You have three doors to choose from. One to freedom, two to certain death. You must choose. However, after you have chosen, your tormentor opens one of the remaining doors unto certain death and asks, 'Do you want to keep your initial choice or change?' You now have two choices left, the one you picked, or the other door.
Common intuition says that originally you had a one in three chance of choosing the right door. Then, after the tormentor's intercession, you have a one in two chance - the door you picked, or the other door. In other words, 50/50.
It turns out that this is not the case. You are much better off switching your choice to the other door. That has kept me awake tonight. What is going on?
The first assumption is correct. You have a one in three chance of choosing the correct door when the 'game' begins. However, by making your choice, you force your tormentor into a tricky position.
There is a 1/3 chance that you chose the right door to start with. If that is the case, your tormentor can open either of the other doors and if you choose to switch, to your death you go. That is, there is a 1 in 3 chance that you chose correctly, and changing your choice will send you to the pits of hell.
However, there is a 2/3 chance that you chose the wrong door to start with. If that is the case, your tormentor has given you the path out by eliminating the only remaining path to death. That is, if you switch, the only choice is life.
Therefore, given the choice, you should change doors. You have a 2/3 versus 1/3 chance of survival by doing so.
History is important. The rules of the game are important. By only looking at the choices in front of you now, you can be gamed. By looking at how you got here, you can make a wise choice. (Somewhere in the back of my mind, this seems to relate to Greece at the moment.) Perhaps I can sleep now?
I'm sure when I awake in the morning this will confuse me again. But now, I'm high on Limoncello and it all makes sense. Except the bit as to why it doesn't make sense without thinking and drinking about it to excess.
Such is life. Good night. And Greece - tell em' to fuck off. They are bad playmates. You'll get over it before they do.
4 comments:
So the strategy is to pick the wrong door originally and change doors.
You don't have that luxury. Pick a door, any door. Then after they show you a wrong door, change picks. That is the strategy!
I knew this would happen! In the morning, my logic was fuzzy again and I couldn't see with the drunken clarity of the early hours. This may help.
In the original game, your choice gave you a 1/3 chance of survival, and a 2/3 chance of death. Let's say your tormentor allowed you the following choice: Keep your initial choice, or choose BOTH the other doors, surviving if EITHER were correct. Obviously (if you have an honest tormentor), this is a better choice - a 2/3 survival rate against 1/3. It is the same as being able to choose two doors initially instead of one. Yet this what the original game is equivalent to, because the tormentor gives you the choice of BOTH other doors by showing you one is a literal dead-end.
Change the rules of the game. You pick a door (any door) - you have a 1/3 chance of surviving. Now, your tormentor shows you one 'death door' - but this time, it can include the door you originally chose. If your initial choice was wrong, the tormentor can show you this, or the other death door. Given these rules, you only have a 50/50 chance of survival (as intuition would suggest). To change choices or not is an even bet.
In the original scenario, the tormentor gives you information (subtlety) by telling you which out of two doors is the wrong choice (you don't know squat about your initial choice, but you are certain that if your initial choice was wrong, what the right choice is, i.e. you don't know if your 1/3 bet was good, but if it isn't, you damn well know your 2/3 bet is). In the the latter scenario, the tormentor gives you information by telling you which out of three doors is the wrong choice (if you find out your 1/3 bet was wrong, you have no information about the remaining two choices. If you don't know if your 1/3 bet was good, you have no information between your initial bet and the other choice.)
Even now, I still don't quite 'get it'. Perhaps that is because all the morning alcohol has run out.
It doesn't matter if originally you pick the right or the wrong door (in fact better the wrong, but that's just stress) - by changing doors you improve your chances to 2/3. You can't argue with NannaCop! and you can't always make the right choices. Fuck, if i could make make the right choices 2/3 of the time, I'd be pretty happy with myself :)
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