Lets put this another way.
You and I are sitting and deciding if this is true. You decide that it doesn't matter, but I know it does. So we go to the game show and do the show a thousand times. At the end of the competition, I will have walked away with 50% winnings and you would of walked away with 33% winnings. That's reality.
The math is an abstraction of that outcome. It's confusing, but it is grounded in reality. The outcome, with enough iterations, is absolute. In fact, if we ran this simulation with 1000 doors, I'd still walk away with 50% prizes while you would walk away with 0.1% prizes.
It's simply visualized  take a piece of paper and do a binomial tree of the possible outcomes.
The reason it is so hard to grasp is because it runs counter intuitive. Your first choice doesn't matter. Think of it as two seperate trials  in the first, you have to pick the winning 1 out of 3 or you have to pick the winning 1 out of 2. You always pick the 1 out of 2, if you want to win, so you essentially repick the only choice that is from the 1 out of 2.
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Thread: Simple puzzles to stump people

09182007, 10:28 AM #41

09182007, 10:29 AM #42

09182007, 10:34 AM #43OberonGuest

09182007, 10:36 AM #44

09182007, 10:37 AM #45
I'm still trying to figure out if the Blue cabbie was drinking heavily the night of the accident because his girlfriend broke up with him, or whether some old lady propped up on cushions dropped her dentures on the floor and accidentally cut him off on the highway.
All right  just so you know, I am in "goofy Ne" mode, so take everything right now with a grain of salt. You can blame Oberon."Hey Capa  We're only stardust." ~ "Sunshine"
“Pleasure to me is wonder—the unexplored, the unexpected, the thing that is hidden and the changeless thing that lurks behind superficial mutability. To trace the remote in the immediate; the eternal in the ephemeral; the past in the present; the infinite in the finite; these are to me the springs of delight and beauty.” ~ H.P. Lovecraft

09182007, 10:38 AM #46
I think I have it now. The problem is the assumption that the choice is based on 3 doors. It only appears to be based on 3 doors at first. It is true that when you first make the selection, any of the 3 doors could potentially contain the car. However, once 1 of the doors is opened, you know that that particular door doesn't contain the car. Now you know that the correct door is either the door you have already chosen, or the door you haven't chosen. All that is certain is that the opened door doesn't contain the car. Are you really going to tell me that it isn't possible for me to have selected the door with the car simply because the opened door didn't contain it? It's clearly the opposite.

09182007, 10:47 AM #47
Up to here, this is the case. So, essentially you have, at this point, even with the other door missing, a 1/3 chance of winning. Why? Because it doesn't matter if you have the right door or not, a door would be removed. IOW, the likelyhood that the right door is still left is 50/50, but your door is still only 1/3  by switching, you ar picking the 1/2 door and not staying with the 1/3 door.
I know, counter intuitive. But think about that configuration  he's not going to take the car away but he is going to take away one of the doors... no matter what. And he can't take your door away, so your door plays by different rules.
All that is certain is that the opened door doesn't contain the car. Are you really going to tell me that it isn't possible for me to have selected the door with the car simply because the opened door didn't contain it? It's clearly the opposite.
(Oberon, I'm just tackling it from different perspectives )

09182007, 10:48 AM #48OberonGuest
Okay, here's the deal:
You make your choice out of three possibilities. When your choice is made, there are two sets of doors to evaluate, those being the set of doors you did pick (comprising one door) and the set of doors you didn't pick (comprising two doors).
Now there are a couple of things you know for sure. The door you picked may be the correct one (a 1 in 3 chance), in which case the other set of doors contains two goat doors, or the door you picked may be the wrong one (a 2 in 3 chance), in which case the other set of doors contains one car door and one goat door.
In any case, the other set of doors contains at least one goat door.
So when Monte opens one of those doors to reveal a goat, nothing has changed. You already knew the set of unpicked doors contained at least one goat door. Knowing which door in the unpicked set was the goat door does not change the basic odds...1 in 3 you picked correctly, and 2 in 3 that you did not.

09182007, 10:49 AM #49OberonGuest

09182007, 10:54 AM #50
Okay, wait a minute.
In the first place, I have a 1/3 possibility of choosing the car. Will you admit that it is possible that I would choose the winning door out of them?
Now, he opens the one of the doors I didn't choose, showing one of them to be a goat. Since the original odds were that I had a 2/3 probability of choosing the goat, and he eliminates one of them, and I switch... then I will win 2/3 of the time, because the chances that I originally chose the goat rather than the car are 2/3.
Did I finally get it? It certainly took me a long time... was I much slower than the average person?
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