Women, math, and the Monty Hall problem

[IndigoChild5559]

Imagine you are on Let's Make a Deal. Carol Merrill shows you three doors. Good ol' Monty Hall explains that behind two of the doors is a goat, but behind one door is the sexiest sports car you've ever imagined. You pick door #1. But before that door is opened, Monty Hall opens door #3, and guess what? It's a goat. Monty looks at you and asks you, Do you want to stick with door #1, or would you like to switch to door #2? If you picked door #2, congratulations, you just doubled your chances of getting the car. HUH?

Like most people in these forums, I have a decent IQ, but I'm no genius. The way it looked to my own mind was this. When all three doors were shut, I could pick any of the three and have a 1/3 chance of being correct. After one door is opened to reveal a goat, and there were two doors left, I figured I had a 50-50 chance of being right, and that it would make no difference if I stayed with #1 or switched to #2.

And I would be WRONG.

Here is the reality: If you switch, you double your chances at getting the car.

When I first saw the answer to this problem and the reasoning behind it, I was like, OMG this is so simple. It is exquisitely beautiful, eloquent, in its simplicity. If you want to know the reasoning, it is at the end of this post.

Now, to the second part of this story.

Although scientists have known for a very long time that switching doubles your odds of choosing the car, they haven't been able to explain it mathematically. It's been one of those famous math puzzles, like Fermat's Last Theorem. Well guess what. It finally did get solved in 1990, and guess who solved it? A woman.

When she first came up with her answer, she was vilified. She received thousands and thousands and thousands of letters telling her she was wrong, including a thousand from PhD's. She was told she misunderstood the problem. She was told she couldn't understand basic logic. She received a lot of ad hominem attacks like, "You are the goat!"

And many of these attacks were blatantly sexist. "Maybe women look at math problems differently than men."

You see, the prevalent theory at that time was that women simply weren't as good at math as men. In fact, even among those scientists who acknowledged that women could be very bright at math, they still maintained that at the farthest end of the spectrum, the math geniuses were men.

As time went on, the evidence became overwhelming that she was correct. We are talking computer simulations. We are talking scientific experiments. Eventually, many of those who had ridiculed her apologized.

And what does she have to say about it? She blames the school system, which teaches kids to memorize answers instead of challenging assumptions and thinking independently.

Okay, I'm sure you are all wondering what her math proof was. I'm going to quote from the article that inspired this thread. Ready?

Let’s break it down. When you first choose a door, you have a 1/3 chance of picking the car and a 2/3 chance of picking a goat. If you’ve picked a goat (which happens 2/3 of the time), switching after the host reveals another goat guarantees the car.​

​

It's a numbers game backed up by MIT supercomputers and the Mythbusters alike:​

​

Pick the car (1/3 chance): Switching loses.​

Pick a goat (2/3 chance): Switching wins.​

​

So, by switching, you win 2/3 of the time.​

OMG is that eloquent or what?

You know how in many of my posts I've have remarked that intuition is adaptive, but it makes a lot of mistakes? This is why the problem is so hard for us to wrap our heads around. Our intuition tells us that after the reveal of door #3, that the odds reset. But that is not how math works in real life. The odds of your INITIAL pick remain at 1/3.

How a woman with the world’s highest IQ shattered a math myth and silenced her critics

Marilyn vos Savant took on 10,000 angry letters to prove her controversial answer right.

www.upworthy.com

 

[Heyo]

You see, the prevalent theory at that time was that women simply weren't as good at math as men. In fact, even among those scientists who acknowledged that women could be very bright at math, they still maintained that at the farthest end of the spectrum, the math geniuses were men.

Blatant disrespect of reality.
While there were mostly men in the field of maths, the farthest end of the spectrum in the 20th c. was Emmy Noether. Noether's Theorem is the basis of the Theories of Relativity. She's to mathematical physics what Marie Curie is to the hard sciences. (Nobel Prizes in physics and chemistry and the first person to ever have got two Nobel Prizes.)

 

[PureX]

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

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. This is why there is no mathematical way of proving your odds increased. Because there was no change in the odds. It was just irrelevant theater. There is only one choice, and two possible outcomes ... therefor 50/50 chance that the outcome will be the car and 50/50 that it will be the goat.

 

[Mock Turtle]

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

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. This is why there is no mathematical way of proving your odds increased. Because there was no change in the odds. It was just irrelevant theater. There is only one choice, and two possible outcomes ... therefor 50/50 chance that the outcome will be the car and 50/50 that it will be the goat.

Not according to the test results, since apparently these coincide with what was presented.

 

[Debater Slayer]

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

Both the math and extensive simulations demonstrate that the odds double by switching doors after one door is revealed to not contain the prize.

Saying that the odds remain 50-50 is like saying that 1 + 1 = 3; the former is just as factually and experimentally incorrect as the latter, albeit far more intuitive.

 

[PureX]

Not according to the test results, since apparently these coincide with what was presented.

Both logically and mathematically the odds are 50/50 for and against choosing the car.

The test results are irrelevant because the odds are equal. Flip a coin 100 times and the "test results" will almost certainly not be 50/50. Nevertheless, the odds remain 50/50 for every coin flip, no matter how many times we test it.

 

[PureX]

Both the math and extensive simulations demonstrate that the odds double by switching doors after one door is revealed to not contain the prize.

No, they don't, because you were never going to choose 1 of 3 doors. That "choice" is just irrelevant theater. And has no bearing whatever on the odds of your actual choice. So your odds did not improve, since they never existed as a reality until there were only two doors, and only two possible results.

Saying that the odds remain 50-50 is like saying that 1 + 1 = 3; the former is just as factually and experimentally incorrect as the latter, albeit far more intuitive.

You have fallen for the theater.

 

[Heyo]

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

And that's where you go wrong.
There is a 2/3 chance that Monty has no option to choose.
When you did pick a goat (2/3) there is one goat and one car left. Monty has to open the goat. That makes it a 100% chance to pick the car by switching.
The relevant information you didn't recognize in your calculation is that Monty knows where the car is. That is the thing that doesn't make it a new situation.

 

[Debater Slayer]

No, they don't, because you were never going to choose 1 of 3 doors. That "choice" is just irrelevant theater. And has no bearing whatever on the odds of your actual choice. So your odds did not improve, since they never existed as a reality until there were only two doors, and only two possible results.

You have fallen for the theater.

The problem with the above is that you're skipping the initial condition where there are three closed doors and jumping to the step where one door is out of the question. In order to correctly calculate the probability of winning in the scenarios of switching and keeping the same choice, the initial condition where each door is closed and has a winning probability of 1/3 has to be taken into account.

This image illustrates the math for all possible scenarios:

No matter which door you pick at the start, switching always doubles the probability of winning due to the fact that two doors have goats behind them, whereas only one has the prize behind it.

 

[Stevicus]

It's still just a shell game. A guessing game.

Besides, as I recall in "Let's Make a Deal," the final three doors usually contained one "zonk," while there was usually a mid-level prize, like a living room set or a refrigerator, while the "big deal" was the car. They also had consolation prizes, usually provided by the sponsors. My mom was on a game show and lost, but she ended up with a year's supply of Rice-A-Roni.

 

[wellwisher]

Say you are playing dice in a casino. You place your bet and you lose. Instead of taking your money, the dealer gives you a second chance to place the same money and bet. You can move your bet or let it ride. Doesn't staying increase your odds?

If I have a coin, over time both sides will equally appear; 50/50. By getting it wrong on the first toss, I was already the farthest away from 50/50 or 0/100. It is in my favor to stay, since the coin toss has to balance out to 50/50, over time.

I am not big into casino math and casino science since it can hide subjectivity, due to not being properly connected to time. The 50/50 coin toss over time, is indeterminate in time, yet appears to be carved into stone.

 

[PureX]

The problem with the above is that you're skipping the initial condition where there are three closed doors and jumping to the step where one door is out of the question. In order to correctly calculate the probability of winning in the scenarios of switching and keeping the same choice, the initial condition where each door is closed and has a winning probability of 1/3 has to be taken into account.

This image illustrates the math for all possible scenarios:

No matter which door you pick at the start, switching always doubles the probability of winning due to the fact that two doors have goats behind them, whereas only one has the prize behind it.

You fell for the illusion that there was a 1/3 - 2/3 option. There never was.

 

[PureX]

Say you are playing dice in a casino. You place your bet and you lose. Instead of taking your money, the dealer gives you a second chance to place the same money and bet. You can move your bet or let it ride. Doesn't staying increase your odds?

No. All that happened is that the dealer removed the consequence of your first bet. Allowing you to repeat it at the same odds.

If I have a coin, over time both sides will equally appear; 50/50.

Actually, it is very likely that they will not come up exactly 50/50 even after many tries. Because the odds are only projections, and the tries are actual events. The real world does not have to obey the projections of our mathematics.

By getting it wrong on the first toss, I was already the farthest away from 50/50 or 0/100. It is in my favor to stay, since the coin toss has to balance out to 50/50, over time.

You are linking all the tries together in your mind. But in reality, every try is 50/50 no matter how many times you try.

I am not big into casino math and casino science since it can hide subjectivity, due to not being properly connected to time. The 50/50 coin toss over time, is indeterminate in time, yet appears to be carved into stone.

Yes mathematics is an (approximate) abstract representation of actuality, like words are.

 

[Revoltingest]

Both the math and extensive simulations demonstrate that the odds double by switching doors after one door is revealed to not contain the prize.

Saying that the odds remain 50-50 is like saying that 1 + 1 = 3; the former is just as factually and experimentally incorrect as the latter, albeit far more intuitive.

I once tried to prove to a friend that switching
doubled the odds. After about a dozen tries,
the proof failed. It was unlikely, but it happened.

BTW, she was criticized for being the "goat".
But this could'a meant "greatest of all time".

 

[Stevicus]

Sometimes it was canned squid instead of a goat.

 

[Alien826]

The problem with the above is that you're skipping the initial condition where there are three closed doors and jumping to the step where one door is out of the question. In order to correctly calculate the probability of winning in the scenarios of switching and keeping the same choice, the initial condition where each door is closed and has a winning probability of 1/3 has to be taken into account.

This image illustrates the math for all possible scenarios:

No matter which door you pick at the start, switching always doubles the probability of winning due to the fact that two doors have goats behind them, whereas only one has the prize behind it.

I have no doubt that what was demonstrated in 1990 is correct. I'm just going to attempt a slightly different explanation from yours (which may not actually be different), to see what you think.

At the middle point, after Monty reveals the goat, you have two possible assumptions (as the diagram illustrates). If you assume you picked the car, you cannot infer anything from Monty's choice, but that doesn't matter, you keep your original choice as the car must be behind the door you chose originally. On the other hand, if you assume you picked a goat door, then once again your choice is determined as you know that Monty wouldn't reveal the car and revealing the goat tells you that the car has to be behind the the other door. In both cases, once the assumption is made, the second choice gives you no choice, you know where the car is. BUT, of course you don't know which assumption is correct. So, it drops back to the original choice, which is that it's more likely (2/3) that you chose a goat. So logically you choose the second assumption, decide on that basis, and switch.

 

[Foxfyre]

Imagine you are on Let's Make a Deal. Carol Merrill shows you three doors. Good ol' Monty Hall explains that behind two of the doors is a goat, but behind one door is the sexiest sports car you've ever imagined. You pick door #1. But before that door is opened, Monty Hall opens door #3, and guess what? It's a goat. Monty looks at you and asks you, Do you want to stick with door #1, or would you like to switch to door #2? If you picked door #2, congratulations, you just doubled your chances of getting the car. HUH?

Like most people in these forums, I have a decent IQ, but I'm no genius. The way it looked to my own mind was this. When all three doors were shut, I could pick any of the three and have a 1/3 chance of being correct. After one door is opened to reveal a goat, and there were two doors left, I figured I had a 50-50 chance of being right, and that it would make no difference if I stayed with #1 or switched to #2.

And I would be WRONG.

Here is the reality: If you switch, you double your chances at getting the car.

When I first saw the answer to this problem and the reasoning behind it, I was like, OMG this is so simple. It is exquisitely beautiful, eloquent, in its simplicity. If you want to know the reasoning, it is at the end of this post.

Now, to the second part of this story.

Although scientists have known for a very long time that switching doubles your odds of choosing the car, they haven't been able to explain it mathematically. It's been one of those famous math puzzles, like Fermat's Last Theorem. Well guess what. It finally did get solved in 1990, and guess who solved it? A woman.

When she first came up with her answer, she was vilified. She received thousands and thousands and thousands of letters telling her she was wrong, including a thousand from PhD's. She was told she misunderstood the problem. She was told she couldn't understand basic logic. She received a lot of ad hominem attacks like, "You are the goat!"

And many of these attacks were blatantly sexist. "Maybe women look at math problems differently than men."

You see, the prevalent theory at that time was that women simply weren't as good at math as men. In fact, even among those scientists who acknowledged that women could be very bright at math, they still maintained that at the farthest end of the spectrum, the math geniuses were men.

As time went on, the evidence became overwhelming that she was correct. We are talking computer simulations. We are talking scientific experiments. Eventually, many of those who had ridiculed her apologized.

And what does she have to say about it? She blames the school system, which teaches kids to memorize answers instead of challenging assumptions and thinking independently.

Okay, I'm sure you are all wondering what her math proof was. I'm going to quote from the article that inspired this thread. Ready?

Let’s break it down. When you first choose a door, you have a 1/3 chance of picking the car and a 2/3 chance of picking a goat. If you’ve picked a goat (which happens 2/3 of the time), switching after the host reveals another goat guarantees the car.​

​

It's a numbers game backed up by MIT supercomputers and the Mythbusters alike:​

​

Pick the car (1/3 chance): Switching loses.​

Pick a goat (2/3 chance): Switching wins.​

​

So, by switching, you win 2/3 of the time.​

OMG is that eloquent or what?

You know how in many of my posts I've have remarked that intuition is adaptive, but it makes a lot of mistakes? This is why the problem is so hard for us to wrap our heads around. Our intuition tells us that after the reveal of door #3, that the odds reset. But that is not how math works in real life. The odds of your INITIAL pick remain at 1/3.

How a woman with the world’s highest IQ shattered a math myth and silenced her critics

Marilyn vos Savant took on 10,000 angry letters to prove her controversial answer right.

www.upworthy.com

As a woman who claims zero aptitude in advanced math--I am very very good at basic math--it requires only basic math for me when it comes to guessing the best choice among three possible choices.

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. And when the game show host gives me the option of changing my choice it really doesn't change the equation. When he allows me to make a different choice it introduces the human element of trust or distrust into the equation and that has nothing to do with math.

 

[Subduction Zone]

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

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. This is why there is no mathematical way of proving your odds increased. Because there was no change in the odds. It was just irrelevant theater. There is only one choice, and two possible outcomes ... therefor 50/50 chance that the outcome will be the car and 50/50 that it will be the goat.

No. That is a misunderstanding of how the system works. Monty always knew which door had a goat behind it. Your claim would be true if Monty did not know and sometimes exposed the car by mistake and that did not count. But since Monty knew which door the car was behind he could always show the contestant a goat. Monty did not make his choice until after the contestant did. One's odds are two out of three if one switches because you are getting one extra chance for free. But by sticking with your original choice you would be limiting yourself to your first choice.

Here is another way of thinking of it. You are given your one out of three choices. You are then given a choice to trade your one choice for the other two doors. That is in effect the choice the Monty is given. I need to emphasize that Monty always knows so when he gives you your choice he will always show a goat first. The set up is that Monty always gives a person a choice, and since Monty never reveals a goat we know that he knew where the car was. You are always switching from what was behind one door to what was behind two doors.

 

[sayak83]

The problem with the above is that you're skipping the initial condition where there are three closed doors and jumping to the step where one door is out of the question. In order to correctly calculate the probability of winning in the scenarios of switching and keeping the same choice, the initial condition where each door is closed and has a winning probability of 1/3 has to be taken into account.

This image illustrates the math for all possible scenarios:

No matter which door you pick at the start, switching always doubles the probability of winning due to the fact that two doors have goats behind them, whereas only one has the prize behind it.

Another easy way to explain is that,
The first choice creates three possible universes. In one universe you have chosen the correct door and switching causes a loss. In the other two universes you have chosen the wrong door and switching causes a win. So the switch let's you win in 2 universes and not switching let's you win in 1 universe. Hence switching gives 2/3 victory probability.
Simple.

 

[Subduction Zone]

As a woman who claims zero aptitude in advanced math--I am very very good at basic math--it requires only basic math for me when it comes to guessing the best choice among three possible choices.

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. And when the game show host gives me the option of changing my choice it really doesn't change the equation. When he allows me to make a different choice it introduces the human element of trust or distrust into the equation and that has nothing to do with math.

And your math is bad. Like most people you forgot that Monty knows. When there are two doors left he can always show a contestant a goat. In effect you are changing from only have one chance to having two.

Do you remember the TV show Mythbusters? They did an episode where they tested this and it works. The problem is a bit misleading because people forget the fact that Monty is not totally honest since he knows.