Double Monty - action at a distance

Simon · 2008-08-01 03:21:41

This may mystify some.

Imagine two Monty Hall games taking place simultaneously on different sides of the world. The two games are completely independent.
Imagine you're a contestant in one of them.

There are three doors - two of them empty, one has a prize.
The host knows what's behind each door.
You pick a door.
The host eliminates an empty door from the remaining two.
You are asked whether you'd like to stick or swap.
You're aware of the familiar solution - that there is a 2/3 chance that the door you chose is empty and that the other has the prize.

Before you make your final decision, a phone call is made. You are about to learn whether the contestant on the other side of the world chose the same as you or the opposite.

Depending on what you learn, does this knowledge in any way affect the odds of your game?

Last edited by Simon (2008-08-02 23:55:41)

John E. Franklin · 2008-08-01 05:06:39

Please clarify what "alternatively" means in this context. Are you suggesting
2 scenarios with the word "alternatively" ?

Simon · 2008-08-01 07:15:17

I simply mean: you are told whether the other contestant picked the same as you or not.

I'm asking if this information changes the odds of your game. In other words:

If he picked the same as you, are the odds for your game affected?
If he picked the opposite, are the odds for your game affected?

Hope that clarifies...

mathsyperson · 2008-08-01 07:34:27

Unless there's something deeper that I'm missing, the probabilities all remain the same as if that other game wasn't happening, in the same way that a dice roll or a coin flip isn't affected by what the dice/coin has previously done.

You yourself say that the games are independent, so I don't see how they can affect each other's odds.

Simon · 2008-08-01 08:19:17

It's a good one isn't it!

Yes, the games are both independent and random.
And yes, the odds of your game are affected by information about the other contestant's choice.

If I'm right, here's the mysterious answer:

I suspect this will be considered impossible!

mathsyperson · 2008-08-01 09:23:58

Could you explain a bit more on how you got to those results?
It seems very counterintuitive to me. (Then again, Monty Hall in general is )

Simon · 2008-08-01 10:24:05

Yes it is conterintuitive, but in fact we can dispense with the Monty Hall elimination of an empty door. The principle remains the same if we just stick to what the contestants are most likely to have chosen.

Does that persuade you?

Last edited by Simon (2008-08-01 10:25:13)

John E. Franklin · 2008-08-01 12:58:48

Can you define what it means to pick "the same as you" please?
Can you define what it means to pick "the opposite of you", given 3 doors, and 1 winning door.
Is the word "winning" part of this?

John E. Franklin · 2008-08-01 14:18:32

I know what you mean by picking the same now.
Picking the same means that you both picked the winning door (chance 1/9) or you
both picked the losing door (chance 4/9).
Picking the opposite means that the American picked the winning door and the Chinese picked the losing door (chance 2/9) or the American picked the losing door and the Chinese picked the winning door (chance 2/9), so in that case it doesn't matter what you decide to do, because it is 2/9:2/9, or 50/50 if you picked oppositely.

Put for picking the same, it is 1/9:4/9, so you should switch 4/5 or 80% of the time.

mathsyperson · 2008-08-03 05:59:09

Oh, now I get it!

I interpreted it as being told whether or not the opposite contestant picked the same door number.
If you're told whether the win/lose status of the other contestant's door is the same as yours, then I agree with your reasoning completely.

Math Is Fun Forum

#1 2008-08-01 03:21:41

Double Monty - action at a distance

#2 2008-08-01 05:06:39

Re: Double Monty - action at a distance

#3 2008-08-01 07:15:17

Re: Double Monty - action at a distance

#4 2008-08-01 07:34:27

Re: Double Monty - action at a distance

#5 2008-08-01 08:19:17

Re: Double Monty - action at a distance

#6 2008-08-01 09:23:58

Re: Double Monty - action at a distance

#7 2008-08-01 10:24:05

Re: Double Monty - action at a distance

#8 2008-08-01 12:58:48

Re: Double Monty - action at a distance

#9 2008-08-01 14:18:32

Re: Double Monty - action at a distance

#10 2008-08-03 05:59:09

Re: Double Monty - action at a distance

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