The puzzle: You build a time machine to travel back to The Coliseum in LA in 1986 to see Notre Dame beat USC 38-37. Unfortunately there's a glitch, and you end up instead in the Coliseum of Rome in 86AD, in the middle of one of Domitian famous games. You are grabbed by a Centurion and brought into the arena, where you are forced to play the following cruel game:
You are shown a wall with three doors. Behind two of the doors there are ravenous lions. Behind the other door is a scroll offering you Roman citizenship, a guarantee of safety. You are asked to select one door. As soon as you have done so, the host of this game, Probabilius Maximus, opens one of the other doors, revealing one of the lions. He then asks you if you would like to switch your choice of door from your original choice, to the other unopened one. At this moment the Centurion whispers in your ear ``you know, he would have opened one of the lion doors for you, no matter what door you selected - he always does''.
Assuming that you desire the citizenship scroll (and not to be eaten by a lion), what do you do? Specifically:
A solution: It is better for you to switch. There are many ways to make the argument; here is what I think is the simplest: 1/3 of all times (on average) that you play this game, you will choose the right door initially, and if you switch you certainly lose. But 2/3 of the time, you initially pick one of the wrong doors, and it this case, because Probabilius Maximus has no choice but to open the other wrong door, when you switch you certainly win. So 1/3 of the time, switching is bad, and 2/3 of the time it is good. On average, it is better to switch.
What's going on? One way to think of it is this: you are genuinely getting some information from Probabilius Maximus when he opens a lion door, at least in a probability sense; with probability 1/3 (when you select the right door) you are not getting any information, because Probabilius Maximus has a free choice of doors to open, but with probability 2/3 (when you select a wrong door) you are forcing his hand and getting him to reveal exactly where the scroll is. Of course, during an actual play of the game, you don't know whether you are in the 1/3 or the 2/3 situation, but at least you do know that you are twice as likely to be in the 2/3 (better) situation than the 1/3 (worse) situation; so if you have to choose, it's better to assume that you are in the better situation.
This puzzle is commonly known as the ``Monty Hall puzzle'', as it it is loosely modelled on a game from Monty Hall's tv show Let's Make a Deal. It was popularized when it appeared in Marilyn vos Savant's column in Parade magazine, in 1990. It generated quite a storm among professional mathematicians, many of whom greatly embarassed themselves. Have a look at this website to see the amazing details of the controversy. Google ``Monty Hall problem'' for innumerable websites discussing the problem and its variants.
Winner was be decided by looking at the current value of the Dow Jones average, as recorded by Yahoo!. It's quoted to two decimal places, so we multiplied it by 100 to get an integer, and looked at the remainder on division by 13.