Wednesday, 25 November 2009

A thought about Monty Hall

There is a classic problem called the Monty Hall problem. (Here you can read about it

Lets explore an interesting twist on this. Lets use three cards instead of three doors. One of the cards is a King. The other two something else. The three cards are on the table showing the back side.

Carol now picks one of the card - but she does not turn it. The Game Host Eric now turns on of the other two. He shows Carol (and the audience!) that it is not the King. Eric then puts the card in his pocket.

Let us assume we have the version of the Monty Hall-problem where the game host always shows a non-winning card (i.e. he knows where the King is and select a non-King to show and put in pocket).
He then asks Carol if she wants to swap cards, i.e. picking the other card on the table. Should she?

Yes, she should swap. When Carol picked the first card probability for her was 1/3 to pick the King. And 2/3 that the King was among the other two. When Eric reveals a non-King nothing changes really. And by swapping Carol will increase her odds to win. (Read more in the wikipedia article!)

Let us also assume that John now enters the room from a short bathroom visit. What he see is a table with one card close to Carol. And another card closer to Eric. John asks the person sitting next to him what the game is about - and neighbor responds: "One of those cards is a King. And Carol can keep her card or switch to the other. If she gets the King she wins."
John do not know that there was a third card. (Now in Eric's pocket)

Carol is now convinced she should swap cards and that the odds for her to win would then be 2/3. (Again: Check the wikipedia article to see why the odds are like that. Remember that the game hosts knows where the King is.)

John on the other hand is convinced that the odds for winning if she swap cards is 1/2. Two cards on the table - one if King. Simple as that.

So, now we have two persons in the same room. Seeing the same table and cards. But they are convinced that probability for win if swapping are different.

Who is right? Are they both right?

If Eric now would tell John about the third card and how Eric put it in his pocket. Should Eric rethink the probability? Will past event influence probability from Johns perspective? Would a hidden card from the past in Eric's pocket change John's calculations?

No comments:

Post a Comment