Quiz

[300CE24]

Dear 2+2ers,

first of all thanks everybody for a great forum.

I have a small but very interesting quiz for you that deals with probability. Some of you may have seen it before, but I'll fire anyway:

You are in a game show and you must choose to open 1 out of 3 doors. You know that behind 2 of the 3 doors there is nothing and that behind 1 door there is $1.000.000,-. If you open the $ door you get the $, if you open one of the other doors you get nothing. You make a choice and are about to open the door. The game show host says: 'wait a minute'. He then opens 1 of the 2 doors you did not choose, and there is nothing behind it. He then says: 'you now have the option to choose again, ie. you can open the door you chose first or you can choose the other closed door'

What do you do? And what is the probability that you get $1.000.000,- by doing so?

[pzhon]

This is the well-known Monty Hall problem. It is a problem of conditional probability. You didn't state the conditions under which the host takes those actions. Therefore, there is not enough information to determine the conditional probabilities. See the rec.puzzles archive (http://rec-puzzles.org/new/sol.pl/decision/monty.hall).

[300CE24]

All,

true I forgot to say that the host does know where the money is.

[pzhon]

[ QUOTE ]
All,

true I forgot to say that the host does know where the money is.

[/ QUOTE ]
I suppose you also mean for the host to offer the chance to switch every time, whether the contestant choose correctly or not. These may seem like little details, but changing them can change the correct decision. Part of understanding the Monty Hall problem is understanding why these conditions are important.

[300CE24]

Dear all,

I read pzhon's link and I realize the problem was well known than I thought in the first place.

Anyway, it's pleasing to see that a majority of people have answered correctly.

BRGDS,

[Danenania]

I just read about this problem in a novel by Mark Haddon, The Strange Incident of the Dog in the Nighttime. Good book and interesting problem. Is that where you got it?

[300CE24]

No, haven't read that book. Simply heard it from a friend.

BRGDS

[M50Paul]

[ QUOTE ]
Dear all,



Anyway, it's pleasing to see that a majority of people have answered correctly.

BRGDS,

[/ QUOTE ]

Not sure I undersatnd why asking for the other door is correct. I didnt, I said keep my first choice. The correct answer is where the $1M is not where the most likely place is so..... Either way I make money with my door or the other door and unless I win the million it is moot. Also there were 10 that wouldnt ask for the other door and 11 that would. How is it that the right answer?

[Nottom]

Think of it this way ...

If your plan is to switch everytime he opens a door then you have a 1-in-3 chance of choosing the winner right off the bat and then switching to a loser. On the other hand, you have a 2-in-3 chance of picking a loser to start with and then switching to the winner.

If you choose to just stick with your original door, you will only have a 1-in-3 chance of being correct.

[meep_42]

I always found it easier to visualize 100 doors, with the host opening 98 losers, leaving 2 doors, your choice and a mystery door.

The chance that you chose correctly initially is 1%, the chance that the other door is correct is now 99%, so you'd be a fool to keep your first choice.

-d

[daryn]

i find it's much easier to visualize 10,547 doors, and then the host opens 10,545 doors,..

[M50Paul]

I have seen the errors of my way. How silly of me. I had misread the quizz ( waht's new ) I thought it said there was $$$ behind 2 of the 3 doors and one got opened.

[brewmeister6]

Why does your original choice remain with such a low probability, even after the choices have been narrowed down. It seems like the statistics open up when you are limiting the number of elements in the set.

And more importantly, how does relate to my low limit poker game /images/graemlins/cool.gif

[tubbyspencer]

[ QUOTE ]
Think of it this way ...

If your plan is to switch everytime he opens a door then you have a 1-in-3 chance of choosing the winner right off the bat and then switching to a loser. On the other hand, you have a 2-in-3 chance of picking a loser to start with and then switching to the winner.

If you choose to just stick with your original door, you will only have a 1-in-3 chance of being correct.

[/ QUOTE ]

Excellent. This is the best simple explanation I've seen yet.

[bigpooch]

The question is only interesting without the stipulation
that the host will always show one of the other doors.

Then you haven't provided an option for the correct answer!

[IlliniRyRy]

FYI...this problem is also mentioned in Market Wizards by Jack Schwager, which is a book comprised of interviews with traders. I know there are tons of poker players who have an interest in the stock market, and being a trader myself, this is my favorite book of all time.

[Rudbaeck]

[ QUOTE ]
Why does your original choice remain with such a low probability, even after the choices have been narrowed down. It seems like the statistics open up when you are limiting the number of elements in the set.

[/ QUOTE ]

Because the host doesn't open the winning door?

[Tuneman100]

Wait, ok I understand that arguement, but why don't the probibility change to 50/50 when that third door is opened? I understand all the arguments, but I can't find an answer to that specific question.

[Rudbaeck]

[ QUOTE ]
Wait, ok I understand that arguement, but why don't the probibility change to 50/50 when that third door is opened? I understand all the arguments, but I can't find an answer to that specific question.

[/ QUOTE ]

Because the host didn't open a random door, the hosted opened a door that wasn't the winner.

You are at 1/3 before the host opens, and your door is still 1/3, but the other door is 2/3 AFTER the host opens a dud.

[Tuneman100]

ok I undersand. thats strange, usually I am very good with probibility and odds, but that one just got me.

[Rudbaeck]

Don't worry, it initially got lots of people. I recall several statistics professors getting it wrong when it was 'news' a decade or so ago.