OT: Statistical question

[Steve Firth]

Roger Mills wrote: [snip]

Chhange your choice. It will improve your odds of winning. By revealing an option the presenter has skewed the odds.

Covered in some detail on Man Lab recently.

[dennis]

1/3 to win by holding onto what you have or 2/3 by swapping.

Because you know more than the other contestant.

If the host removed one door *before* you picked you would have a 50:50 chance too.

[Martin Brown]

The odd one of immense complexity has been later found to contain errors or implicit assumptions that could not be fully justified or derived from the starting axioms. Sometimes simple typographical mistakes can occur too and remain unnoticed for a long while.

Everything that humans do is susceptible to error, but a rigorous mathematical proof has generally withstood the best minds on the planet making serious efforts to find faults and break it.

The point remains that mathematical truth is absolute when starting from a given set of axioms. You can either derive a self consistent formal proof of correctness or a counter example to show it isn't true.

You can take a pretty good bet for example that any alien civilisation that gets as far as being able to receive radio will have discovered mathematics of prime numbers, 0, -1, e and pi for example although they will very probably have given them different names.

The formal proof is the scaffold that underpins any theorem.

A theorem must have *at least* one formal proof.

[Andy Burns]

One time out of three, your initial pick would win if you didn't change later, in that case it doesn't matter which door the host leaves closed, in the other two times out of three, the host has no choice which door he leaves closed it must be the winner.

The stranger doesn't know which of the unopened doors was your random selection and which was the one the host deliberately left closed, therefore he has less information and for him it would be a 50:50 choice, if you chose to disregard that information you are lowering your own chances.

[Andy Burns]

[Dave Liquorice]

Because you have already picked a door and know which one it was.

[Martin Brown]

A priori if *he* has no other information about it then yes.

Because you also know that there were originally *three* doors and that one losing door was taken away by the host collapsing two losing states for you into certain wins if you swap to the other remaining door.

2/3 > 1/3 so you should always swap to exploit the degeneracy factor.

[meow2222]

This is a trivial statistical problem surrounded by a great deal of confusi on.

When looking at 3 doors, the odds of picking a winner are 1 in 3.

When one door is opened, you now have 2 doors with no data at all as to whi ch is the winner, thus the odds are now 0.5 for each door. Thus it makes no difference whether one switches doors or not. Statistical probability figu res are nothing more than reflections of the data they're based on, and ope ning one door changes the available data. Missing that point is where so ma ny are going wrong.

Other points where people go wrong on this are:

A computer simulation proves nothing because we don't know what was program med into it - the output is merely a reflection of the views of the program mer.

The IQ and experience of a mathematician proposing an answer also does not establish it to be true, everyone makes mistakes. It seems to be a popular myth that high IQ and specialist subject knowledge immunise people from tha t.

In the end the problem is simple O level stuff.

NT

[Andrew May]

Whilst I agree with all the conclusions I am not entirely convinced of this step. Suppose that there were only two doors. You choose one. There is therefore a 1:2 chance of being right. The host opens the other one to reveal a goat. Does the door you chose still have a 1:2 chance of being right?

[Andrew May]

Because you have additional information that the other contestant does not. Namely which door you chose the first time around.

[Man at B&Q]

MBQ

[Man at B&Q]

You failed at he most fundamental hurdle, sorry, but you haven't won anything.

MBQ

[Martin Brown]

That depends on whether the object of the game was to win a goat!

[Huge]

No you aren't. You have more information - to whit that the door the host opened was a losing one. The host will never open a winning door. That information changes the odds.

[Huge]

Precisely.

[Andy Burns]

and he'll never open your door.

[Roger Mills]

I couldn't have put it better myself!

By removing a losing door, the game has changed to a straight choice between two equally likely/unlikely doors.

[John Rumm]

You should *always* change your choice...

Its one of the standard Game Theory scenarios.

[Roger Mills]

No you don't. You know that there are now two doors, one of which is a winner. The fact that there was originally another losing door which has now been eliminated doesn't come into it.

In effect, he *has*. You are now faced with a new choice between just 2 doors - to stick with your original choice or to swap. Each is equally likely to be correct, just as it is for the substitute contestant.

[Roger Mills]

The only thing you know which the stranger doesn't know is which door you originally picked. But that doesn't help you! You still don't know whether you picked the winner or not - all you know is that one or other is the winner, so you're in the same position as the stranger.