Because you are being offered the chance of switching, it's like taking a fresh start with two doors. You can stick with what you chose, or switch. One door has the car behind it, the other has the goat. 50/50, isn't it?

 

I knew it seemed too easy, and if it's a famous puzzle I'm willing to accept I'm wrong. But if you claim you understand this puzzle, you should be able to explain it. Picking the correct answer because you know it is not good enough.

 

-Sale

[Curious]

Ill take my own stab at it:

 

Its because at the beginning, 2 goats and one car = 2/3 chance of a goat, the chances are higher that you will pick a goat first time, this is the reason it is not "a fresh start" when the next round begins, it is not a 50/50 chance either way, the odds say you picked a goat. Because it is more likely that you already have a goat, it is better to switch.

 

 

 

Ruchiks explanation was also pretty clear - its a "brute force" approach. as he goes through all 3 pssible scenarios, resulting in 2/3 times = you win by switching - we know that the host does know whats behind each door, its in the question.

 

A = Car

B= Goat

C= Goat

 

The below works with whatever behind the doors.

 

Well, in that case, I still say switch.

 

Let's say you pick door A with the car behind it, and the host opens door B with a goat behind it, then switching to door C will mean that you lose.

 

But if you pick Door B, and the host opens door A and you switch, you win.

 

If the host opens door C, and the host opens door B and you switch, you win again. Y

 

our chances of winning double by switching, from 1/3 to 2/3.

 

Then again, it'll only work if the host knows what's behind each door to begin with.

<{POST_SNAPBACK}>

[Chimpy]

I already explained in my previous post.

 

They aren't mutually exclusive events.

 

The host will always open a door with a goat behind it. <--- this is the super important bit. If the host didn't do this and could reveal the car then that changes the probability to make it equally likely.

 

The chances of you being right by your first selection is 1/3. Opening a door to show a wrong selection doesn't alter that chance. Thus the chances of the door you haven't selected being right are the remainder which is 2/3.

 

If we do a truth table (your selection, hosts selection, last door):

 

Goat 1 | Goat 2 | Prize

Goat 2 | Goat 1 | Prize

Prize | Goat 1 | Goat 2

Prize | Goat 2 | Goat 1

 

Now that looks like a 50:50 chance right there? Well you are right with this data set it is. But have we got all the combinations? No. There are another two combinations where the host would have to show the car but is actually now forced to show a goat:

 

Goat 1 | Goat 2 | Prize

Goat 2 | Goat 1 | Prize

Prize | Goat 1 | Goat 2

Prize | Goat 2 | Goat 1

Goat 1 | Goat 2 | Prize

Goat 2 | Goat 1 | Prize

 

So by changing after the host revealing the goat it doubles the chances of you selecting the car. This is due to the host having two doors to choose from but only ever being allowed to select one if the other contains the prize!

[Sale]

Hmm... Think I got it now. The possibility of the first door being correct is 1/3, that part I understood. So the possibility that it's wrong is 2/3.

 

Especially the part of the host being able to choose of the two doors, but only allowed one door if the other one contained the prize, was helpful. So basically if you select the door with a goat behind it (initially) and you don't lose at this point already, then the host would open the other goat door (instead of the car door) and the last one would have to be the car.

 

Gotcha. I wasn't sure if you would lose immediately if your first choice was the goat.

 

-Sale