21 was a movie based on a book about the MIT Blackjack Club, a group of students who were trained by a professor to count cards at Blackjack tables.
It’s a true story. They made lots of money until they were caught.
The movie has a scene in which Professor Kevin (shown to the left) figures out that a student of his is a mathematical genius and therefore worthy of a spot on his team.
Something about that scene has been bothering me since I saw it.
I started to look up the answer, but I thought it might be interesting if I wrote what I was thinking before I looked it up.
I believe the scene where the professor demonstrates “variable change” in probabilities is complete crap.
Here’s what happens:
The professor says to the class to pretend they’re on a game show and the host presents them with three doors. The good prize is behind one of them and there are goats behind the others.
He says that the host asks you to make a choice.
So you choose door 1.
Then the host says to make it interesting, he’s going to open door #3 — it’s a goat.
Then the host asks if you want to change your mind.
The student and hero of the story says, “YES.”
The professor asks why.
The student says, “To account for variable change.”
The idea is that by opening the 3rd door, that the host changed everything and now it needs to be figured out again.
I believe this is complete bullshit unless you are going to take into account why the host opened the 3rd door.
If it’s probability, then the chance on the first try is 33% each door. On the second try, it’s still 50-50.
If you want to consider something like, “Why did he open that door? Does he know I have it? Does he want me to change? Maybe I should stick? Wait, he wants me to think I should stick, so I should change.”
Psychology aside… The location of the good prize DID NOT CHANGE. There was no variable change, so the position is static, and unknown, and it’s behind one of two doors… FLIP A COIN!!!
** OK, NOW I AM SEARCHING THE INTERNET FOR THE ANSWER **
Ok, here is the answer from Yahoo Answers.
The key is we need to understand that the host knows where the car is. Ben picks door number 1, so at that point he has a 33.3% of being correct (so odds are against him because he has a 66.6% chance of being wrong). The host knows whether he is right or wrong, but he reveals one of the doors that was definately wrong (door 3).
Now it does seem like regardless to what the movie said there is a 50% chance of him getting it correct now.
But remember, before door 3 was revealed there was a 66.6% (in favor) that the car was in door 2 or 3. The host HAD to pick a wrong door out of 2 and 3, and since it is MOST LIKELY in either door 2 or 3 (before 3 was revealed), by eliminating door number 3 the chances are greater that is in door number 2 because ORIGINALLY the chances were better (66.6%) that it was in door 2 or 3.
If you still don’t understand, look at it this way:
Bens pick – Door 1 (33.3%)
Other pick – Door 2 and 3 (66.6%)
Now say your a bystander, are you going to bet that Ben is right? or that its in either door 2 and 3?
Your going to go with the door 2 and 3 because chances are better. The host CANT REVEAL Ben’s pick obviously, and the host CANT REVEAL the car, so he shows you that door number 3 doesn’t have the car. So theres still a 66.6% chance that Ben’s choice is wrong.
If the host didn’t know what was behind the doors and randomly picked number 3 and it happened to be a goat, then yes he would have a 50% of being right. Totally different story.
This is still complete CRAP. This is exactly what was explained in the movie, but they’re just wrong.
It makes no difference that the host knows whether the choice was right or wrong.
** UPDATE HERE IS A LONG DISSERTATION ON WHY THEY SAY I’M WRONG ON WIKIPEDIA: http://en.wikipedia.org/wiki/Monty_hall_problem
I still don’t believe it. YES. It makes sense mathematically. I must be missing something here. I understand the host is going to know where the car is. I know the host is going to reveal a goat.
Does the host ALWAYS give the choice to switch? If so, then it doesn’t change ANYTHING.
Oh wait… It’s starting to sink in.
Wait for it…
Wait for it…
I’ve got to think about it backwards.
Ok, fine.
I was getting ready to write a computer program to prove they’re wrong, but it’s going to prove they’re right over a number of tries.
When picking randomly one out of three choices, there is a 2 out of 3 chance you are wrong. To the prize is most likely behind one of the other two. Then they eliminate one of them, so it’s most likely behind that remaining one… you should switch.
66% of the time the car is behind the other door. Switching would cause then to win the car.
So if you run trials in your head, you see that 66% of the time you win by switching.
You should switch.
I’m sorry, Mr. Scientist.
I’m going to steal your girlfriend!!
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