Brain Teasers

[pbaldy]

Chipper, I think Brian's statement was based on the original premise that door 3 has been revealed and does not have the prize. Thus it has to be either behind 1 or 2, thus a 50% chance either way. I'm with Brian on this one. At the point the "switch" decision is made, it's a coin flip.

 

[ChipperT]

You have 2 doors left, 1 has the prize, that's the scenario, if none have it you have already lost.

Brian

But nothing in the scenario given guarantees that one has the prize. The only stipulation in the scenario was that one of THREE doors had the prize, you choose a door, the host gives you a chance to switch to one of the other two. That does not remove the possibility that the door you orginally chose does not have the prize. That is why I have so much difficulty with this question in all the numerous times I have encountered it.

 

[ChipperT]

Chipper, I think Brian's statement was based on the original premise that door 3 has been revealed and does not have the prize. Thus it has to be either behind 1 or 2, thus a 50% chance either way. I'm with Brian on this one. At the point the "switch" decision is made, it's a coin flip.

In that case if the door is revealed BEFORE you make your choice you might as well stay with your original choice since you will still have a 50% chance of being right. If it is revealed AFTER then when you are given the opportunity to switch, you still have a 33% chance of being right WHEN you make the choice to switch or not. Right?

 

[Vassago]

But door3 has gone for the final choice. you cannot include it in the 2nd choice scenario.

Brian

It's an overall scenario... ignoring the original riddle. Honestly, he shouldn't have mentioned a specific door that was opened, only that the host opened one that was not a winner.

Okay, let's look at it like this...

Let's say instead of opening one of the doors, the host tells you you can either stick with the door you chose or take the OTHER TWO doors instead of the one you chose. That's essentially what he is doing. Does that make more sense?

 

[pbaldy]

Sure, but the original premise has the switch decision being made after door 3 is revealed to be a loser.

 

[chergh]

I would like to see the proof, tho' I doubt I'd understand it.

I think that you have now got a new choice, door1 or door2, it doesn't matter what you chose before its a new choice. So 50 -50 stick or move.

Brian

The important point in this puzzle is that the host knows which door the car is behind, so the door with the car will never be eliminated. So the outcomes can be seen in the following table:

Winning Door              	Door Chosen	If you switch do you win?
1	                               1	                      No
1	                               2	                      Yes
1	                               3	                      Yes
2	                               1	                      Yes
2	                               2	                      No
2	                               3	                      Yes
3 	                               1	                      Yes
3                                      2	                      Yes
3	                               3	                      No

So with 6 outcomes in your favour if you switch and 3 outcomes that are bad you are more likley to win if you do switch.

 

[Vassago]

Here's a single game simulator...

http://www.theproblemsite.com/games/monty_hall_game.asp

Here's a multigame simulator...

http://dl.dropbox.com/u/548740/www/DoorKeeperGame/index.html

It's hard to explain, but I see it in my head so clearly. Argh! lol

 

[pbaldy]

This backs you guys up, but I'm going to have to ponder it:

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

 

[Alc]

The important point in this puzzle is that the host knows which door the car is behind, so the door with the car will never be eliminated. So the outcomes can be seen in the following table:

Winning Door                  Door Chosen    If you switch do you win?
1                                   1                          No
1                                   2                          Yes
1                                   3                          Yes
2                                   1                          Yes
2                                   2                          No
2                                   3                          Yes
3                                    1                          Yes
3                                      2                          Yes
3                                   3                          No

So with 6 outcomes in your favour if you switch and 3 outcomes that are bad you are more likley to win if you do switch.

I wish I'd never started reading this one

The host has already shown you that door # 3 definitely isn't a winner? I can see that, at that point, if you take all three doors into account, he's offering you the option of one door or two (one of the two being a definite loser). However, the fact that the third door exists at all is now irrelevant, as you're never going to pick it.

That would immediately reduce your options to the winning door being 1 or 2. Since you would now only have a choice between door 1 or door 2, surely the odds are 50/50?

 

[chergh]

I wish I'd never started reading this one

The host has already shown you that door # 3 definitely isn't a winner? I can see that, at that point, if you take all three doors into account, he's offering you the option of one door or two (one of the two being a definite loser). However, the fact that the third door exists at all is now irrelevant, as you're never going to pick it.

That would immediately reduce your options to the winning door being 1 or 2. Since you would now only have a choice between door 1 or door 2, surely the odds are 50/50?

The odds of an item being behind one of two objects is 50/50 but thats not the point.

There are nine total scenarios which can occur in the game. If you were to switch in all nine scenarios you would win six of the nine. That is the important part.

 

[Vassago]

I wish I'd never started reading this one

The host has already shown you that door # 3 definitely isn't a winner? I can see that, at that point, if you take all three doors into account, he's offering you the option of one door or two (one of the two being a definite loser). However, the fact that the third door exists at all is now irrelevant, as you're never going to pick it.

That would immediately reduce your options to the winning door being 1 or 2. Since you would now only have a choice between door 1 or door 2, surely the odds are 50/50?

The odds are only 50/50 if the host doesn't know where the car is. You must assume he does since he's obviously not going to open the winning door.

If you stay where you are, you're in the exact same scenario you started with. You still have 1/3 of a chance of winning, even one exact door that contains a goat. The reason why is because you already knew one of them would. Knowing which one doesn't change anything. It's like if you were never given an option to switch...

However, if you switch, you'll have the remaining 2/3 chance of winning, because obviously you're not going to choose Door #3. Basically, it's now like this...

Door #1 - 1/3 probability...
Door #2 - 2/3 probability...
Door #3 - 0/3 probability... since you know it's a goat

It still have to come out to 100%.

 

[Brianwarnock]

I have read the wiki and your answers and am going out on a limb to say that it is all smoke and mirrors. Door3 is kept in in all discussions whereas it is now eliminated.
Whether the host knows or not is irrelevent, you now have 2 doors 1 contains the car one doesn't, you have a choice, how does that differ from if that had been the original choice?

Brian

 

[Alc]

The odds are only 50/50 if the host doesn't know where the car is. You must assume he does since he's obviously not going to open the winning door.

If you stay where you are, you're in the exact same scenario you started with. You still have 1/3 of a chance of winning, even one exact door that contains a goat. The reason why is because you already knew one of them would. Knowing which one doesn't change anything. It's like if you were never given an option to switch...

However, if you switch, you'll have the remaining 2/3 chance of winning, because obviously you're not going to choose Door #3. Basically, it's now like this...

Door #1 - 1/3 probability...
Door #2 - 2/3 probability...
Door #3 - 0/3 probability... since you know it's a goat

It still have to come out to 100%.

I get what you're saying, but it still assumes that the door that everyone knows to be wrong would still be included in the equation. By proving it to be wrong, the host has removed it from any future consideration.

You may as well say that there are 1,000 doors, 998 of which are open and can be seen to have a goat behind them. That doesn't make your odds of winning 1 or 2 in 1000, it's 50/50, since you've only realistically got a choice of two. The open doors are never in the equation.

 

[Vassago]

I get what you're saying, but it still assumes that the door that everyone knows to be wrong would still be included in the equation. By proving it to be wrong, the host has removed it from any future consideration.

You may as well say that there are 1,000 doors, 998 of which are open and can be seen to have a goat behind them. That doesn't make your odds of winning 1 or 2 in 1000, it's 50/50, since you've only realistically got a choice of two. The open doors are never in the equation.

I have read the wiki and your answers and am going out on a limb to say that it is all smoke and mirrors. Door3 is kept in in all discussions whereas it is now eliminated.
Whether the host knows or not is irrelevent, you now have 2 doors 1 contains the car one doesn't, you have a choice, how does that differ from if that had been the original choice?

Brian

It's not out of future consideration though, that's what people don't seem to understand. Just because the host showed you what was behind it, did not remove it from the equation. I don't know how to explain it any easier than the post I made above... It's still there as an option, but obviously your change of winning with it becomes 0 out of 3.

 

[Vassago]

Okay, how about this. You choose Door number 1, and I DON'T show you what's behind one of the doors. You know for a FACT that at least ONE of the remaining doors will have a goat, correct? I tell you that you can trade your one door for BOTH of the other doors, would you do it? You still know for a FACT one of them is a goat.

Knowing which one is a goat, DOES NOT MATTER! You'd have a 2/3 chance of winning.

 

[Alc]

Okay, how about this. You choose Door number 1, and I DON'T show you what's behind one of the doors. You know for a FACT that at least ONE of the remaining doors will have a goat, correct? I tell you that you can trade your one door for BOTH of the other doors, would you do it? You still know for a FACT one of them is a goat.

Knowing which one is a goat, DOES NOT MATTER! You'd have a 2/3 chance of winning.

I'm going to just admit here that you're clearly far better at understanding mathematical proofs than I, Gunga Din.

I do, however, have a ticket for last weekend's lottery that I can let you have cheap, if you're interested? It didn't win, but I'm sure the odds can be shown to stil be good.

 

[PNGBill]

Is the issue here on when the decision is made to switch?
ie if you decide before the game that you will switch your choice when one of the Goat doors is revealed then the chance of that 2nd choice is 50/50 whereas your first choice is 1/3rd chance.

But, if you only make the choice after the door is revealed, then this is a new game with 50/50 chance and you either pick the other door or stay with your current door, which effectively is a choice in itself.

Your original choice 1/3rd automatically becomes a 50/50 choice even by not switching which is a choice. - I think

 

[Vassago]

Is the issue here on when the decision is made to switch?
ie if you decide before the game that you will switch your choice when one of the Goat doors is revealed then the chance of that 2nd choice is 50/50 whereas your first choice is 1/3rd chance.

But, if you only make the choice after the door is revealed, then this is a new game with 50/50 chance and you either pick the other door or stay with your current door, which effectively is a choice in itself.

Your original choice 1/3rd automatically becomes a 50/50 choice even by not switching which is a choice. - I think

Nope, it becomes 2/3, not 50/50.

 

[PNGBill]

Isn't your 2/3rd using history to support the number. What if another person had to make the 2nd choice?

What if you took Paul the octopus along? how would that effect the probability?

 

[Vassago]

Isn't your 2/3rd using history to support the number. What if another person had to make the 2nd choice?

What if you took Paul the octopus along? how would that effect the probability?

It's not history... the choices haven't changed. He's only shown you what's behind one, that door is still there. Since you can remove the probability that that door is a winner, the remaining 2/3 must automatically revert to the other option.