Women, math, and the Monty Hall problem

[PureX]

Monty's telling us which door in the unchosen
group is a goat isn't new information?

It's irrelevant information because it's not part of the ACTUAL choice.

For some reason many of you cannot seem to grasp the simple fact that there is only ONE ACTUAL CHOICE being offered. And it only involves two doors, and two possible results.

 

[Revoltingest]

Two points:

1 - I'd be happy to try this experiment with you, 100 times for $5 a go?

Why bother, eh.
I know the outcome (with 99% confidence).

2 - Go back to the OP. After you make your guess, Monty opens up 50 doors to show 50 goats... So, initially you had a 1/52 chance of being correct. You make your guess. Switching back to using a deck of cards... at this moment, you had your pile of one card and there was another pile of 51 cards. Do you agree that until a card is turned over the odds are 51/52 that the bigger pile contains the winning card? Now, Monty turns over 50 of the 51 cards leaving 1 from the bigger pile...

I'm only claiming that your phrasing was awkward.
Have you not noticed my showing the math behind
the advantage of switching?

 

[Revoltingest]

It's irrelevant information because it's not part of the ACTUAL choice.

For some reason many of you cannot seem to grasp the simple fact that there is only ONE ACTUAL CHOICE being offered. And it only involves two doors, and two possible results.

Show me the error in the decision tree in post #106.
It shows all possibilities.
Add up the results.
Switching wins 2/3 of the time.
This is because Monty offered new information.

 

[It Aint Necessarily So]

It there are three choices, I have a one in three chance of making the best choice. Take away one choice I now have a one in two chance of making the best choice. If one of two choices is the correct one, I have a one in two chance of making the correct choice. And that is regardless of anything Monty says that influences my choice.

That would only be true if YOU had chosen which curtain to pull back after making your initial selection not knowing which of the three curtains concealed the car when you made your initial choice and when you chose a curtain to reveal. If YOU make the decision - say to reveal door number 3 - you have a 1 in 3 chance of exposing the car. If you expose a goat, THEN you have a 1 in 2 chance of having chosen correctly initially and a 1 in 2 chance of being wrong. NOW, a trade is a 50-50 proposition.

But when somebody who knows where two goats are chooses and ALWAYS chooses a goat, your odds of having picked correctly in the first place don't change.

Maybe this will help: YOU make the choice of the reveal and expose the car behind a curtain that you DIDN'T chose. Now what are your odds of winning the car if you trade your door - say door number 1 - for the other concealed one, door number 2 (you opened door 3 and there was the car)? What if you don't trade?

Answer: 0% chance of winning whether you trade or not. This is an entirely different scenario than the one where Monte chooses which door to reveal, where your chances never fall to 0%, since he never shows you the car first.

Why? Because unlike as was the case when YOU chose, which was a random choice, your odds don't ever change once you've made your initial selection even after Monte shows you a goat. You know he has one or two of them after you've chosen, and he knows which and where, so HIS choices don't affect the odds that YOU chose correctly or incorrectly initially.

Once again from a slightly different perspective: How often does Monte have two goats to show you after you've selected and how often does he have a car and a goat to show you? Only one day out of three will the contestant have chosen the car and left Monte with two goats, and 2/3 of the time, they will have guessed incorrectly leaving Monte with the car with one goat to choose from to reveal to you.

The gist of this is that changes to 50-50 when YOU choose which of Monte's doors to reveal, because since YOUR choice is random, YOU might choose the car. But he never will, so his revealing a goat from either a car and a goat or two goats changes nothing for you.

One more point. If I haven't convinced you to always switch when Monte makes the choice, there's this: From your perspective, you should recognize that you might be right and you might be wrong. If you're right, it doesn't matter whether you swap or not. Your chances are 50-50 either way.

If you're wrong, it does matter, and you should definitely swap to double your chances from 1 in 3 to 2 in 3.

That's a different math problem from the Monte Hall problem now, but it leads you to the same conclusion by a different path: switch.

Does that help you to decide what to do? By switching, either your chances improve if you are wrong that your odds improve from 1 in 3 to 50-50 after a goat reveal, or are unaffected if you are correct.

Does the phrase everything to gain and nothing to lose mean anything to you? We ALWAYS take bets like that. It's like being able to hit a blackjack hand and if you bust, the hit doesn't count. Wouldn't you ALWAYS hit any hand under 21, even a 20, if offered it at no risk? You have a small chance of improving your position when your hand is close to 21, but no chance of worsening it. That's exactly analogous to this: everything to gain and nothing to lose.

It's also false. The odds remain 50/50 regardless what door you choose, or what door Monty opens.

You have an intuition there, not a mathematical proof. You think that you know the answer but cannot argue for being correct. You can only repeat your intuition. This is why intuition, or as others call it, a gut or visceral feeling or a feeling to the bones, is unreliable as a path to truth or knowledge.

The 3 doors are just theater because one is eliminated before you choose. And the one eliminated is always a goat. So the proposition was always 2 doors, and one with a car.

Nope. The proposition was always three doors and one with a car. The illusion is that that changes after a goat is revealed by the host the way it would if the contestant made that choice as explained above, but that's just not the case.

 

[PureX]

The other angle to this is that, while deciding whether to switch doors or to stick with their original choice, the contestant is in a room full of noisy people yelling out trying to influence their choice, while Monty Hall is pressuring them by offering them cash to switch. There's also a time limit in play where the contestant does not have much time to think or consider the odds.

Of course. It's all theater to raise the excitement of the audience and to confuse the thinking of the contestant. But in the end, it's two doors and two possible outcomes. (Unless he opts to take the cash buyout.)

 

[PureX]

Show me the error in the decision tree in post #106.

There is no "decision tree". The odds based on a choice that was never actualized are irrelevant. Why is it so difficult for you to acknowledge this? The only relevant odds are the odds on the choice that is actualized.

 

[Revoltingest]

There is no "decision tree".

Yes there is.
Post #106.
It shows every possible outcome for every choice.

The odds based on a choice that was never actualized are irrelevant. Why is it so difficult for you to acknowledge this? The only relevant odds are the odds on the choice that is actualized.

Math is hard, eh.

 

[PureX]

If you say so....

....and who would know better, after all? Certainly not all those mathematicians --- how could they know nearly as much as you?

Math is just a language. It will reflect the story that you make it tell. Same as any other language will.

 

[Koldo]

Math is just a language. It will reflect the story that you make it tell. Same as any other language will.

Then take up my offer and pick 10 numbers between 1 and 100.

 

[Revoltingest]

Math is just a language.

Wrong.
It's a logical structure that uses a language to express it.

 

[PureX]

You have an intuition there, not a mathematical proof.

There is no mathematical proof for the odds of an outcome that was never going to manifest. Which is why the tree-door scenario is just irrelevant theater. And is why any mathematical "proof" that includes it is nonsensical.

 

[Evangelicalhumanist]

If anyone disagrees with the OP, I'd be more than happy to set up a test of 100 tries, let's say for maybe $5 a try?

I already provided an on-line test in Post #44. So far, I don't think anybody has even tried it.

Try running the test 500 times (at "instant" speed), once choosing "don't switch" and again choosing "switch." Report back what your results were.

 

[Revoltingest]

I already provided an on-line test in Post #44. So far, I don't think anybody has even tried it.

Try running the test 500 times (at "instant" speed), once choosing "don't switch" and again choosing "switch." Report back what your results were.

Fake news math!

 

[Evangelicalhumanist]

Fake news math!

It's actually quite unbelievable! People have been shown the math. They've been shown links to sites that describe the problem in detail. They've been provided with simulations they can run in seconds.

And still they stick with their incorrect "belief." Such is the nature of religion, I guess. It's not an edifying insight into human nature.

 

[Revoltingest]

It's actually quite unbelievable! People have been shown the math. They've been shown links to sites that describe the problem in detail. They've been provided with simulations they can run in seconds.

And still they stick with their incorrect "belief." Such is the nature of religion, I guess. It's not an edifying picture of human nature.

Belief will tends to trump reality.
Some of the dismissals are imaginative, eg,
it's unknowable because it'll never happen,
a choice between 2 alternatives always has
50/50 odds.

 

[Foxfyre]

Of course. It's all theater to raise the excitement of the audience and to confuse the thinking of the contestant. But in the end, it's two doors and two possible outcomes. (Unless he opts to take the cash buyout.)

Correct. It doesn't matter what the first choice was or that the curtain was pulled back from one of the three original choices. Once that curtain is pulled back the contestant is left with two choices period. And one of those two choices is a car, the other a goat. He/she has a 50/50 or 1 in 2 chance of choosing the one with the car and the one with the goat. It no longer matters what the equation was when there were three choices. Now the only non mathematical factor is the psychological one and the probability that the contestant will stay with his/her first choice which is a different factor. It does not change the equation that he/she has two choices, one good, one bad.

 

[Evangelicalhumanist]

Correct. It doesn't matter what the first choice was or that the curtain was pulled back from one of the three original choices. Once that curtain is pulled back the contestant is left with two choices period. And one of those two choices is a car, the other a goat. He/she has a 50/50 or 1 in 2 chance of choosing the one with the car and the one with the goat. It no longer matters what the equation was when there were three choices. Now the only non mathematical factor is the psychological one and the probability that the contestant will stay with his/her first choice which is a different factor. It does not change the equation that he/she has two choices, one good, one bad.

And on that note, I give up. One can lead a horse to water, but one cannot make it drink.

 

[It Aint Necessarily So]

Here is an actual simulation that you can try for yourself.

Monty Hall Problem Online. Run the monty hall game and simulation , over and over to understand the probability of this problem.

Monty Hall Problem --a free graphical game and simulation to understand this probability problem.

www.mathwarehouse.com

I ran two simulations of 100 games and outcomes (one each way) at your link, Monty Hall Problem Online. Run the monty hall game and simulation , over and over to understand the probability of this problem. When I traded - i.e., chose "Change Choice" - I won 68 out of 100 times. When I didn't trade, I lost 65 out of a 100times.

For those who base their decisions in evidence, the choice is clear - trade.

There is no mathematical proof for the odds of an outcome that was never going to manifest. Which is why the tree-door scenario is just irrelevant theater. And is why any mathematical "proof" that includes it is nonsensical.

And that is how gut feeling works.

You will go on holding to your intuition as if it were fact and ignoring the arguments and the iterative demonstration above.

If you ever get a chance to put it to the test, though you might get lucky after not trading the first or second time (you have 1 chance in 9 of doing that successfully two times in a row), but it will cost you over the long run.

 

[Alien826]

Well, it's a little bit of both, actually -- because there is a change of reference between the contestant's first and second choice. The first choice is:

Guess which door the prize is behind -- this choice has a 1 in 3 chance of being correct.

The second choice is very different: it's not, now guess which (of 2) doors the prize is behind, because your first guess hasn't been answered yet -- it's still in play. No, your second choice is: Guess whether your first guess was right or wrong. That is a very different thing.

And, of course (as you pointed out long ago in your OP, the odds that your first choice was wrong is 2/3 -- and absolutely nothing has happened since to change that. Nobody has moved the car or the goats, so 2 out of every 3 times, if you don't switch your answer, you will be wrong.

No, there's no psychology (which I take to mean emotion based guessing) involved at all. In the real world, it's very different of course, as you are expected to decide quite quickly and Monty has done his best to confuse you. And as many have pointed out the game itself didn't go exactly like this. But that doesn't matter. This not the real world, it's mathematical problem with distinct parameter sand can be solved as such, with mathematics. In this puzzle form of the game, the parameters are fixed. You can rely on them to be as stated. In addition you have as much time as you wish to spend thinking about it. The choices at both stages can be based on mathematics and logic with no doubt whatsoever about the second choice (that is which is more likely to succeed).

 

[Foxfyre]

And on that note, I give up. One can lead a horse to water, but one cannot make it drink.

Oh good. I was tired of repeating myself. But again we can agree to disagree.