Monty Hall Problem... For Newbies

[boooeee]

The problem is made to appear more difficult by having only three elements. If it had more, I think it would be more obvious. You choose one out of a hundred doors. You're then offered the option of abandoning your one door and choosing instead, all 99 others at once. It's the same problem.

This explanation has never worked for me.

An alternative way to look it is that if you switch each time, you'll always get the car if you happened to initially pick a goat, and you'll always lose the car if you initially picked the car. You have a 2 in 3 chance of picking a goat first, and a one in three chance of picking the car. So your chances of getting the car after switching are 2 in 3 and the chances of not getting it (losing it) are 1 in 3.

This one does.

Why are Monty Hall threads so contentious? Because the stakes are so low.

 

[lionking]

In the standard formulation of the problem, you are correct.

Isn't this what the OP is about? The standard proble?

 

[boooeee]

Isn't this what the OP is about? The standard proble?

Yes. And the rest of my post, which you snipped, explained why it's relevant to the standard problem.

The ground rules are crucial to understanding the problem, and one of the reasons people argue over the "right" answer.

 

[bruto]

See, for some reason, that does not help me to see what's going on.
This is why I feel the more ways something can be explained , the more people will find one that works for them.

Oddly, one aspect nobody ever seems to mention re the MHP is the psychology of linking the best strategy to the need to change the original choice. How many folk choose to stick, simply because they feel swapping involves coercion?

Well, here's another way to look at it. You make an initial choice, and that choice is, by definition, a single door. The opening of doors after that is a red herring. The offer that follows is: stick with your original choice (a single door), or change to "all the rest." Unless there were only two doors at the start, "all the rest" will always be a better bet.

 

[lionking]

Well, here's another way to look at it. You make an initial choice, and that choice is, by definition, a single door. The opening of doors after that is a red herring. The offer that follows is: stick with your original choice (a single door), or change to "all the rest." Unless there were only two doors at the start, "all the rest" will always be a better bet.

Yes. This was the "a ha" moment for me.

 

[Brian-M]

Nope. There are now 2 doors. One is the right choice, one isn't. Your odds just changed.

The odds of the remaining door having the car behind it just changed from 1/3 to 2/3. The odds of your door having the car behind it haven't changed at all.

There was a 2/3 chance of the car being behind one of those two doors to begin with. The fact that you now know that the car isn't behind one specific door doesn't change that.

In the standard formulation of the problem, you are correct. And it's crucial to why the odds for switching are 2/3. Which is why it's important to acknowledge that when Monty doesn't know where the car is, AND just picks a door at random, AND happens to pick a goat, you've got a 50/50 shot of having the car, whether you switch or not.

No. As long as he didn't show the car, it doesn't matter whether or not he picked the door at random. The fact that you only have a 1/3 chance of having the car behind your door and 2/3 chance of having the car behind a different door is unchanged.

 

[Brian-M]

You are right that if we had to pre-commit to a strategy before the game started and we chose to switch whatever happens (to the car if he shows the car, to the 3rd door if he shows a goat) then you would be succesful 2/3 of the time. 2/3 is the correct probability for that strategy when viewed from before the moment Monty has made his choice.

Yes, it doesn't matter whether or not he's picking at random or deliberately picking a door with a goat behind it. There's always a 2/3 chance that the door we initially pick is wrong, so switching will get us the car 2/3 of the time.

If he shows a goat, then our probability of getting the car if we switch is now 1/2 (as I showed before).

I don't see how you showed this.

Since there was only a 1/3 chance that the door we first picked was right, until we know where the car is there's always a 2/3 chance that the car is behind another door.

 

[UKBoy1977]

No. As long as he didn't show the car, it doesn't matter whether or not he picked the door at random. The fact that you only have a 1/3 chance of having the car behind your door and 2/3 chance of having the car behind a different door is unchanged.

No, as I showed on page 3, if he chooses randomly and reveals a goat then the chance of your original pick being correct increases to 1/2. To see why, imagine after he has randomly picked a goat you shuffled the 2 doors around and asked him to pick again, and again he got a goat. And then again, and again. Eventually you would start to think to yourself that you must have the car, else how could he keep randomly picking a goat? That is, randomly picking a goat supports the hypothesis that you have the car more strongly and so increases the probability of it.

 

[Squeegee Beckenheim]

Nope. There are now 2 doors. One is the right choice, one isn't. Your odds just changed.

Look at it this way - if you're going to stick with your original choice, then Monty may as well not bother opening a door and asking you if you want to swap. I think you'd agree that if the problem was simply picking a door and then opening it your chances of winning the car would be 1 in 3. Then you need to ask yourself what about opening a door magically changes that 1 in 3 chance to a 1 in 2 chance.

 

[AbleSugar]

I am trying to make it clear that there is no problem here.
You can choose a car or one of two goats. You can then choose to stick or swap. If you chose a goat and swap you win, if you chose a car and swap you lose.
2 out of every 3 swap choices gets you a car. (Dammit, I just typed "goat").
What further analysis do you feel is required?

Click! The light bulb over my head just illuminated. Thanks, I get it again.

 

[Brian-M]

It occurs to me that the Monty Hall problem can be applied to other game-shows.

Take "Who Wants To Be A Millionaire?"... if you have absolutely no idea which option is the correct one, you can still get it right 62.5% of the time on a 50/50.

Just (mentally) pick one of the four options at random, then ask for a 50/50. If the option you picked is eliminated... well, you still have a 50% chance of guessing right. But if the option you picked remains, then you have a 75% chance of guessing right by picking the other option. So that averages out to getting it right 62.5% of the time.

(Not that it would be much of a help to any particular contestant. I don't watch the show, but I assume that you can only use the 50/50 lifeline once.)

No, as I showed on page 3, if he chooses randomly and reveals a goat then the chance of your original pick being correct increases to 1/2.

Okay, I see that you're right. If we throw out the situations where he picks the car by accident, the odds become...

1/3 Original door
1/3 Remaining door
1/3 Redacted

So the odds of it being behind the remaining door are the same as it being behind the door you originally picked.

 

[Mongrel]

The odds of the remaining door having the car behind it just changed from 1/3 to 2/3. The odds of your door having the car behind it haven't changed at all.

This is what did it for me, you pick your door and you have a 1 in 3 chance of being correct, nothing changes this.
When the goat door is opened my original pick is still 1 in 3 chance.

The maths fairy can't magic away the goat door to get to 50/50, you still have three doors -all that's changed is that you are aware that there's a goat over there and your door still only has a 33% chance of being correct.

 

[Ray Brady]

At the start of the game, there are only 9 possible setups.

The car is behind Door A. I pick Door A.
The car is behind Door A. I pick Door B.
The car is behind Door A. I pick Door C.
The car is behind Door B. I pick Door A.
The car is behind Door B. I pick Door B.
The car is behind Door B. I pick Door C.
The car is behind Door C. I pick Door A.
The car is behind Door C. I pick Door B.
The car is behind Door C. I pick Door C.


If I stick with my original choice, I win in three of these cases. If I switch, I win in six. Easy peasy.

 

[hgc]

You don't need to. He does, in case he opens the wrong door and shows you the car.
What counts is that you can only choose the car one way, but a goat two ways. Either way you choose one goat, Monty reveals the other goat. If you stick in either of those cases, you get the second goat.
The only other possibility is that you choose the car.If you do, Monty can open either other door , showing a goat. If you switch IN THIS 1 CASE, you will lose, because you chose right first time. In either of the other two cases, a switch wins.

Of course Monty (or someone whispering in Monty's ear) knows what's behind which doors. That's the only way he knows what to do after you make your initial selection. It's baked into the premise -- after your first pick, Monty always shows you a goat.

 

[AdMan]

Well, here's another way to look at it. You make an initial choice, and that choice is, by definition, a single door. The opening of doors after that is a red herring. The offer that follows is: stick with your original choice (a single door), or change to "all the rest." Unless there were only two doors at the start, "all the rest" will always be a better bet.

I think this is a really useful way to look at the problem which I hadn't considered, but it makes a lot of sense. Thanks.

 

[GlennB]

It occurs to me that the Monty Hall problem can be applied to other game-shows.

Take "Who Wants To Be A Millionaire?"... if you have absolutely no idea which option is the correct one, you can still get it right 62.5% of the time on a 50/50.

Just (mentally) pick one of the four options at random, then ask for a 50/50. If the option you picked is eliminated... well, you still have a 50% chance of guessing right. But if the option you picked remains, then you have a 75% chance of guessing right by picking the other option. So that averages out to getting it right 62.5% of the time.

That looks devilish cunning, but wouldn't help you. It's the equivalent of Monty opening your chosen door when there happens to be a goat behind it, leaving you a 50-50 guess between the others. In fact when Monty plays by the standard rules he gives you extra information that makes your original choice less attractive.

(Not that it would be much of a help to any particular contestant. I don't watch the show, but I assume that you can only use the 50/50 lifeline once.)

Yep.

 

[UKBoy1977]

It occurs to me that the Monty Hall problem can be applied to other game-shows.

Take "Who Wants To Be A Millionaire?"... if you have absolutely no idea which option is the correct one, you can still get it right 62.5% of the time on a 50/50.

Just (mentally) pick one of the four options at random, then ask for a 50/50. If the option you picked is eliminated... well, you still have a 50% chance of guessing right. But if the option you picked remains, then you have a 75% chance of guessing right by picking the other option. So that averages out to getting it right 62.5% of the time.

Alas, no. For your second scenario to work then Millionaire would have to work like Monty Hall i.e. they would need to know your guess and then, from the other 3 choices, intentionally eliminate two wrong answers. Then it would be 75% likely the other remaining one was correct.

What actually happens is that you mentally choose one (probability 1/4 of being right) and then they randomly eliminate 2 of the 3 wrong answers. If yours is not one of the two eliminated this provides evidence for yours being the correct answer (as they could have eliminated yours but didn't). Your probability now increases from 1/4 to 1/2. So it's still 50/50 whatever happens.

 

[UKBoy1977]

Interestingly, ddt hasn't posted for a while in this thread. I wonder if that's simply because he has realised he was wrong, or whether it's more through embarrassment at his inability to argue in a civil manner - variously accusing me of sneakily moving the goalposts, erecting strawmen and "treading close to wilfully lying". ddt, if you're still reading, have a little think about how you might do things differently next time ;-)

 

[Brian-M]

That looks devilish cunning, but wouldn't help you. It's the equivalent of Monty opening your chosen door when there happens to be a goat behind it, leaving you a 50-50 guess between the others. In fact when Monty plays by the standard rules he gives you extra information that makes your original choice less attractive.

Yes, thinking it through I see that it is still a 50% chance.

25% of the time you mentally pick the right answer and get it wrong when you switch.
25% of the time you mentally pick a wrong answer that gets eliminated, but guess incorrectly between the remaining two choices.
25% of the time you mentally pick a wrong answer that gets eliminated, but guess correctly between the remaining two choices.
25% of the time you mentally pick the wrong answer which doesn't get eliminated, and get it right when you switch.

It's only if they deliberately refrained from eliminating your initial choice that the odds would favor switching.

 

[Squeegee Beckenheim]

Interestingly, ddt hasn't posted for a while in this thread. I wonder if that's simply because he has realised he was wrong, or whether it's more through embarrassment at his inability to argue in a civil manner - variously accusing me of sneakily moving the goalposts, erecting strawmen and "treading close to wilfully lying". ddt, if you're still reading, have a little think about how you might do things differently next time ;-)

Not my fight, so I'm not going to get in to it beyond this, but he's not wrong - you are. His initial post made one statement that you responded to by saying "I disagree", and for which you provided incorrect reasoning. It was pointed out to you that your reasoning didn't support your "I disagree" and you changed what it was you were arguing against to something that ddt hadn't claimed to be the case.

If you do disagree with what ddt actually said, then you have yet to explain why what he said was wrong. If you don't disagree with what ddt originally said, then you need to retract your original disagreement and admit that what ddt said was correct.

As far as civility goes, if you have an issue, then report it to the mods. If you think this latest post of yours puts you on any kind of higher moral ground, then you are mistaken. That's my last word on the subject.