Speaking of Monty, here’s a little problem to keep your head spinning in the world counterintuitivity.
http://www.cut-the-knot.org/hall.shtml
Here’s a much better simulation than the one above:
I can’t decide whether to go with link number one or link number two. I’m hoping for a giant Radio Flyer red wagon.
I went with link 2. Interesting simulation. Looks like Marilyn and her prodigious brain were right. I still don’t get it though.
Ok, I played the Three door Monty, 11 games played, switched every game, won 9 times out of 11.
Then I played 11 more games and stayed every time , and lost 8 out of 11 times.
I understand completey, that if I pick a door and stay with it, I have a 1 in three chance of winning, my actual number came out to 27%, which in the short run I did I figure is close enough. The switching though I don’t get, if I switch every time I win consistently higher.
pause for another trial
Three door Monty here http://www.shodor.org/interactivate/activities/monty3/index.html
Ok, I just now played 30 games of 3 door Monty and won 20 games of 30 by switching every time, which is a resounding 66% winning ratio. It would seem that by switching every time that one should only have a 50/50 chance of winning right? If you stay you should only wing 33% of the time which seems to be the case.
I haven't yet figured out why switching every time should increase your odds to a 66% winning ratio. :astonished:It makes absolutely no sense. Once a door has been opened, the probability should switch to 50/50, as if that third door had never existed.
If it’s not behind door A, it’s behind either B or C. Question has been reduced to an either/or. Equal chance of either.
How can this not be the case?
and yet, unless these simulations are rigged…it’s not the case…
Right you are Emmline, this definitely requires a little more research, but right now, off to practice the banjo. 
Actually, it makes perfect sense if you learn to think probability, which admittedly is sometimes counterintuitive.
Here’s how to understand it. The chance of winning if you never switch doors is simply the chances that your original pick is the winner, which in the case of 3 doors is 1/3. Since the total probability for all outcomes must add up to 1, the chances of winning if you always switch is 2/3. The fact that you’re shown the non winning door does nothing to change the original probabilities.
Bob
Here’s the way I think of it:
1/3 of the time, you pick the right door at first. Monty will open one of the other two empty doors, and ask if you want to switch. If you switch, you will always lose.
2/3 of the time, you pick the wrong door at first. Monty opens an empty door, meaning the last door (the one you didn’t pick at first) must be the winning door. If you switch, you will always win.
Therefore switching means you win 2/3 of the time.
Another way to think about it: Monty always opens an empty door. If you pick an empty door, that means he’s forced to open the only remaining empty door (which gives away which is the winning door). But if you pick right, he can open either of the two empty doors, which doesn’t give away anything. So picking the WRONG door at first forces his hand. That’s what tips the scales, and makes it not random any more.
Yeah, like…what’s the probability that practice will make a banjo sound better? 
That seems pretty counterintuitive.
I understand the mathematics and it is clear empirically that Marilyn was right. But I can’t SEE it.
Dale
This explanation works. Weird though.
no, nor I. I understand the above explanations, but my brain still thinks it’s logical for the probability to start from scratch when one door is eliminated.
Well actually the banjo sounds better than my singing. But look at it this way Darwin, if it weren’t for the ugly sounding instruments, the beautiful sounding ones would have nothing to compare themselves to.
Banjos aren’t really meant to sound beautious, they’re meant to sound cool man! 
Oh yeah, Bone Quint’s explanation works for me also, nothing wrong with Norseman’s explanation, (except that I didn’t get it) 
Deep, man, deep…
Banjos aren’t really meant to sound beautious, they’re meant to sound cool man! >
I played 5-string for many years, but it never quite reached the cool stage. Weekender had a mighty cool-sounding 6-string banjo–ancient skin head and nylon strings–at the West Coast Gathering. (You can see a good shot of it at http://www.tinwhistletunes.com/clipssnip/Chiffgathering05.html )
Oh yeah, Bone Quint’s explanation works for me also, nothing wrong with Norseman’s explanation, (except that I didn’t get it) >
Well, you know what they say–if people had an intuitive grasp of probability, Las Vegas would just be a wide spot in the road.
Hiya, here’s a good way to explain intuitively why this works:
Imagine you have 99 doors. You pick door #1. Then I open all the other doors, except for door #37, to show that they’re empty. Now it may seem like a good idea to switch, right?
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.”
So now you’re given this choice between sticking with door #1, or all the doors #2-#99. The same is the case for 3 doors, but it is not so intuitively clear.
Caj
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.
I love the banjo.
Robin
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
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/clipssnip/Chiffgathering05.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 rcrd*r players that have owned up to it on this forum. 