anyone fancy a gamble

[jimmer]

ok, so i'm on a quiz show and to win the star prize of say 20 quid i have to guess which door its behind. there are three doors and i have two chances. my first guess, door number one. The quiz show host opens up door number three which is empty and gives me the chance to stick with door number 1 or change my mind and select door number two. what should i do?

 

jim

[stevie21]

The table at the bottom of this page is informative. I'd have thought that you had a 50:50 chance at the 2nd stage but apparently it is in your favour to switch.

http://www.faisal.com/docs/monty.html

[fantastical]

ok ok, I get it now. I was also talking to some friends about this question today, and they told me a true story. It goes like this.

Some people at one of my friends work was sorting through this problem by example. Similar to the 52 card illustration - which makes perfect sense - they used like 6 "doors" (or upside down plastic cups) Anyway, the girl who chose the one cup from the six happened to pick the right cup (1 in 6 odds) the very first try. This happened on two consecutive occassions. So in her case, switching would fail the test.

I think the hard thing to get over is the fact that we're not dealing with 52 doors, just 3. But...I stand corrected. When talking with my friend, I first deduced that you stand a 25% better chance at switching in the second round (numbers were dancing in my mind), but then he said that he read you stand a 17% better chance. Anyone care to take a stab?

[Wong]

To Ansteyg,

Please don't tear your hair out over this one. Either he gets it now or he never will. Probability is a subject that will always be open to speculation however mathematical it is. There must be loads of other people out there awaiting your expertise.

[helghast]

you dont better chances by switching.  there are three doors withh a one in three chance of picking the right one. you pick door one which may be right but so could door two. just because the host opened door three doesn't mean its behind door two. it could be behind door one and because you picked it he can't open it. therfore he randomly picked one of the two other doors to open, leaving you with a one in two chance whichever door you open.

[bernardo]

This is what mathematicians call "counter-intuitive".  The way I thought of it was to image 100 doors instead of 3.  Suppose you choose door 1, and then the host opens doors 2 to 37 and 39 to 100.  He then gives you the option of switching from door 1 to door 38.

 

This is based on the host's knowledge that the car is behind door 38 (or possibly, door 1).  But when the host has opened all the other doors, there is only a 1-in-100 chance that the car is behind door 1 (the one you chose in the first place).  There must therefore be a 99-in-100 chance that it is door 38, not a 1-in-2 chance.

 

The whole point about this is that the host knows where the car is, and you atre therefore not being given a random choice with your option to switch.

[jimmer]

no no no bernardo. there is no car, and there are not 100 doors. there are three doors and its a twenty quid prize. that aside, a very good explanation.

 

jim

[bernardo]

yes yes yes jimmer.  It doesn't matter whether there are 3 doors or 100 doors.  The point is that the host knows which door has got the armadillo behind it, and that the choice of whether to switch from one door to another is therefore not random.

[jimmer]

the point is not whether the host knows which door the 20 quid is under - although being the host he may well know -  it is whether the 20 quid stays behind the same door. if the 20 quid doesn't move then the contestant is stuck with the original probability until he switches doors. the host has no effect on the outcome.

 

jim

[helghast]

ansteyg you are wrong!!!!!!!!!!!!!!!!!!!!!!!!!!

 

why should the probability be better or worse for door 1 than door two?

[Peter Pedant]

a related q is the Villa Serbelloni problem so called as it almost wrecked a conference held there in 1966

Matthew Mark and Luke are in prison. Two of the three are to die but they dont know which. They know when: the next day. On the last night Mt goes to a gaoler and says one of the other two has to die, it doesnt matter which to me but can you tell me who it is?

The gaoler says Mk. Mt goes to bed much happier because before he asked the question, he had a 2 in 3 chance of dying, but after the answer, it is now 1 in 2.

Is Mt right to feel happier?

From Math ideas in Biology first ed 1969, John Maynard Smith.

 

[Peter Pedant]

hey that faisal.com page - see above - is FANTASTIC !

Everyone who doesnt understand that you must switch should read it......

[Wong]

Ansteyg,

If you're not a teacher you ought to be. I warned you to no avail - you have the patience of a saint or maybe you're just bored.

[Yeah Bee]

Statistic tests show that you should change. Think about the possibilites. When there are three doors 1/3 shot of beinbg correct. two doors 1/2. Don't know why don't know when but it is true,

ALWAYS CHANGE DOOR

[Yeah Bee]

Sorry for stating an already given and correct answer I only read the two top ones, don't chop my head off.

PS like the card game