10-14-2003, 07:13 AM
|
|
|
Confirmed User
Join Date: Jul 2002
Location: Magrathea
Posts: 6,493
|
Quote:
Originally posted by punkworld
<b>The Game Show Dilemma - Multiple Stages</b>
There are 4 doors, behind one of them lies a prize, behind the others nothing. After your initial pick and your first chance to switch, a door with nothing behind it that you didn't pick will be revealed. After your last chance to switch, the prize will be revealed... if you got the right door, you will get it.
This little piece assumes you fully understand the original one-stage dilemma and it's solution.
<b>First scenario: Pick and stay</b>
(your choice is indicated by a *, after each door follows it's chance to contain the price)
You make a choice, namely door A. Here are the chances of each door:
Door A: .25 *
Door B: .25
Door C: .25
Door D: .25
The host removes a door, but you stay with your initial choice, door A:
Door A: .25 *
Door B: .375
Door C: .375
Door D: 0
The host removes another door, but you still stay with your initial choice:
Door A: .25 *
Door B: .75
Door C: 0
Door D: 0
Clearly, this doesn't work well. How about another strategy?
<b>Second scenario: Pick, switch and stay</b>
(your choice is indicated by a *, after each door follows it's chance to contain the price)
You make a choice, namely door A. Here are the chances of each door:
Door A: .25 *
Door B: .25
Door C: .25
Door D: .25
The host removes a door, and you switch to C:
Door A: .25
Door B: .375
Door C: .375 *
Door D: 0
The host removes another door, and you stay with C:
Door A: 0
Door B: .625
Door C: .375 *
Door D: 0
Looking better, but still not great. How about yet another strategy?
<b>Third scenario: Pick, switch and switch again</b>
(your choice is indicated by a *, after each door follows it's chance to contain the price)
You make a choice, namely door A. Here are the chances of each door:
Door A: .25 *
Door B: .25
Door C: .25
Door D: .25
The host removes a door, and you switch to C:
Door A: .25
Door B: .375
Door C: .375 *
Door D: 0
The host removes another door, and you switch again, to B this time:
Door A: 0
Door B: .625 *
Door C: .375
Door D: 0
Now we're getting somewhere. But wait, what if we stay first, and then switch?
<b>Fourth scenario: Pick, stay and switch</b>
(your choice is indicated by a *, after each door follows it's chance to contain the price)
You make a choice, namely door A. Here are the chances of each door:
Door A: .25 *
Door B: .25
Door C: .25
Door D: .25
The host removes a door, and you stay with A:
Door A: .25 *
Door B: .375
Door C: .375
Door D: 0
The host removes another door, and you switch to B this time:
Door A: .25
Door B: .75 *
Door C: 0
Door D: 0
|
Hehe, thanks, but that's not quite what I meant. Like I said, I understand the principle of what you said. I was looking for more a discussion on the theory of the problem than the actual numbers.
It doesn't matter, anyway. It's an interesting dilemma.
SpaceAce |
|
|