Let's Make A Deal

Last night Monnie and I were watching Deal or No Deal. I had planned on blogging to you guys about how the game works - it's a combination of math and luck (who wouldn't like that?) and I recommend checking it out, but when I went to look up a bit more info on the game, I came across a problem that's always been one of my favorites - how big of a math nerd are you when you can call a problem "one of your favorites"?

It's commonly referred to as "The Monty Hall Paradox". If you ever watched "Let's Make a Deal" you're familiar with corny gameshows... another thing you're familiar with though is a game played on the show. You get called up to the front to play and are presented with three doors. Behind one of the three doors is a fabulous prize. Opening either of the other two doors reveals a goat. I'm not sure if you actually get to keep the goat - but the idea is that you lose.

Anyway, you select one of the three doors. Instead of opening your selected door, the host opens one of the other doors - behind which is always a goat. He then asks you: Would you like to stick with your original door? Or would you like to switch to the other door that we haven't shown?

The question is:
Do the odds say that you should stick with your original door?
Do they say that you should switch?
Or do the odds say that it's all on you and there's no difference?

The answer that most people come to initially is: it doesn't matter. There are 2 doors left. 1 has a goat and 1 has a prize: you have a 50/50 chance and switching is irrelevant. If that was your answer, you were... WRONG!

The odds say that by switching doors: your odds of getting the prize double!

The most brief explanation possible is that there are three possible outcomes when you switch and 2/3 cause you to win:
* You pick the prize right away - switching makes you LOSE.
* You pick goat 1 right away - switching makes you WIN.
* You pick goat 2 right away - switching makes you WIN.

If you don't switch, there are three outcomes. Only 1/3 make you win:
* You pick the prize right away - staying makes you WIN.
* You pick goat 1 right away - switching makes you LOSE.
* You pick goat 2 right away - switching makes you LOSE.

If I tried to explain it further I would basically be copy/pasting from here, so just check there for the details and reasoning. It takes a while to get your brain around it, even after thinking about it for a while. but that site explains it in several ways, so eventually it'll have you convinced! ... and even if you read all their logic and still don't agree: they have results from a computer simulation that demonstrates it also!