The Monty Hall Problem

[emmline]

Ah so, Caj. That makes sense. Still, even with 99 doors there's a part of me wishing to switch the problem, at that point, to a 50/50.

But the simulations clearly bear out what you're saying.

[spittin_in_the_wind]

I love the banjo.

Robin

[Caj]

BTW, what makes this problem so interesting is the ugly responses sent to Marilyn Vos Savant, over this puzzle and some other puzzles she printed. She received hundreds of condescending and insulting letters telling her that her column was "obviously" wrong, and that she should take remedial mathematics. This is the world-record holder for the highest recorded IQ---and of course, her column was right.

The level of animosity aimed at Vos Savant is kind of scary. On Sci.math, we have to deal almost monthly with people who storm in to complain about one of her puzzle columns with a counterintuitive answer. People just assume she's wrong when a puzzle has a surprise answer, and they're always trying to prove that she ain't that smart. There's even a "Marilyn is Wrong" website, which once sold "Marilyn is Wrong" t-shirts. The alpha-male who runs the thing never really finds any real mistakes, but pounces on nit-picks that don't really matter in story problems. He seems to get pretty angry sometimes.

Caj

[Monster]

Hey Darwin, what do you figure the probablility is for turning this into a banjo thread?

I looked at the Weekender's banjo http://www.tinwhistletunes.com/clipssni ... ing05.html ) and man does he look happy playing that thing, just scroll down to the very last pict. and you'll see what I mean.

Sorry about the deep ruminations about how some instruments compare themselves to others, if I've upset anybody I apologize.

Five string is good for bluegrass of course, but actually I play four string, plectrum style and tuning.. (like a five string except the little tiny fifth string is absent, also the low D string is tuned to a C instead heh heh) Plectrum banjo is sometimes used in Dixie-land bands. Eddie Peabody was the sole promoter of the instrument, really can't think of anyone else except some guys in St. Louis that I used to play with that ever would attempt such a thing. Oh, just remembered, Buddy Wachter is the instrument's spokesperson thesedays.

Any other banjoist's out there lurking in the shadows? Banjoist's come forth! Do not be afraid, all makes and musical styles accepted! Banjo is a noble light-hearted and fun instrument. Beware of the naysayers from the dark side that will try and trample over your banjo playing will. Rember now, there are even r*c*rd*r players that have owned up to it on this forum.

[Monster]

I love the banjo.

Robin

[Darwin]

On Sci.math, we have to deal almost monthly with people who storm in to complain about one of her puzzle columns with a counterintuitive answer. People just assume she's wrong when a puzzle has a surprise answer, and they're always trying to prove that she ain't that smart.

I probably shouldn't admit how much time Army time I spent trying to trisect an angle using just a compass, when I shoulda been defending my country.

There are lots of interesting people (notice how bravely I refrained from saying "nutcases") who show up on sci.lang with all sorts of original ideas about languages, and then abuse the resident professionals when their ideas are refuted. Some of them have become institutions in their own rights (when they should probably just be institutionalized).

[BoneQuint]

That's the trick. By opening all the other doors but one, I am really saying, "if it's behind any of these other doors from #2 to #99, then it's behind this door."

Yeah, that's good. Here's another intuitive way to understand it.

Say you pick a door, and Monty asks if you want to switch, but doesn't open one of the other doors or give any other information. In that case, the odds obviously don't change, any one of the other two is just as likely as your original pick.

But now imagine he asks you if you want to switch, but adds, if you switch, and happen to pick an empty one, I'll let you have what's behind the third door. So you have two shots at it, clearly a better deal than not switching.

Revealing the empty door is exactly the same as that deal.

[MarkB]

I don't think that it has anything to do with probability. I really think that Dale or a family member is going to Los Angeles to be on the Price is Right show

MarkB

[tcm]

I understand the mathematics and it is clear empirically that Marilyn was right. But I can't SEE it.

Dale

I "visualize" the problem this way: Because Monty always opens a non-winning door, I can't ever switch from a losing door to another losing door. So, switching is equivalent to saying "I think my first choice was not the winner". If I stay with the door I originally chose, I am betting "my original choice was the winning door" (a 1/3 chance). If I switch, I am betting "my original choice was not the winning door" (a 2/3 chance).

Tom

[lyrick]

I think of it similar to Tom (tcm). My original choice of doors has a 1/3 chance, the other two doors combined have a 2/3 chance. Once Monty eliminates one of the other two choices, that 2/3 chance is now totally in the remaining door, so if I switch I have a 2/3 chance, if I stick with my original choice I've only got a 1/3 chance.

[Dale]

Yeah. I keep thinking about it like this. Assuming the host doesn't know where the prize is, at the point of being offered the chance to switch, I "know" that the chances are 50/50 that the prize would be behind my original choice vs. the unrevealed door. So, it doesn't matter whether I switch.

I know that's wrong. The simulations have convinced me. It's just the way I think about it.

In a real life appearance on Let's Make a Deal, I'd switch every time because I would figure that the show knows where the prize is which, of course, changes everything.

Dale

[emmline]

The thing here is that the host does know.

I always yelled at contestants, on "Who wants to be a millionaire" who, when stuck on a question, would say aloud..."Well Regis, I'm certain the answer is either a)stewed prunes or d)apple butter...I just can't decide betwixt'em! Guess I'll use my 50/50 life line."

Guess which choices are then removed? Of course, b)marshmallows, and c)necco wafers, which the silly fool wasn't considering anyway!

[emmline]

BTW...I've enjoyed and appreciated all the contributions to this thread which have helped me grasp how the probability works here.

I tend to trust the mathematical explanation, even before being able to get it visually. I envy people who can think mathematically. I first noticed a discrepancy in 2nd grade, when other kids were able to compute, eg, 4 x 8 considerably faster than my clunky method of breaking it down into 8 4's, which I would then painstakingly add together.
I think numeric thinking is to me as literacy is to a dyslexic. I can do math correctly and competently, but it takes a slower, more convoluted pathway for me to arrive at the answer.

But it's a cool language, and I wish I spoke it more fluently. I find it remarkable that numerous pieces of the particle physics puzzle have been proposed based solely on mathematical likelihood...proposals that weren't born out by experimental evidence until years later.

I ignored the graphing calculator on my Mac for a long time, having little use for it, until I started fooling around with it once, and discovered that a very simple looking equation, when plotted on a 3D grid, can create a complicated flower, or other shapes which occur organically in nature.

Being dysnumeric, I couldn't have gone into this as a field of study, but I find it exquisite that numbers can so accurately describe the universe, and cool that there are human brains capable of thinking in that language.