Do you know the correct answer?

[Gordon Bartlett]

Wow…. 20 pages.

I’ve been out for the last half dozen pages, so I’ll take a swing.

No one reads my sh!t anyways.

Nick,

If Monty was selecting from his two doors at random and happened to select a goat, then you would be correct, the odds of either door containing the car would be 50/50.

But he’s not. Monty knows the locations of the car and goats from the outset. The draw is only random for the contestant.

Once Monty makes the informed choice to remove a goat from his two, the remaining door now has the odds are locked in a 1/3 and 2/3.

This is 100% correct.
If Monty randomly flung open either of his two doors, the two remaining doors would have identical odds. That would be 50/50 if the door revealed a goat, and 0/0 if the door revealed the car.
But there is nothing random about Monty's choice of door. If the car is to the right, he opens the left. If the car is to the left, he opens the right. The only time he can pick a door randomly is if the contestant has picked the car.
The whole concept of the "always switch" argument revolves around this introduction of Monty's non-random choices to the mix.
If everything remained completely random, there would be no advantage in switching.

[Dave Wright]

I don't believe that it is counterintuitive.

You can put a gold coin underneath one of three identical cups. You switch them around a bit.

Then you ask a child, " Do you think you can guess which cup the coin is under." The child usually answers yes, if I'm lucky!"

You then ask the child, "If I let you turn over two cups rather than just one do think you might have a better chance of finding the coin?

A normal child gets the correct answer very quickly.

The Monte Hall depiction of this problem is full of goats, doors, and other confusing nonsense. Some people are bamboozled with bullsh!t. The real solution is intuitive when the problem is presented simply and unadorned.

[Tom Wilkinson]

^
No. I believe those who believe the choice comes down to a 50/50 proposition are sincere in that belief. Any error must be in the assumptions behind their calculation.

If it were me, and most people were telling me I was wrong, I would test my belief with a real-world test. That is in fact what happened with me. When I first encountered the problem I thought there was no advantage to switching. But upon investigating further I found an overwhelming number of sources providing logical explanation and statistical calculation that changed my mind.

Nevertheless the solution to the problem certainly is counter-intuitive. Actually playing the game 100 times eliminated any remaining doubt I had regarding the advantage in switching. The critics claim that my results were due to being unable to randomly shuffle a deck of cards. I would play it again using three marbles. But I do not have marbles. Are they even commonly found for sale these days? In any event I suspect some other objection would discount the result of that form of the game as well.
.

So you’ve lost your marbles. No wonder after this thread.

I agree with Dave Wrights assessment above (825). The distractions are throwing people off as they are intended to do.

[Gordon Bartlett]

Too many people are ignoring the difference between a randomly opened door vs a selectively opened door. That's the key to the whole thing.
Randomly opened door = even odds.
Selectively opened door = 66%/33%.

[SchoonerRat]

Originally Posted by coelcanth

no that's not it

as long as the contestant has to pick first, their door will always have worse odds than Monty's two doors

Monty is not cheating at the game and he's not 'removing' the goat from play
even with a 1 in 3 chance and without switching doors, the player will still win the prize sometimes, there's just a smaller chance
it's never 50/50

"It's never 50/50"?????????

For God's sake, man! I just tried to find 1 winner from a pool of 50 possible choices. What are my odds of success? I give you that my odds are not now 50/50. What do you think they are? 1/3, 1/5, 1/10. I have 4 years of undergraduate study in Math. Somehow, during that time I was able to convince the powers that be in the school of letters and science that I had earned a bachelor's degree in mathematics. Even ignoring the 3 or 4 semesters I spent doing grad work in math, I think I've earned enough cred to believably posit that my odds are 1/50. Can we agree on this? I can't clearly hear your answer over the internet, so just nod or shake your head.

Okay, now my new best friend Monty has now given me a second chance to choose a different door. I say that this is a completely new choice that is totally unrelated to the first...except that both choices involve goats and car. I now have new info on which to base my decision. Monty now has 1 door that either does or does not hide my shiny new toy. I have 1 door which either does or does not contain a goat that I wouldn't know what to do with if I won it. How in the world can you look at this new choice and still say that it is not now 50/50. There is no place, other than my door or Monty's door where my prize can possibly exist.

NO PLACE!

I sincerely apologize that the hollow bit of bone which resides upon my cervical vertebrae does not contain sufficient gray matter to more plainly explain my position on this. I'll do my best to sum it up.

I chose door #42 using the well-known mathematical principal of "Eenie Meenie Meiney Moe". The odds of my success are 1/50.

Mr. Python has now given me a new choice. I now know, without a doubt, that either my door, or door #13 is correct. When I originally chose #42, #13 was no more than Eenie to my Moe. Odds for #42, 1/50. Odds for #13, 1/50. But I now have more info that gives me a 50/50 chance of being correct no matter whether or not I choose to go with my first gut feeling or to consider changing my mind. There has been nothing revealed to me in the new info available to me that would lead me to believe that my odds are now any better than 1/50.

I now have a 50/50 choice before me. One of those 1/2 choices has no better than a 1/50 chance of being the right choice. The chances for #42 seem to be less than 1/2. I think this makes the chances for #13 somewhat better than 1/2.

Well, after all of this pontificating, I have to admit that your are right. "It's never 50/50. Changing your mind is better than 50/50. No matter which way you go, there's always a good chance that you're gonna lose. If you accept Monty's offer to change your mind the odds will be in your favor. Every time you play. Without Fail. You might lose, but rest assured, you made the correct choice.

[twodot]

How in the world can you look at this new choice and still say that it is not now 50/50. There is no place, other than my door or Monty's door where my prize can possibly exist.

Sorry, if you read the thread then you will see that you are wrong. Out of 50 doors your door choice has a 1/50 chance of having the car. After Monty opens 48 doors to reveal goats, leaving two doors, your door will still only have 1/50 chance of being the correct door.

It has been stated many times before, but you can easily convince yourself if you imagine that you have skin in the game: You put 100 dollars in a bag and I put 100 dollars in the same bag. A third neutral person places the bag randomly behind one of 50 doors. You choose a door, and the third person then opens (knowingly, not randomly) 48 of the remaining doors, showing them empty of bags. This leaves your unopened door and...my unopened door. It is my door because in this version of the game you can't switch, and the other door is mine.

I will play that game with you as long as you wish, 100 times even. You won't play it once, because you will realize that I have a 49/50 chance of taking your money, every time.

This is the argument, stated many, many times on this thread, that Nick and CW won't address.



.

[David G]

Sorry, if you read the thread then you will see that you are wrong. Out of 50 doors your door choice has a 1/50 chance of having the car. After Monty opens 48 doors to reveal goats, leaving two doors, your door will still only have 1/50 chance of being the correct door.

It has been stated many times before, but you can easily convince yourself if you imagine that you have skin in the game: You put 100 dollars in a bag and I put 100 dollars in the same bag. A third neutral person places the bag randomly behind one of 50 doors. You choose a door, and the third person then opens (knowingly, not randomly) 48 of the remaining doors, showing them empty of bags. This leaves your unopened door and...my unopened door. It is my door because in this version of the game you can't switch, and the other door is mine.

I will play that game with you as long as you wish, 100 times even. You won't play it once, because you will realize that I have a 49/50 chance of taking your money, every time.

This is the argument, stated many, many times on this thread, that Nick and CW won't address.

.

But what if it was 1,000 doors? That'd change things, right? And if you're only left with two doors to pick from... stick, and switch... then your odds are 1/2, right? It only stands to reason.

[David G]

Never kid a kidder, as my mom would say when we were young, you are just trying to push the post count past 1000. It won't happen, probability speaking.

What? How dare you!!! Are you accusing me of trollish, nefarious, Plessish, mopery and dopery? You shall hear from my barristers, sir!!!

[Tom Montgomery]

But what if it was 1,000 doors? That'd change things, right? And if you're only left with two doors to pick from... stick, and switch... then your odds are 1/2, right? It only stands to reason.

You know better. Shame on you.

[David G]

You, good sir, are a Plessian rapscallion.

You keep escalating like that... and you'll hear from my seconds, not my barristers!!!

[peb]

Epoxy is crap

[Paul Pless]

​....

[George Jung]

Ah, hell.

'I double-dog dare ya!'

Now, that's escalation!

[Canoeyawl]

a Plessian rapscallion.

Plessian rapscallion

3FCEA88D-D175-4200-ADBD-0F1281BC7069.jpg

[David G]

Ya think that critter takes after Paul?

Well.... he does have that whole 'Roman Nose' thing going on strongly...

And there's something about the chin <G>

[Chris Woodward]

A + (B + C) = 100
where A= the original pick; B and C are the other 2 choices.
C = the opened door.
A is 1 out of 3 (1/3) or 33.33%
C is wrong and has 0% chance of winning.

33.33% + (B + 0%) = 100%
Solve for B
B+0% = 100% - 33.33%
B = 66.67%

[Jimmy W]

Sonny & Samantha.jpg

[David G]

Sonny & Samantha.jpg

I do love Nubian goats.

[Decourcy]

A + (B + C) = 100
where A= the original pick; B and C are the other 2 choices.
C = the opened door.
A is 1 out of 3 (1/3) or 33.33%
C is wrong and has 0% chance of winning.

33.33% + (B + 0%) = 100%
Solve for B
B+0% = 100% - 33.33%
B = 66.67%

Bam!

[Ted Hoppe]

careful kids, there is no such thing as free candy.

[David G]

[miketaylor]

But what if it was 1,000 doors? That'd change things, right? And if you're only left with two doors to pick from... stick, and switch... then your odds are 1/2, right? It only stands to reason.

What if there were 1000 doors and Monty opened 998 of them to reveal goats?

Contestant originally had a 1:1000 chance of winning. Now with only two doors available are you guys seriously telling me that he will stilll have only a 1:1000 chance??

I can't get my head round this.

Mike

[oldcodger]

Imagine a Lucky Dip where there are 1000 packages one of which contains the keys to a car.

The organizer has put a secret mark on the winning box so he can see which it is but you can not.

You pick one and ensure no one can tamper with it.

It is a 1 in 1000 chance that it has the keys.

The organizer sifts through the remaining boxes until he has one left, be it empty or has the keys.

He then presents his unopened box to you and asks if you want to switch.

Nothing he has done changes the box in your possession therefore the odds are still 1 in 1000 that you have the keys and the organizer's odds are 999 in 1000. So switching is the right strategy.

I don't know if you had the chance to view the video by the professor in statistics at Harvard that I posted previously.

If you have not time to view the first 25 minutes at 21.00 he addresses multiple choices. It will only take 4 minutes.

[miketaylor]

Imagine a Lucky Dip where there are 1000 packages one of which contains the keys to a car.

The organizer has put a secret mark on the winning box so he can see which it is but you can not.

You pick one and ensure no one can tamper with it.

It is a 1 in 1000 chance that it has the keys.

The organizer sifts through the remaining boxes until he has one left, be it empty or has the keys.

He then presents his unopened box to you and asks if you want to switch.

Nothing he has done changes the box in your possession therefore the odds are still 1 in 1000 that you have the keys and the organizer's odds are 999 in 1000. So switching is the right strategy.

Thanks Old Codger.

I think that is something I can get my head round.

Mike

[Gordon Bartlett]

A + (B + C) = 100
where A= the original pick; B and C are the other 2 choices.
C = the opened door.
A is 1 out of 3 (1/3) or 33.33%
C is wrong and has 0% chance of winning.

33.33% + (B + 0%) = 100%
Solve for B
B+0% = 100% - 33.33%
B = 66.67%

I'm not happy with this equation because it does not reflect that the opened door was purposefully chosen to reveal a goat, and only a goat.
If the door was opened randomly and happened to reveal a goat by chance, it would imply nothing about the likelyhood of the location of the car. Doors a and b would be 50/50. After all, we could have just as easily drawn the parentheses around a and b, or around a and c.
The only thing that makes b and c special is that Monty is looking behind them and opening only a door with a goat.
Many card games are played "by the odds", but the odds are not the same if one player is looking at his cards while the other guy's cards are face-down.

[peb]

What if there were 1000 doors and Monty opened 998 of them to reveal goats?

Contestant originally had a 1:1000 chance of winning. Now with only two doors available are you guys seriously telling me that he will stilll have only a 1:1000 chance??

I can't get my head round this.

Mike

This is what should make it clear. You can have no doubt that your pick was 1/1000 chance. And if you know that Monti has to leave the car hidden, yet reveal 998 goats, then you have to realize that the door Monti did not open almost has to have the car.

Now contrast that to Monti randomly opening 998 doors. Alost every time, 499 out of 500, he will eventually reveal the car, and you will NEVER be surprised that when the car appears. If you happen to get through 998 cars, you would consider that extremely lucky. At that point, it would be 50-50 between the two doors. But its easy to see that it would almost never happen. As opposed to him following the rule that he has to leave the car hidden. Then it would be extremely rare for the car to be beind the door you picked.

Again, it may be more beneficial to think of Monti's rule as not that he must reveal a goat, but that he must keep the car hidden.

[SchoonerRat]

Sorry, if you read the thread then you will see that you are wrong. Out of 50 doors your door choice has a 1/50 chance of having the car. After Monty opens 48 doors to reveal goats, leaving two doors, your door will still only have 1/50 chance of being the correct door.

It has been stated many times before, but you can easily convince yourself if you imagine that you have skin in the game: You put 100 dollars in a bag and I put 100 dollars in the same bag. A third neutral person places the bag randomly behind one of 50 doors. You choose a door, and the third person then opens (knowingly, not randomly) 48 of the remaining doors, showing them empty of bags. This leaves your unopened door and...my unopened door. It is my door because in this version of the game you can't switch, and the other door is mine.

I will play that game with you as long as you wish, 100 times even. You won't play it once, because you will realize that I have a 49/50 chance of taking your money, every time.

This is the argument, stated many, many times on this thread, that Nick and CW won't address.



.

Sorry, but if you read my post a little more carefully, I think you'll see that that's pretty much exactly what I said. I apologize again for not owning enough gray cells to properly explain myself. Let me try one more time.

48 open doors. 2 closed doors. I own the contents of 1, Monty the other. The only way accepting Monty's offer of a switch will be the wrong choice is if I chose the correct door the first time. 1/50!

This is how I should have summarized my last post. I guess I got so hung up on the 50/50 comment that I kinda lost my way. I think that we are in complete agreement on this.

Can we agree to agree?

[twodot]

Sorry, but if you read my post a little more carefully, I think you'll see that that's pretty much exactly what I said. I apologize again for not owning enough gray cells to properly explain myself. Let me try one more time.

48 open doors. 2 closed doors. I own the contents of 1, Monty the other. The only way accepting Monty's offer of a switch will be the wrong choice is if I chose the correct door the first time. 1/50!

This is how I should have summarized my last post. I guess I got so hung up on the 50/50 comment that I kinda lost my way. I think that we are in complete agreement on this.

Can we agree to agree?

OK, sorry, I read your post as an argument for not switching.

[SchoonerRat]

OK, sorry, I read your post as an argument for not switching.

Thank you. I hate being told I'm wrong by somebody who is wrong about my being wrong. It's even worse when I agree with somebody who agrees with me but still can find a way to think I'm wrong.

I am still capable of logical thought. I've never been very good at logically expressing such thoughts.

[David G]

Thank you. I hate being told I'm wrong by somebody who is wrong about my being wrong. It's even worse when I agree with somebody who agrees with me but still can find a way to think I'm wrong.

I am still capable of logical thought. I've never been very good at logically expressing such thoughts.

Sounds logical to me!

[Canoeyawl]

These are all pretty good reasons to avoid gambling, generally

[Tom Wilkinson]

These are all pretty good reasons to avoid gambling, generally

The Harvard professor had an interesting talk on The Gamblers Ruin. Very good reason not to gamble.

[Canoeyawl]

The Harvard professor had an interesting talk on The Gamblers Ruin. Very good reason not to gamble.

From the perspective of someone that fixes aircraft, "gambling" with odds is probably not even on your list of things to consider.

[David G]

The Harvard professor had an interesting talk on The Gamblers Ruin. Very good reason not to gamble.

It depends.

If you're gambling at a casino - the house will, in the end, get their cut. Odds of you coming out ahead are vanishingly small. That said, I used to play blackjack in Vegas when I was there on business. Always set aside a set amount, lost it, and quit. Sometimes, though, I played for quite a while for emptying my poke. Then, one magic evening, I won big. I could feel the bubble burst when my streak was over, cashed in, and have never been back to the tables. Rare, though. I'd never repeat it if I continued. So I quit while I was ahead. Gave me a GREAT deal of satisfaction.

And then there's gambling socially. I made beer/spending money playing draw poker in high school & college.

[Tom Wilkinson]

From the perspective of someone that fixes aircraft, "gambling" with odds is probably not even on your list of things to consider.

I’m not much of a gambler at all.