Monte Hall Problem

Izzy and Tom were discussing the Monte Hall problem of game theory when I walked by today.

The Monte Hall problem is based on the show Lets Make a Deal. Some poor slobs come forward in strange costumes and are offered a choice of three doors. Behind one is a fabulous prize. Behind the other two are pieces of crap referred to as "goats".

The slobs picks a door. Monte then reveals a goat behind one of the doors they did not pick. He then asks whether they would like to change their mind and switch to the other door. They have the option of sticking with their original choice or switching to the other door.

Should they stick or switch?

Think.
Think.
Think.
I'm writing these "think"s so I don't automatically give the answer away by the way.
Think.
Think.
Think.

The answer is that they should always switch. By showing the goat behind one of the unpicked doors, Monte aggregated the probability of both unpicked doors into the remaining "switch" door. So the switch door is twice as likely to be right as the "stick" door. If you didn't get it, don't worry I didn't the first time either.

To better illustrate why, imagine Monte had a million doors. Thats 1,000,000 just so we don't get confused with the british million. You pick a door. Monte then opens 999,998 goat doors. He could do it in sequence or not. It doesn't matter. What is the chance you picked the correct door on the first chance? One in a million. So the other door is almost certainly the prize. You should obviously switch.

The difference between this example and the Monte Hall problem is only one of degree.

This is not to be confused with the Monte "Haul" problem in role-playing. The Monte Haul problem is a question of how much you give the players as a reward for completing the quest. Too much and they get too powerful. Too little and they lose motivation. It does not have as elegant a solution.