The Monty Hall Paradox Revisited...Part One

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The overwhelming (to me, at least) paradox of "the laws of chance" dictate that on each selection you have an equal chance of making the correct (or incorrect) choice.
I see the logic involved in changing from 1 out of 3 to 50%.

Let's try this little thought experiment:

We have three cards. An ace and two deuces. I shuffle the cards around and lay them out on the table. You choose one card that you think is the ace. What are your chances of being right?

If you said 1 in 3, congratulations, you're correct.

Now, I (knowing which card is the ace, because I peeked) touch one of the other two cards. What are your chances of having selected the ace on your first try?

If you said 1 in 3, congratulations. Nothing has changed, so your chances are still the same.

Now, I turn that card over, showing a deuce. What are your chances of having been correct on your first try?

If you said 1 in 2, please explain what has changed. Remember, I can always turn over a deuce. If you said 1 in 3, congratulations. Nothing has changed except you now know that one of the cards you didn't choose was a deuce. The chances of your first guess being correct are still 1 in 3. If you have two cards to choose from and one of them has a 1 in 3 chance of being the ace, what are the chances of the other being an ace?

The answer is left as an exercise for the student.

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Because you now have a 2 to 1 chance of picking the right door. If you chose the wrong door in the first guess (a 2 to 1 chance) by changing you will get the right door.

Huh? My point?
Look Faldage, I don't want to call attention to your inarticulate sentence structuring but this is my point: Just what the heck does your sentence mean? Changing will not get you the right door; changing will only give you a 66.6% probability of getting the right door.

Geez! What you meant is not what you said. Apologize.

Wait! Now I get it. You are omnipotently proclaiming the knowdledge of which is the right door. Sure, Faldage, now I understand your words as they are so...so...you. Good. Now you are ready for Monty Hall Part Two. Oh boy, this should be fun.

Inasmuch as the Monty Hall Paradox works and can be tested it seems likely that we humans might have incorporated this manner of decision making in the course of our evolution. In order to explore this possibility let us re-examine the Monty Hall Paradox, but this time let's use one hundred doors.

This time you select a door from a group of doors numbering from 1 to 100 in order to win a car. Monty Hall will then open a door with only a sack of dog food there and ask you if you would like to change your selection. You say "yes" and so Monty then opens another door and again asks you if you would like to change your selection, you say "yes" and etc..

So the question is...how does the percentage rate of likely success increase as you and Monty Hall work your way towards one hundred.

You have exactly one seventh of a sennight to register your answer.

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Because you now have a 2 to 1 chance of picking the right door. If you chose the wrong door in the first guess (a 2 to 1 chance) by changing you will get the right door.

Huh? My point?
Look Faldage, I don't want to call attention to your inarticulate sentence structuring but this is my point: Just what the heck does your sentence mean? Changing will not get you the right door; changing will only give you a 66.6% probability of getting the right door.

Wull, I spose it might could be inarticulate if you don't understand the basics of logic and if/then statements. What's that? You don't see no "then" in there? OK, if you cain't read between the words here it is:

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If you chose the wrong door in the first guess (a 2 to 1 chance) then by changing you will get the right door.

I redded up the "if", too, so you could see it. Parboly you missed it the first three, four times through the sentence.

YAWN

The most convincing and intuitively palatable approach to this problem IMO is to make diagrams of your choices and the possible outcomes. While one may argue the philosophical point that nothing has changed between the first choice and the first reveal, making a diagram of each strategy makes it clear that a strategy of switching after the reveal will result in a successful outcome (in this case, apparently, some interaction with the comley Ms. White) two-thirds of the time vs sticking with the original choice.

Short of the diagrams, here is my best explanation� of why switching works: When you first picked, you had a 1/3 chance of picking the winner and a 2/3 chance of picking a loser. Once shown one of the losing options, the strategy of switching will only result in a bad outcome in the event that you had picked the right one in the first place. And since you originally had only a 1/3 chance of winning, you now have only a 1/3 chance of losing if you make the switch, vs a 2/3 chance of winning.

Marilyn Savant's solution�that choosing a new door increases your odds from 1:3 to 1:2�doesn't seem very controversial to me. In a single instance it probably doesn't make a great difference, but I think if you played this game a number of times, the decision to change doors would yield better results.

This problem divides people into lateral thinkers (who choose to change doors) and vertical thinkers (who stick with their first choice). Lateral thinkers view the problem in time and in terms of two discrete sets of initial conditions and reason: "Under the first set of initial conditions my first choice yields a 1:3 probability, under the new set of initial conditions a change yields a 1:2". Vertical thinkers view the problem in space and say, "There was a 1:3 chance that my first choice was correct. Now there is a 1:2 choice that my first choice was correct�after all, either choice will be 1 of two doors!"

Personally, I would change my first choice.

Consider instead a bifurcation in time, which gives you two sets of doors. Set A consists of three doors (two bags of dogfood and one girl). Set B consists of two doors (one bag of dogfood and one girl). Which set of doors yields the greatest chance of getting the girl? Clearly, Set B. Making a decision under condition B yields a greater chance of getting the girl. If you allow this fact to have retroactive effect on the first choice (which was out of three doors) it is simply a case of choosing bewteen a 1 in 3 chance and a 1 in 2 chance.

Wonderfully clear, Alex, you too, Hydra, and Faldage...I heard that!

Now to the point. If in fact a counter intutive or a seemingly trans-logical method of decision making exists such as that which we have demonstrated here, then the processes of our evolution must have incorporated those methods within the circuits of our brains. (You known like a woman's intuition; or a woman's prerogative to change her mind,)

As an example of how the Monty Hall Paradox could be utilized to assist in our survival I offer this hypothetical as a case in point...

Al Gore was right. Global Warming has come and gone, leaving only four people alive and who are camped out in the top floor of the Empire State Building eagerly awaiting the comming Ice Age.

They are a throughly modern crew -- one woman and three New York Metrosexuals -- their mission is to re-populate Earth.
The woman, a good Christian lady from Alabama, must choose between the three men -- Freddy, Perk, and Tony -- but all are equally dull and she can't decide, so (what the hell) she flips a coin and selects Tony.

Bad luck. Before she can tell Tony of his fate by chance she intercepts a love letter writtten by Perk to either Freddy or Tony. (New York has a 2006 law that decrees a person's biological sex is officially immaterial, anyone can be whichever sex they want to be as long as they openly declare it so) Hmmm?

Now your question: Does the nice woman from Alabama now change her choice from Tony to Freddy or stay with Tony?

Take your time. Only the fate of all future mankind awaits your answer.

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...(New York has a 2006 law that decrees a person's biological sex is officially immaterial, anyone can be whichever sex they want to be as long as they openly declare it so) Hmmm?

Then, logically, I can be whatever Height I wish to be. The physical attribute is immaterial. Better yet, I can be whatever WEIGHT I desire. WOOOHOOO!! I am no longer overweight!!

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...(New York has a 2006 law that decrees a person's biological sex is officially immaterial, anyone can be whichever sex they want to be as long as they openly declare it so) Hmmm?

Then, logically, I can be whatever Height I wish to be. The physical attribute is immaterial. Better yet, I can be whatever WEIGHT I desire. WOOOHOOO!! I am no longer overweight!!

Sorry, ParkinT, that is not the way the world works in wambly New York.
The bill up for vote in the New York legislature is an attempt to circumvent any judicial ruling against marriage between the same sexes.
You will be required to document the sex that you wish to be in writing and then wait two years, then the state will go back and change the sex listed on your birth certificate if it differs from the sex that you want to be.

Soon, in a world gone mad, being overweight in New York will land you in jail.

Strange World.

Whatsamattau, big boys, have you all no balls for contesting the fact that the paradoxical nature of the question that I have posted has had a postive impact upon the thinking processes of all humanity?

What? Don't you understand the implications?

You do? Well, has the cat's got your tongue?

Some of us ain' had no time to plow through the logical fallacies left unexpressed. The unknown knowns, if you will.

I think under the circumstances it will be easy for the woman to convince all three men to help her "get with child" since the alternative is the extinguishment of humanity. Then again, the human species of the future in this scenario are to be descendants of three NY metrosexuals and a woman from Alabama, so what hope is there anyway?

W V O Quine, J Searle, and D Hofstadter walk into a bar. One of them orders a twelve-inch pianist. Who was it?

You boys really don't understand the precis, do you?

Oh well, I won't bother you about this anymore.
You fellows go back to looking up amusing words in your books.

This concept as set out in the game example is just B unk S ee, did not really put what you expected
So what if the odds become 50:50 by showing one of the wrong doors? That is not going to change which door is the 'one', even if you were "more likely to be wrong on the first pick". Picking the same one again still has 50% chance, but the act of choosing has some mysterious power over the hidden result? B uffoonery .

the *kicker in this problem is Monty Hall (the rat). He knows which is the right door, and will always reveal a wrong one. In doing so, he "freezes" the odds on your first choice at one in three! but the remaining door obviously has odds of one in two of being right!!

and the real gut-shot is that if you get down to the last two cases in Deal or No Deal, it's just a coin flip.

edit: not that I expect this will sway any doubters..

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This concept as set out in the game example is just B unk S ee, did not really put what you expected
So what if the odds become 50:50 by showing one of the wrong doors? That is not going to change which door is the 'one', even if you were "more likely to be wrong on the first pick". Picking the same one again still has 50% chance, but the act of choosing has some mysterious power over the hidden result? B uffoonery .

The whole point is the odds aren't 50-50. As tsuwm pointed out, they stay 1 in 3 for the first door you picked. Then you must proceed to ignore where he said:

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but the remaining door obviously has odds of one in two of being right!!

since it obviously has odds of two in three of being right.

As to Milo's fair damsel, my prediction is that she will mate with all three in whatever order they choose. The joke is on them, though, since recent studies indicate that the last guy is the most likely to have offspring as a result of the encounter.

>since it obviously has odds of two in three of being right.

this is where y'all lose me (but I'm sure only momentarily). isn't the door revealed by Monty out of play, leaving just the two - one good and one not so?

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>since it obviously has odds of two in three of being right.

this is where y'all lose me (but I'm sure only momentarily). isn't the door revealed by Monty out of play, leaving just the two - one good and one not so?

Ok, back to my thought experiment. You've chosen one of the cards and have a one in three chance of being right. Now I put a finger on one of the other cards. Have the odds of your being right changed? If so, why?

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Ok, back to my thought experiment. You've chosen one of the cards and have a one in three chance of being right. Now I put a finger on one of the other cards. Have the odds of your being right changed? If so, why?

okay, drag away the dead horse.

I "get" the logic of it -- once you've picked a door, with your 1:3 chance, the odds of the other two total 2:3. Now Monty obligingly removes one of those, but the odds on the remainder? 2:3, et viola!

but I just can't get [s]over[/s] around the notion that there are two doors extant, and (intuitively) it's a coin flip.

Hmmm...we're getting no where with Joe using words, Mister Faldage.
What say we dazzle Friday with numbers?

Hey Joe, like you, a guy on the net couldn't see beyond the gumshoe logic that is endemic to those who are blessed with being average. But he was a hardheaded fellow and was dogged to try.
These are his results....
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He did the sequence of two steps a billion times.
And here are his results:

"Number of times I stayed with the choice in step (2): 499,990,387
Number of times I chose to switch: 500,009,613
Number of times I was right when I stayed: 166,585,940
Number of times I was right when I switched: 333,391,697
Fraction right when I stayed: 0.333
Fraction right when I switched: 0.666."

Yet he still can't intuit the process.
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like I said, them, I follow the logic (even *with your overkill of that dead horse); it's just so dern counterintuitive (sympathy for those who still doubt).

"You should"

Go to http://geocities.com/elbillaf/gameshow.html

Because it explains the answer to this problem in several different ways.

Let's try looking at the problem backwards to see if we learn anything.

Say we have 2 cups. One of the cups is on your right and one on your left.

Prior to your walking into the room, I have placed a gold coin beneath one of the cups. I tell you that l rolled a "fair" die and if it came up 1 or 2, I put the coin under the left cup; otherwise, I put it under the right one.

So if
L = coin is under left cup, and
R = coin is under right cup, then we conclude:

P(L) = 1/3 and P(R) = 2/3.

Question 1: Does this conclusion make sense? If not, then don't proceed to the next part.

Note: If you have gotten this far, it means you realize very clearly that the probability of the coin's being under each of the two cups is not the same.

Unfortunately, while I was distracting you with that question, someone decided to be clever. He moved the cups around and around and around. You turn your head back suddenly and you see the cups are jostled from their original positions.

You can no longer tell which cup is which.

What now is the probability that the coin is under the left cup?

I assert that P(L) = 1/2 and P(R) = 1/2.

Question 2: What is different about this situation from the initial state?

Answer: We have lost information.

Conclusion: Our calculations of probability are based on how much information we have about the situation.

Gaining or losing information changes the probability.

In that example we have lost information and the probablities (may) have changed. In the three door monte problem we (may) have gained information. But we must remember, Monty can always open a door with a non-prize behind it. We don't know if he had a choice in which door to open.

And if I keep beating this dead horse it's only because it insists on getting up and whinnying all the time.

"But we must remember, Monty can always open a door with a non-prize behind it. We don't know if he had a choice in which door to open."

Very astute observation - and this was one of the reasons that MVS had such trouble. People did not assume this game was exactly like the game show. In the game show, Monty Hall has perfect knowledge and we KNOW that he never turns over the grand prize - only the goat!

If MH's guess is random, the problem doesn't work out like this. It depends on the fact that his guess is not random. If his guess *IS* random, then the probability is what everyone else thinks it is.

But if Monty opens a door with the prize behind it it is eliminated from the universe of discourse since the problem states that he reveals a non-prize when he opens the door.

"But if Monty opens a door with the prize behind it it is eliminated from the universe of discourse since the problem states that he reveals a non-prize when he opens the door. "

Precisely. That is why it is prohibited in the correct statement of the problem that he should do so.

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If MH's guess is random, the problem doesn't work out like this. It depends on the fact that his guess is not random. If his guess *IS* random, then the probability is what everyone else thinks it is.

If MH shows a door with the prize behind it then the contestant would pick that door! What makes the problem intersting is the counter-intuitive nature of the result favored by most mathematicians. The facile case in which MH reveals the prize thereby giving it away to the contestant (who after all may pick any door they wish) has no interest. Why do people bring it up?

If Monty doesn't know and *does open the door with the prize and then those instances are eliminated, you *do have a fifty-fifty chance. If you assume he *does know then half the time he will have a choice of what door to open and half the time he won't and you're better off switching.

So, in a sense, either side of the argument is correct, depending on which assumption you make about Monty's knowledge.

What if the contestant is a dog and after choosing a door where he hopes to find a bitch in heat he gets a whiff from an up-wind door (that he didn't select) and he smells only dogfood? In his eagerness will he now switch directions and go directly to the door that he had not considered?
He should... if he is in a hurry.

You see, my search is to find a situation in nature (outside of quiz shows) where the mathematical switch is a shortcut to the positive results of the step-by-step elimination process.

HINT!!!!

okay, here's your chance to play Deal or No Deal

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okay, here's your chance to play Deal or No Deal

I dunno, tsuwm, the only two that I would select would be Anya and Brooke. All the other chicks seem boringly cloned.

Especially "Alike"... and Lauren.

Deleted by Hydra.