h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I've recently again read about a problem that baffles a huge number of people, including actual mathematicians and statisticians. I'd like to try it on my fellow-h2g2ers. Here it is.

It's a game show. There are three doors. Behind one of the doors there is a fancy car; behind two of the doors there are goats. The game show host knows what's behind each door but the contestant doesn't. The idea of the show is to win the car.

The contestant picks a door, any door.

The host then opens one of the two doors the contestant hasn't picked - *she* picks a door that has a goat behind it. So now the contestant knows at least that that particular door doesn't hide the car. But now he's left with a choice of two remaining 'unknown' doors: the one he originally picked, or the other one, that the host hasn't opened.

He now gets a chance: he can keep the door he picked first, or he can switch his choice to the other door, the one the host hasn't opened. Should he switch - does he improve his chances if he does?

Can you answer this - and if you can, can you give a simple and clearly understood explanation of why? Because apparently this baffles a heck of a lot of people, and I think I can give a clear and concise explanation of the solution.

Hi Willem,

There is no skill involved. It's just a game of chance.

To heighten the tension for the viewers the player is given the 'chance' to change his option and risk losing what might be a winning guess.

Most TV game shows are run along the same lines.


lil x

I have read statistic analysis about this, but I can't remember where, and I can't remember what they said.
One line of reasoning, I think, had it that it was a better option to change selection, but it doesn't really compute for me...

Hi Milla and Lil! The answer *is* online but I'd like for you not to look it up but to try and work it out. I see that many people cannot understand what is going on and I want to try and see if I can explain it so people actually understand why. Lil, it is a game of chance - but there are two different strategies and the one works twice as well as the other! So if you understand what's going on, you can still not get it right all of the time, but you can double your chances of getting it right. This is a thought experiment, not a real game show - but there are people who have made it into a real experiment to see if in actuality it works out the way it does when reasoned out, and it does. Many people don't understand why it works like that even after having performed the experiment and having proved that it does work! Again, I want to see if I can give an easily understood explanation ... but still before I do that I'd like people to think about it a bit.

Off the top of my head. To begin with the contestant randomly chooses a door. There is a one third chance that behind that door lies the car. That probability stays the same as an original selection.

The host opens a door behind which is a goat. Now what remains are two doors. But the chances of finding the car behind the remaining closed doors is higher: there is now a one half chance of finding the car behind each door on the NEXT selection ...

... EXCEPT there is still only a one third chance of finding the car behind the ORIGINALLY selected door (which is 33%) ...

... so the contestant MUST change selections to IMPROVE his/her odds of finding the car (which is 50%).

The reason why the contestant must change selections is because the host's choice is not random - the host will avoid the car and open the door where there is a goat. The above I think is logically correct but mathematically not quite right ...


A better way of looking at it. After the contestants original choice (1/3 probability of car) there is a 2/3 chance of finding the car behind one or the other unselected doors. You know the host will open a door containing the goat ... but there is still a 2/3 chance of finding the car behind the unselected remaining door --> so in this perspective the contestant will double his chances by selecting the other door .., and I think this is mathematically correct (?)

Yes I think the second way of looking at it is mathematically better - because Willem says by changing strategy you can double your chances.

I disagree, while there is only a 33% chance of picking the door at the start, there is a 100% chance of not picking a door with a goat! One of the doors has no statistical bearing on the final result. I has always been a 50-50 chance. The only advantage might be in carefully watching the body language of the host and staff of the show. They know the correct answer and you don't.

F S

OK so if everyone has thought of this ... Stone Aart is right and gives a good explanation. Florida Sailor disagrees! So anyways ... the challenge for *me* now is to try and give an explanation that as many people as possible will understand. The reason I'm doing this is to try and exercise my own 'explanation muscles'. But there's of course a big goal behind this. It is a problem of understanding of the nature of probability. This is mathematics but it doesn't need big numbers or long equations with strange symbols. Instead you need to think about the different ways in which things might be. Now if I can explain this to just one or two folks it would make me very happy about my own powers of explanation!

Anyone else want to try explaining this problem before I give it my go?

OK here's the answer and explanation.

If you've chosen a door and the host opens another door, showing the goat behind it, you should switch to the remaining door, because that will increase your chances from 1/3 to 2/3.

Most people see it like this: when you have 3 doors, and the car can be behind any one of them, you obviously have a 1 in 3 chance of picking the car, if you pick any of the doors.

Now suppose the host now opens one of the other doors - knowing that there's a goat behind it. So now you know the car is *either* behind the door you picked first, *or* behind the remaining door than neither you nor the host have picked. So it would seem you have a 50/50 chance of picking the car. So in this view, switching makes no difference.

But actually it doesn't work like that! See - when you first picked a door, any door, your chances of hitting the car was 1 in 3. The host opening another door doesn't change this, cannot change this. So when the host opens the other door to show the goat, the chance of the car being behind the door you picked first cannot suddenly jump from 1 in 3 to 1 in 2.

No - it remains 1 in 3! So you have 1 door left that you can choose. Since the odds of the car being behind the first door you picked is 1 in 3, and the odds of the car being behind the door the host opened is 0/3 (there is a goat behind it) - and the car has to be somewhere so the total chances must be 3/3 - it means the chance that the car is behind the remaining door is 2/3.

Does anybody understand that?

Sorry for bumping in here (hi everybody) , but I don't think that is correct, even if mathematicians say it. The maths may be right, but the view on the whole thing is wrong in my opinion.

Of course at first the chance to pick a car is 1/3 and the chance to pick a goat is 2/3. No matter if you picked the car in the first place or picked a goar the game master will always open a goat door.

After this it is essencially a *new* game that starts, which has nothing to do with the choice made before. You now have the chance to pick the door you picked before or you pick the other one. One is the car, one is the goat. So that's 50:50.

In fact the first choice is absolutely irrelevant because there will always be a goat behind the opened door, no matter which door you choose. Only the 2nd choice is relevant and has any influence on the outcome. The first choice is completely irrelevant for winning or loosing.

I understand you both! But my head spins, and I can't say who is right!

I have to say that I find Tav's explanation convincing.

I remember, rather horribly, watching exactly this process on US TV growing up. It was a game show called 'Let's Make a Deal'.

This is the Monty Hall problem

Tav's explanation is not right, as it is not a new game that is starting after one door is chosen - the opened door is not chosen at random, but is selected because it is known that there is a goat behind it, and if you initially chose a door with a goat behind it, then the door with the car behind it *cannot* be opened, so the other one must be opened instead.

There is an article http://www.jstor.org/discover/10.2307/2684453?uid=3738032&uid=2&uid=4&sid=21102830823093 which indicates that Willem's explanation is also not quite right.

It is a complicated problem, though, and depending on how the question is asked, the probability will be different. It is always better to switch, though.

I saw this on TV (James May's Man Lab) and he did the test 100 times and won 60 times (if the probability were 2/3, then it would be expected that he won 66 times). I think Stone Aart's first explanation agrees with the explanation on Man Lab - the probability that the car is behind your first choice is 1/3, and the probability it is behind the other remaining door is 1/2. I don't know exactly what argument was used by the Man Lab to justify the result that it is a 60% chance of winning by switching, but I noted that 1/3 + 1/2 = 5/6 and 1 = 5/6 x 6/5, and 1/2 x 6/5 = 3/5 =0.6.

Thanks, Sasha. I'll come back and read that, when I get time. Maybe I can manage to understand it. It's hard, I'm almost innumerate. But I'll give it a try.

OK I cannot read the entirety of that article, and I know the problem can change based on the way it is formulated but I formulate it like this:

1. There is a car behind one of the doors.
2. There are goats behind two of the doors.
3. The contestant doesn't know what is behind any of the doors and she can choose any of the doors with equal probability.
4. The host knows what is behind each door.
5. The contestant makes a choice.
6. Out of the two doors NOT chosen by the contestant, the host opens one - one with a goat behind it. If there is just one unchosen door with a goat behind it, then the host has to open that door, but if there are goats behind both unchosen doors, the host can open any one of them.
7. After the contestant sees the goat behind the door opened by the host, she is given the chance to change her choice to the other door - the one she didn't choose and the one the host didn't open.
8. There is absolutely no interference or influence from the host with the contestant's choice ... this is for science not for profit!

I think as formulated above this problem is mathematically rigorous and can be easily analysed. HOWEVER I am well aware that even very learned people keep arguing about it! So you're free to think I am wrong about it!

Yeah, unfortunately I could only access the first page of the article as well. It does show, though, that even though the problem is quite straightforward to state, analysis is not as straightforward as it might appear at first, which is fascinating

Yes, many arguments about it have emerged over the years, and it's not my field of expertise, so that's all I can come up with...

http://www.youtube.com/watch?v=tvODuUMLLgM I enjoyed Man Lab, anyway

Willem, you are 100% right. The reason why the question confuses people is as much psychology as anything else. They fail to take into account the fact that the game show host KNOWS WHERE THE PRIZE IS. That renders all of the carefully considered probabilities a nonsense.

Might have to link to this, though:

http://xkcd.com/1282/

RE Post 12

'Let's Make a Deal' was a bit of a different problem, they had a great prize like a car, a medium prize like a new room of furniture and a 'Zonk' like the goat - something nobody wanted.

If you selected the car they could show either the furniture or the goat.

If you selected the furniture they had to show the goat, as the car would never be left out of the choices.

if you picked the door with the goat they had to show the furniture.

So if you see furniture you should switch, if they show you the goat stand pat.



In the example given both other choices are equally bad and one will be reviled no matter what your choice was. The object behind the doors has not changed so the odds remain equal

If you pick a goat the host knows which door to open without any thought. If you pick the car the host has to decide which door to open and this could cause hesitation. A good host would always hesitate, but we are talking real world.

RE post 18 point 8

You can not isolate personal observation from a scientific study. Science has to consider all the factors that could influence the result of an experiment.

F S

'They fail to take into account the fact that the game show host KNOWS WHERE THE PRIZE IS. That renders all of the carefully considered probabilities a nonsense.'

That's exactly my point. The first choice is completely irrelevant because the host knows where a goat is and opens that door. Afterwards you have a 50:50 choice, no matter what the host knows.