# let's make a deal

2006.07.04
Boingboing linked to a new free eBook on probability... in their writeup, the mention the Monty Hall problem. 3 doors... goats are behind 2 of them, and a new car behind the third. The player picks a door. Monty then opens one of the doors, revealing a goat. Should the player switch doors? (Assume an equal chance for the 3 doors to hold a car, and that Monty will pick one of the 2 goat doors at random if the player has already picked the car.)

The article mentioned that Marilyn vos Savant "encouraged her readers to simulate the game and draw their own conclusions"... Well, here's a simulation! You can modify the speed to run lots of simulations, "Wargames"-finale style. You can select always switch, never switch, or some probability of switching.

 events: switch: never 1 in 3 50/50 2 in 3 always results: speed: stop step slow fast zoom

2020 UPDATE: Jim Holt's "When Einstein Walked with Gödel" provides one of the best summaries of why you should switch, and I feel I "get" it now in a way I don't remember if I did when I wrote this simulation:
Counterintuitively enough, the answer is that you should switch, because a switch increases your chance of winning from one-third to two-thirds. Why? When you initially chose door A, there was a one-third chance that you would win the car. Monty's crafty revelation that there's a goat behind door B furnishes no new information about what's behind the door you already chose--you already know one of the other two doors has to conceal a goat--so the likelihood that the car is behind door A remains one-third. Which means that with door B eliminated, there is a two-thirds chance that the car is behind door C.
2024: let me try and reframe again. If there was no door-switching, the game would be easy to figure out: 1/3 chance you win, 2/3 chance you lose.

So when you "stay", it's STILL that game. Monty Hall can open that curtain or not - nothing has changed. 1 in 3 chance you win, 2 in 3 chance you lose. It's a one round game, and showing you a losing curtain changes nothing, because you're not changing your vote.

But when you switch? It's a new round - but the win/lose result will be *EXACTLY* OPPOSITE of what it was if you stayed. If you picked the car at first as round 1, the new round switches you to being a loser. But if you picked the goat in Round 1 - and remember there was a 2 in 3 chance you picked a goat... in round 2 you are a winner!

It feels like Monty Hall's reveal should change the game to 50:50 or something (which it would be if you started from scratch at Round 2) but instead it is still based on the odds of Round 1, but what was one of 2 losing pick now always wins, and the single old winning now loses.