Monday, 13 July 2015

Monty Hall at War

I am unsatisfied by my previous understanding of the Monty Hall dilemma. I can understand why the solution I put forth (not my own, but someone else's) is correct, but I don't clearly understand why common intuition is so wrong on this point. My concern is, if faced with a similar dilemma, I would not be able to convince myself or anyone else that common sense was, in this case, wrong.

So I'm going to re-work the problem using a slightly different metaphor.

During the American-Vietnam war, there was a village in South Vietnam called Lưu Nó. They receive a message saying the B52s are on their way and that they have to evacuate. There are three paths out to other villages: A, B and C. They know that two of the paths out of town are heavily mined, and one is free of mines. They don't know which one is free of mines. They hastily call a village meeting to decide which path to take. Given their lack of knowledge, the choice is for all intensive purposes is random - let's say A.

Given what they know, they have a one in three chance of getting out alive (staying in the village gives zero chance of living). Suddenly one villager remembers they have a radio with which to call central command. Perhaps they have more information about where the mines are?

Option 1: They radio central command, and a nice, but not well informed minion informs them that he knows path A is mined. He has no information about the other two paths. The villagers have gone from certain death, to a 50/50 chance of death. (They know either B or C is mined, but not which one.) If this minion had informed them that either B or C were mined (but had no further information) they would have gone from having a 1 in 3 chance of survival to a 1 in 2 chance.

Option 2: They radio central command, and a completely informed minion answers, but he is a dickhead. He knows exactly which roads are mined, but doesn't want to tell them. But like most dickheads, he is also a smart arse. He knows they have chosen path A, and let's them know that he won't provide any information about that path. However, he does say, "I can tell you that path B is mined." The villagers know two two things: i) path B is bad, and ii) the minion is a dickhead and knows the situation about path A, but is not going to tell them.

They call another meeting. Someone stands up and says, well, we know B is bad, and that leaves A or C. It's a 50/50 choice, so let's go with what we decided upon. But then the local school teacher stands up and says, hold on! Before we had three choices, one path, A, B, or C was safe - each equally likely. That dickhead knows which is safe, but isn't going to tell us about the state of path A. That gives us some leverage.

If A is safe (1 in 3 chance before we called the dickhead), he would have told us either B or C was dangerous (though both would be dangerous)
If A is not safe (2 in 3 chance before we called the dickhead), he would have to tell us that B was not safe either, and by default, that C IS safe.

Given the above, C is dangerous 1 in 3 three times, and safe 2 in 3 times!

The school teacher, though not quite sure of her logic, walked along path C with her students, and by chance had it, survived. The rest of the villagers being stubborn, walked along path A and were never heard of again.

The moral of the story is, always listen to dickheads. They give stuff away by being smart arses.

1 comment:

Scratchindog said...

Another comment on this problem can be found here: