## Tuesday, April 8, 2008

Human intuition is a very bad at reasoning about certain problems. For example, consider the classic "Monty Hall" problem: there is a game where there are three doors. One of the doors has a prize behind it, and the other do not. You pick one door, but don't open it. The host of the show then opens another door that 1.) is not the one you picked and 2.) is not the winning door. So now you've picked one door, and another door is now revealed as not the winning door. There are now two unopened doors, the one you picked and another one. The host offers you the chance to switch your choice to the other unopened door. Does it make a difference to your chances of winning if you do so? The surprising answer is yes. You go from having a 1/3 chance of winning to having 2/3 chance of winning.

Now consider a slight variation on that. You've made your initial choice of door, when an earthquake hits and happens to open a door that 1.) is not the one that you picked and 2.) is not the winning door. The host then offers you the choice: do you want continue to bet on the door you initially picked, or switch to the other unopened door. Does it make a difference to you chances of winning? Perhaps even more surprisingly, no. Your odds are 50/50 either way!

Anyway, it took me a long time to see why the first is the case, although I was able to finally convince myself by argument why it was the case, instead of by brute mechanical force. However, to figure out the answer to the earthquake scenario, I actually plotted out on a piece of paper all the possible outcomes. Only after doing that was I able to reason out why it was the case, working backwards (I ended up also plotting out the possible outcomes of the first case (the "classic" Monthy Hall problem, too, in order to compare them.)

Oh well, I just wasted like an hour and a half just thinking about the second problem. (And about as long thinking about the first one, but that was about a year ago).