A fun probability thread - the Monty Hall Problem

I think we've done this one before here. I always love these arguments, especially because this one has a demonstrably correct answer. 

On stage, there are 3 closed doors. Behind 2 of the doors, there is a first generation Prius. Behind 1 of the doors, there is an FM Miata with an LS7. There is a judge, who can see behind the doors. The judge does not lie. You get to pick one of the doors, and the car behind it is the car you have to drive for the next 5 years. 

You pick door 1. The judge then opens door 3 to reveal one of the Prius's. He gives you the option to switch to door number 2. 

Do you switch? 

Mythbusters did that one:

When presented with the Monty Hall Problem, people tend to stick with their first choice.

CONFIRMED

After they built a game show mock-up set at a local theater, Adam acted as a game-show host and had 20 volunteers play a game of “Pick a Door”. Once a player chose a door, Jamie opened an empty one and Adam offered the player a chance to switch; all 20 stayed with their original pick, many of them believing that they had a 50-50 chance to win at this point.

When presented with the Monty Hall Problem, people would be more likely to win if they changed their decision.

CONFIRMED

They built a small-scale simulator to do 50 trials each, with Adam always switching his choice and Jamie never switching. Adam won far more often than Jamie did, and Jamie explained the reason: because the player has a 2/3 probability of choosing a losing door at first, switching turns the odds in his favor.

I leave sad knowing I have not won the Prius. 

The Miata can't do what I need the Prius to do and the Prius does it well. 

In reply to mtn :

This is the classic Monty Hall problem, and the answer is to switch.

If you don't switch, you have a 1/3 chance of picking the correct door. If you switch you have a 2/3 chance of picking correctly, because effectively your first guess is, "I'm guessing that this is *not* the winning door."

Unless of course, you'd rather be driving a 1st gen Prius for 5 years. It would arguably be a more practical vehicle.

John Welsh said:

I leave sad knowing I have not won the Prius. 

The Miata can't do what I need the Prius to do and the Prius does it well. 

We can switch it to 2 Del Sols and a brand new Prius V Wagon if you want...

"Are you trying to Monty Hall me? It's unbelievable. I don't need Monty Hall ruining my place of work when Monty Hall has already ruined my home life."

For you weirdos who'd rather have the Prius: 

In reply to mtn :

Haha. I like your goat. 

The part that screws up the math in this one is the humans. 
 

Monty Hall knows the right answers, and will never reveal the car. So, he is changing the odds with his knowledge.

If you picked the right door originally (which is only a 1 in 3 chance), then he eliminates one of the wrong ones and your odds remain the same. 

If you picked the wrong door originally (2X as likely), Monty eliminates the other wrong one, and your odds improve.

Is there a dead parrot in one of the cars?  Oh wait, that's the Monty Python problem.

I'll show myself out. 

No need to perform experiments or assume Monty is rigging the game. Just draw a tree diagram. It's totally counter-intuitive but it's real. 

This is obviously a thought experiment. We never built a Miata with an LS7. 

Monty's not "rigging" the game intentionally, but he is removing one of the 2 wrong options halfway through. 

SVreX (Forum Supporter) said:

The part that screws up the math in this one is the humans. 
 

Monty Hall knows the right answers, and will never reveal the car. So, he is changing the odds with his knowledge.

If you picked the right door originally (which is only a 1 in 3 chance), then he eliminates one of the wrong ones and your odds remain the same. 

If you picked the wrong door originally (2X as likely), Monty eliminates the other wrong one, and your odds improve.

Yep... that's the big question. Do the rules require Monty to open a door with a low-valued prize after the initial guess, or does Monty have the choice to open any door or no door as he pleases? If it's the latter, this turns into one of those "I know you know I know" situations. Does Monty think you are the sort who would switch, or does he think you're the sort who would stay with your first choice?

I switch! I switch! I switch!

I'm literally teaching a stat class today and the topics are data distributions & process capability analysis.

I'll be taking a dozen engineers through a simulation of the Monty Hall problem in a few hours...I'll post a pic of my set up later (class is starting now).   

...what are the odds

In reply to SVreX (Forum Supporter) :

Ah, yes, if you put it that way. He prunes the tree. 

In reply to MadScientistMatt :

If Monty ever opened a door with the car, there would be no need to switch. Unless you happened to like the other goat better!  (and the odds would ALWAYS remain 1 out of 3). 
 

Monty always reveals a goat. That's the "trick". 

I always watched Monty Hall......and never figured it out.....

well it was lunch time and my toasted peanut butter sandwich was ready :)

Monty knows where the "goat" is and also the big prize is , Montys job is to make you change to the lesser prize .

has anyone gone thru the old shows and see how many switched ?

Cheech & Chong - Let's Make a Deal

In reply to californiamilleghia :

That makes it sound like it's rigged, and Monty "coerces" you into choosing the lesser prize. 
 

It's not rigged. The rules never change. And Monty doesn't have to know where the big prize is. He just has to reveal 1 lesser prize, and the odds of sticking with your original choice go down. 
 

Human nature... we THINK the odds didn't change, and that the odds are now 50/50 (they are not).  We also THINK someone is trying to get us to change our mind, so we don't.  Which leaves us choosing to stick with our original choice (which is now statistically less probable). 

I think doing some datamining on the old shows would be really interesting. And was this show really just a long-form psychology experiment?

In reply to Keith Tanner :

Absolutely. 
 

But the real psychology experiment was not with the contestants. It was with the viewers. 
 

We all understand now the value of product placement. Monty and friends developed the idea in the first place. 

We still love "free" stuff, and will give away all kinds of freedom and personal info just to get some "free" stuff.

My mother won a car on a show like this. 
 

She watched it for months, and figured out her best odds of being selected as a contestant would be if she was pregnant. 
 

She waited until she was 8 months pregnant with my little brother, and came home with a car!

SVreX (Forum Supporter) said:

In reply to californiamilleghia :

That makes it sound like it's rigged, and Monty "coerces" you into choosing the lesser prize. 

well  the best outcome as far as a TV show is that  the player changes his pick and "loses" the car , 

I wonder if there was ever a book  telling what limits Monty had  and if he knew which door the "grand prize' was behind , plus other  "rules"

This was on after the scandel of quiz game fixing , so there must have been rules .

Keith Tanner said:

This is obviously a thought experiment. We never built a Miata with an LS7. 

This is the biggest oversight in the thread, Keith.