The Monty Hall Paradox

Foxfyre wrote:

Whether you change your initial choice or decide to change your initial choice, how at that point are your odds any other than 50-50 of making the right choice?

Because you have more information than that there are two doors with one prize behind one door. If you had no more information you could do no better than to calculate 50-50 odds.

So, if you saw two doors with a prize hidden behind one and have no further information you can only calculate the odds of one of your selections having the prize at 50%. So you pick, say, the one on the left. Now I open the doors and you see the car behind the door on the right and I offer you the opportunity to change your decision.

Once again, you are confronted with a choice of two doors. But do you really think you are now faced with a 50-50 decision? No, of course not. You have additional information that allows you to calculate your odds with a lot more precision.

I made this example very obvious, and the logical puzzle you are stumped on is a little more complex. So let's move it a little closer to the puzzle.

Let's say the host picks at random and doesn't open the door. In that scenario you pick a door, the host randomly picks a door and then you have the opportunity to switch to the remaining door. In this scenario changing your decision does not increase the probability of picking the prize.

But that's not the scenario. You know that the host always picks an empty door and this is additional information. Here is how you use it:

Your initial decision without any additional information gives a 1/3 chance of picking the prize and a 2/3 chance of not doing so.

Next, your host will show you an empty door. In the cases you already picked the prize on the first try (which we can only calculate to 1/3 odds) the remaining door is empty. In the cases in which you did not pick the prize on the first try (which we can only calculate to 2/3 odds) the remaining door has to contain the prize.

So switching doors gives you a 2/3 chance of picking the prize while not doing so gives you a 1/3 chance.

Now I was going to write a program to quickly show this to you, but I realize they already did and you suspect it of being rigged. So I thought I'd give you the code and let you see for yourself that it's not rigged but I'd likely do it in PHP and you'd need a server to run it. So it isn't an easy way to show you. JavaScript, however, runs in your browser without necessarily needing a server. So I looked for a JavaScript example (since I am no good at JavaScript myself) and found one here:

http://www.cut-the-knot.org/hall.shtml

Anyone on the internet can look at the page source and read the JavaScript to see if it's rigged.

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The House always wins... :wink:

RH

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When I first heard of this problem, I pondered it for a few minutes before saying lets supersize it and see if the conclusions to draw are more obvious.

You're a visitor from another galaxy on an intergalactic game show.

There are say 200 billion planets (guess) in our galaxy and the host ask you to pick one.

Now our game show host will eliminate every other planet in the entire galaxy bar one and tell you - one of those two choices has intelligent life on it - do you want to go with your initial choice or the one he has selected?

Do you think you have a fifty-fifty chance of being right if you don't change choices - because he used information asymmetry to eliminate bad choices, or do you think you have a 1 / 200 billion chance of being right with your original choice and a massive chance of being right if you go with his alternate?

Its kind of obvious what the solution is and why it works when you think of it that way!

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Robert Gentel wrote:

Foxfyre wrote:
Whether you change your initial choice or decide to change your initial choice, how at that point are your odds any other than 50-50 of making the right choice?

Because you have more information than that there are two doors with one prize behind one door. If you had no more information you could do no better than to calculate 50-50 odds.

So, if you saw two doors with a prize hidden behind one and have no further information you can only calculate the odds of one of your selections having the prize at 50%. So you pick, say, the one on the left. Now I open the doors and you see the car behind the door on the right and I offer you the opportunity to change your decision.

Once again, you are confronted with a choice of two doors. But do you really think you are now faced with a 50-50 decision? No, of course not. You have additional information that allows you to calculate your odds with a lot more precision.

I made this example very obvious, and the logical puzzle you are stumped on is a little more complex. So let's move it a little closer to the puzzle.

Let's say the host picks at random and doesn't open the door. In that scenario you pick a door, the host randomly picks a door and then you have the opportunity to switch to the remaining door. In this scenario changing your decision does not increase the probability of picking the prize.

But that's not the scenario. You know that the host always picks an empty door and this is additional information. Here is how you use it:

Your initial decision without any additional information gives a 1/3 chance of picking the prize and a 2/3 chance of not doing so.

Next, your host will show you an empty door. In the cases you already picked the prize on the first try (which we can only calculate to 1/3 odds) the remaining door is empty. In the cases in which you did not pick the prize on the first try (which we can only calculate to 2/3 odds) the remaining door has to contain the prize.

So switching doors gives you a 2/3 chance of picking the prize while not doing so gives you a 1/3 chance.

Now I was going to write a program to quickly show this to you, but I realize they already did and you suspect it of being rigged. So I thought I'd give you the code and let you see for yourself that it's not rigged but I'd likely do it in PHP and you'd need a server to run it. So it isn't an easy way to show you. JavaScript, however, runs in your browser without necessarily needing a server. So I looked for a JavaScript example (since I am no good at JavaScript myself) and found one here:

http://www.cut-the-knot.org/hall.shtml

Anyone on the internet can look at the page source and read the JavaScript to see if it's rigged.

Good grief man, I've just now arrived at the point that I recognize a computer when I see one two out of three times, and you want me to read a computer Java script; evenmoreso understand it?

I am already way outnumbered on the theory end by a lot of folks, yourself included, who know a whole lot more about this stuff than I do and, without any additional information, will bow to the opinions of the majority.

I will say that I enlisted a guinea pig to try the bean under the cups experiment however....result by staying with her original choice 6 misses, four successes. But I do accept the explanations of why removing one door from the equation does not change the odds.

It's sort of like this old classic math problem:

Three men go to stay at a motel, and the man at the desk charges them $30.00 for a room. They split the cost ten dollars each. Later the manager tells the desk man that he overcharged the men, that the actual cost should have been $25.00. The manager gives the bellboy $5.00 and tells him to give it to the men.

The bellboy, however, doesn't know how to divide $5 among three men so he gives each man one dollar and pockets the other $2.00.

Now each man has paid $9.00 to stay in the room. 3 x $9.00 = $27.00. The bellboy has pocketed $2.00. $27.00 + $2.00 = $29.00 - so where is the missing $1.00?

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I did the bean in the cup experiment BTW, but I took 2 styrofoam cups and drew and X on the inside bottom of one to represent a bean, so I wouldn't hear/feel it rattling around, and also because the BBQ sauce on the bean was messy :wink: The third cup that Monty picked stayed off to the side.

I shuffled while reading something else, flipping them, rotating them, stacking them...believe me, I had no idea each time which was which when I finally chose.

I picked a cup, then put down, and switch to the other.

I did this 30 times.

Results?

19 out of the 30 times, I ended up switching to the cup with the X.

Pretty close to 2/3's I'd say.

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Suppose a 3 horse race over jumps for 3 miles.

All 3 horses are 2-1 at the off.

There is a betting market in running.

One of the horses falls at the first fence.

The two left are evens the both.

Why would changing your mind give you a better chance.

I would always switch in g__day's example because the host has to choose a planet with life, unless you picked correctly out of 200 billion, in order to provide a live choice.

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spendius wrote:

Suppose a 3 horse race over jumps for 3 miles.

All 3 horses are 2-1 at the off.

There is a betting market in running.

One of the horses falls at the first fence.

The two left are evens the both.

Why would changing your mind give you a better chance.

I would always switch in g__day's example because the host has to choose a planet with life, unless you picked correctly out of 200 billion, in order to provide a live choice.

Nice of you to post an example that highlights the fault in your argument perfectly.

When you go to the track and bet on that race, when do the betting windows close?

The odds are based on what happens before the race starts - not half way through it. The same applies to the question at hand here. The odds are set on whether or not you are better off changing your original pick from the start - not half way through the game.

If a horse falls down half way through a race they don't reopen the betting windows and allow you to change your bet, do they? That is, in effect, what you are trying to do with your 50/50 argument.

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Foxfyre wrote:

Now each man has paid $9.00 to stay in the room. 3 x $9.00 = $27.00. The bellboy has pocketed $2.00. $27.00 + $2.00 = $29.00 - so where is the missing $1.00?

Each man has paid $9 to stay in the room. 3 x $9.00 = $27.00. The bellboy has pocketed $2.00. $27.00 - $2.00 = $25.00 -- which is in the cash register.

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fishin-

I did say that there is a betting market in running and there is in every race in England. The Tote closes at the off but not the bookies.

I know you only have a machine on course in America like our Tote which is not all that popular.

In a 3 mile chase with 3 horses all at 2 to 1( which they wouldn't be otherwise the bookies couldn't live) an early faller would start a new market.

g__day has shown me where I went wrong. I didn't see that in the original post.

If the contestant chooses the car door the host has to open a goat door and then switching loses and staying wins. If the contestant chooses a goat door the host chooses the other goat door to keep the car hidden and thus points to it 2 in 3 times i.e. contestant choosing a goat door 2 in 3 times.

I see where I went wrong. The choice is 50-50 but the strategy is not. The host has given you a clue 2 in 3 times. And not done 1 in 3 times.

Your strategy increases your chances. Gee! I learned something. I am an ID-iot after all. The Sergeant Major was right. I'm still here though and some of his "Golden Wonders" are not.

Would you bet at 10 to 1 on a coin toss with Mr Gates who insisted on a minimum stake of $100,000.

How much would you pay to switch if the car was worth $100,000?

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Foxfyre wrote:

Good grief man, I've just now arrived at the point that I recognize a computer when I see one two out of three times, and you want me to read a computer Java script; evenmoreso understand it?

Well even if you can't the fact that anyone can see the code and millions of people can understand JavaScript makes it a poor choice of technology to use to rig the test.

I'm hoping you feel less suspicious about a script whose every line of source is available for anyone to see.

Quote:

I will say that I enlisted a guinea pig to try the bean under the cups experiment however....result by staying with her original choice 6 misses, four successes. But I do accept the explanations of why removing one door from the equation does not change the odds.

You should really do it at least 100 times if you want to use the results for anything. I once had a bad run in poker, in which a situation that's only supposed to happen about 1 in 3 times happened to me every single time the situation came up (at least once a day) for about 6 weeks with the exception of two or three times.

It was a monster bad-run over hundreds of hands. A bad run across 10, 20 or even 30 tries is not going to be that uncommon.

Quote:

It's sort of like this old classic math problem:

Other than that they both are famous puzzles that confuse a lot of people with above-average intelligence I don't see the similarity.

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Robert Gentel wrote:

Foxfyre wrote:

Good grief man, I've just now arrived at the point that I recognize a computer when I see one two out of three times, and you want me to read a computer Java script; evenmoreso understand it?

Well even if you can't the fact that anyone can see the code and millions of people can understand JavaScript makes it a poor choice of technology to use to rig the test.

I'm hoping you feel less suspicious about a script whose every line of source is available for anyone to see.

Quote:
I will say that I enlisted a guinea pig to try the bean under the cups experiment however....result by staying with her original choice 6 misses, four successes. But I do accept the explanations of why removing one door from the equation does not change the odds.

You should really do it at least 100 times if you want to use the results for anything. I once had a bad run in poker, in which a situation that's only supposed to happen about 1 in 3 times happened to me every single time the situation came up (at least once a day) for about 6 weeks with the exception of two or three times.

It was a monster bad-run over hundreds of hands. A bad run across 10, 20 or even 30 tries is not going to be that uncommon.

Quote:

It's sort of like this old classic math problem:

Other than that they both are famous puzzles that confuse a lot of people with above-average intelligence I don't see the similarity.

I agree that probability is an unreliable advisor except on a large and broad scale.

As for the similarity between the two math puzzles, I see that both are a paradox in which working the problem one way produces a different result than working the problem another way; yet both ways seems to be logical at first blush.

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I give up.

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raprap wrote:

It's not a paradox

I agree. It's a known replicatable quantifiable demonstratable probability.

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Foxfyre wrote:

As for the similarity between the two math puzzles, I see that both are a paradox in which working the problem one way produces a different result than working the problem another way; yet both ways seems to be logical at first blush.

Er... neither is a paradox. One is an example of counter-intuitive probability, the other is deliberately misleading.

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ros gives up.

Hang on--no he doesn't.

That's an unknown replicatable quantifiable demonstratable probability.

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Foxfyre wrote:

As for the similarity between the two math puzzles, I see that both are a paradox in which working the problem one way produces a different result than working the problem another way; yet both ways seems to be logical at first blush.

Oh good heavens, that's not a paradox.

It's working the problem incorrectly.

Say good night Gracie.

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I found the thought process here interesting...

First, a question is asked (no fault there, lots of people don't understand things and ask questions).

But then the theory itself is challenged:

Foxfyre wrote:

So gut level, Joe, do you think the on line game was rigged to demonstrate "proof" for the theory of probability here? Or is the theory of probability the real deal?

Followed quickly by suggesting "design" in the results:

Foxfyre wrote:

Coincidence? Fluke? Or by design?

It just reminded me of the same thought process ID'ers use when trying to understand evolution.

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Quote:

How much would you pay to switch if the car was worth $100,000?

Does a bit of thought on that get you into how these dealers on the floor of the Stock Exchange work? Imagine your permission to switch was tradeable in a market along with millions of other permissions to switch with cars of many values. Imagine the price of an item as the exchange rate between that item and the $, and thus to all other currencies.

Then link it up to psychological states of mind, harvests, media ownership, etc, etc. Become humble in the face of irreducible complexity and shove all your silly opinions up the chimney.

Monte's game becomes a serious learning experience. For those with an open mind I mean.

As good as Going For Gold.

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ros wrote-

Quote:

It just reminded me of the same thought process ID'ers use when trying to understand evolution.

What about the thought process AIDs-ers use when trying to understand science and evolution on the basis of having read some stuff in the papers.

My last post was an example of the thought process ID-ers use in trying to understand everything.

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Spendi..

Give it up.. You can't even decide if the odds change or not after one of the doors is revealed. Even if shown the math you waffle and stammer and slobber all over yourself. One is left to wonder if you are ever sober.

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The Monty Hall Paradox

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