Statistical Mythology - TRUE! - Nothing to See Here
Jul. 24th, 2008
09:45 pm - Statistical Mythology - TRUE!
We're watching the movie "21". It seems like it has the potential to be an interesting story.
But then ... the story starts out with the main character's induction into a group of blackjack card counters at his university, MIT, because he wows his professor by first spouting history about Newton-Rhapson method (which apparently people in the US refer to only as Newton's Method! Huh!), which is fine,
but then by spouting a STUPID logical fallacy, which the professor agrees with him on.
Well, the question is:
You choose a door.
The host opens one of the other 2 doors, showing nothing.
He offers you the option of changing your guess.
Should you change it?
... so the one you chose first was 33.3%, which means the other remaining one has become 66.7%!
I've heard this fallacy often enough, back in college at least.
It makes me really angry that a movie which a lot of people who don't know any better will see depicts a supposedly top-flight MIT student and an MIT professor proclaiming this as statistical fact. I hope I can enjoy the movie in spite of this early huge screw-up.
So, yeah, I was wrong. Argh.
I take it back. Nobody has ever succeeded in explaining this to my satisfaction. It's always sounded like hand-waving (hm, how zen). The statistical explanations have always sounded like voodoo.
I just worked it out on paper, and it actually ... does work that way. WEIRD. But true. I shall explain. Perhaps my reasoning (since it's the only reasoning which has persuaded me) will help persuade others
I've heard people talk of the fact that the host is an intelligent agent, and therefore not subject to statistical analysis and so on and so forth, but none of this has ever pointed out the real reason why switching is best:
In 2 out of 3 cases, you have forced the intelligent agent to take one, with no option. He only has a choice 1 out of 3 times!
When you have forced the agent's hand, the remaining card is it. When he had a choice of which card, the one you picked first was it.
Weird? Yes, but true. I came to this realisation after I jotted down the possibilities.
I = your initial pick
E = choice eliminated by host
S = the switch option
|Door Layout||Dealer Option 1||Dealer Option 2||Initial Outcome||Switch Outcome|
|1 0 0||1(I) 0(E) 0(S)||1(I) 0(S) 0(E)||always win||always lose|
|0 1 0||0(I) 1(S) 0(E)||--||always lose||always win|
|0 0 1||0(I) 0(E) 1(S)||--||always lose||always win|
(You can re-order things so that you don't always choose the first door, but that is statistically irrelevant, assuming the prize door is properly randomized).
So, ur, wow. 2/3 of the time, the agent was forced to point out the right door, effectively.
Weird, but no longer pure voodoo. If someone had shown me what I worked out, I would have believed them. Other people's explanations now make some sense, in hindsight, but really didn't in the absence of this realisation.
I guess I owe the makers of the movie an apology!
[Edited to fix my mismatched legend and use of symbols! Oops!]