## Follow the Maths

### Follow the Maths

I'd like to hear your take on the classic "Monty Hall" problem. Here are three videos to tell you what the problem is and the math behind it. Do you agree with the math and why? Do you disagree with the math, and why?

The original Monty Hall problem.

https://www.youtube.com/watch?v=4Lb-6rxZxx0

The "explanation" of why the maths. You watch the vids and tell me if you agree with your original assessment or agree with the explanation.

https://www.youtube.com/watch?v=7u6kFlWZOWg

Another explanation, may or may not change your mind

https://www.youtube.com/watch?v=mhlc7peGlGg

Here's a movie clip, maybe this will help make sense of the problem?

https://www.youtube.com/watch?v=Zr_xWfThjJ0

The original Monty Hall problem.

https://www.youtube.com/watch?v=4Lb-6rxZxx0

The "explanation" of why the maths. You watch the vids and tell me if you agree with your original assessment or agree with the explanation.

https://www.youtube.com/watch?v=7u6kFlWZOWg

Another explanation, may or may not change your mind

https://www.youtube.com/watch?v=mhlc7peGlGg

Here's a movie clip, maybe this will help make sense of the problem?

https://www.youtube.com/watch?v=Zr_xWfThjJ0

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### Re: Follow the Maths

Well I think the math is right even though I have never thought of it in those terms. I think most people would say after the first "goat" is revealed that their chances are 50/50 not 33/66. Very interesting thing look at.

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### Re: Follow the Maths

It's hard to understand, but I accept its truth!

### Re: Follow the Maths

I'm of the 50/50 camp.

Let's suppose you walk into a room and are given a choice between two doors, there are only two door, your odds of winning are 50/50. There is no third door, it never existed.

Now, if you walk into a room with three doors and one of those doors is revealed to you to be fake, you STILL only have two working doors to choose from, do you not? So, how have your odds changed simply because a fake door was revealed? If all doors are left in play, sure, maybe the odds move around a bit, but this is simply not the case, there are only two doors in play.

Let's suppose you walk into a room and are given a choice between two doors, there are only two door, your odds of winning are 50/50. There is no third door, it never existed.

Now, if you walk into a room with three doors and one of those doors is revealed to you to be fake, you STILL only have two working doors to choose from, do you not? So, how have your odds changed simply because a fake door was revealed? If all doors are left in play, sure, maybe the odds move around a bit, but this is simply not the case, there are only two doors in play.

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### Re: Follow the Maths

I would assume they have all the episodes of this game show on tape somewhere. And it should be a large enough sampling size to draw some conclusions. This may hold up. I've seen the new revised show a couple of times just to see if they changed things up and oddly this situation is one of the things they did change. Now you only get one pick at a door. They don't let you switch. My conspiracy side tells me that they've looked at enough samples of the original show to know that this actually holds up in the real world thus the change.

### Re: Follow the Maths

I was for a really long time but I think I get it now. I'll try and persuade you.

Yes, that would be 50/50 for sure. Two doors, one choice.

But you got the order wrong. You need to make the choice of which of the three doors is fake BEFORE it is revealed to you which of the two remaining doors are fake, then you can elect whether to stay with or switch your choice.

If one door is revealed fake BEFORE you make your choice, that revealed door becomes mathematically irrelevant among the remaining two doors.

For example, pretend it's 100 doors instead of just three. If 98 of them are revealed to be fake before you make any choice, then you are just left with two doors. Same thing.

So going back to 100 doors. You pick Door 1. Then doors 2-33 are revealed fake, and 35-100 are fake. Do you stay with Door 1 or do you declare Door 34 to be fake?

### Re: Follow the Maths

The problem is, once a door is revealed, it removes it from play, it becomes irrelevant and can no longer be factored into the equation. The choice is narrowed to 2 doors, both have an equal chance of having the prize.

Part of the problem is that virtually any viewpoint can be supported by math, basically, math can be twisted to create the results the observer chooses. IMHO, that is what is going on with this scenario.

Part of the problem is that virtually any viewpoint can be supported by math, basically, math can be twisted to create the results the observer chooses. IMHO, that is what is going on with this scenario.

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### Re: Follow the Maths

I found another video by the same guy explaining the problem a different way.

https://www.youtube.com/watch?v=C4vRTzsv4os

I can see how the 33/66 camp arrives at their statistical advantage, unfortunately, they're still wrong!

https://www.youtube.com/watch?v=C4vRTzsv4os

I can see how the 33/66 camp arrives at their statistical advantage, unfortunately, they're still wrong!

### Re: Follow the Maths

This just hurt my brain a little. Like that time Shannon made me calculate the hypotenuse of a triangle for the first time in about a decade to figure out dimensions for my future shed. You trying to keep us sharp, Spruce?

### Re: Follow the Maths

He was teasing you, triangles don't have hippopotamus'!

Nah, just wondering what your perspective of the game problem is.

Nah, just wondering what your perspective of the game problem is.

### Re: Follow the Maths

No! It's not irrelevent at all. That reveal is very important because it completely eliminates any probability *within* the 2/3 probability. Keep with me...

So you have three doors. You pick one. There's ONLY a 1/3 chance it's the good door, right? That means there is a 2/3 chance the two remaining doors you did not pick are good doors. Still with me?

A 2/3 chance is better than a 1/3 chance. The other two doors have a greater possibility of being a good door than just the one door you picked.

So the game host reveals one of the two remaining doors in the group of 2/3 probability that one of them is a bad door. That still does not change the outcome of the other door he did not reveal that you did not initally choose--it still has that original 2/3 probability that it had after you made your first choice!

It's counterintuitive, but it makes sense mathematically.

Well you have to put on your math cap and really suppress your intuition to understand this problem. It helps to think of it at the much larger scale of, say, a million doors.

Say you pick Door #1 out of one million doors. The host reveals 999998 other doors are bad except Door #227443 which you still don't know is bad or good.

Are you going to stick with your Door #1, or switch to Door #227443?

You would be a fool not to switch. Because the first time you made your choice, your chances were a razor thin 1/1000000 it was a good door, and ONE *of* the doors among Door #2 through Door #1000000 are 999999/1000000, or nearly 99% chance it's a good door.

If all EXCEPT ONE of those OTHER doors are eliminated, there is still that 99.99% chance that Door #227443 is the good door, and only the vanishingly small 0.0001% chance your first choice is correct.

If you can apply that same logic to just 3 doors instead of a million doors, it begins to make sense.

Last edited by Aaron on Thu Oct 12, 2017 2:47 pm, edited 1 time in total.

### Re: Follow the Maths

I'm still chewing on this . . .

### Re: Follow the Maths

Does the millions of doors make sense? You pick one, and then all but another one are eliminated so there are two left--just the first one you picked and one other?

### Re: Follow the Maths

I am not sold on the idea just yet, however, I do believe that I have found the fly in the ointment that is creating the discord.

I am looking at the problem from the perspective of choice, in which case, it is 50/50 as to whether or not you have chosen the right one. You are coming in from the math side, which is indicating the higher probability that the "unchosen" door has the statistical advantage of winning.

I'm still chewing on it though.

I am looking at the problem from the perspective of choice, in which case, it is 50/50 as to whether or not you have chosen the right one. You are coming in from the math side, which is indicating the higher probability that the "unchosen" door has the statistical advantage of winning.

I'm still chewing on it though.

### Re: Follow the Maths

No, it's never 50/50.A. Spruce wrote: ↑Thu Oct 12, 2017 3:25 pmI am not sold on the idea just yet, however, I do believe that I have found the fly in the ointment that is creating the discord.

I am looking at the problem from the perspective of choice, in which case, it is 50/50 as to whether or not you have chosen the right one.

Your first choice is 33.33/33.33/33.33. Then it's 33.33/66.67 after all the bad doors among your unchosen (which happens to be just one) are revealed. You should switch to the door on the 66.7 side of the probability, rather than remaining on the 33.3 side.

But of course that could be the wrong choice if you luckily picked the good door in the first place despite your odds. Switching would then be a mistake, and you should stay. The host will never tell you that, obviously, and he will reveal at random just one of the two remaining bad doors. But statistically you are always better off switching in the long run.

To be crystal clear, it's a greater probability ONE of the doors AMONG the unchosen doors is a good door. Whether those unchosen doors number two or more (millions), the group of unchosen doors has a higher probability of containing ONE GOOD DOOR than just the door you picked BEFORE all the BAD DOORS among the unchosen are revealed. That is the key.You are coming in from the math side, which is indicating the higher probability that the "unchosen" door has the statistical advantage of winning.

If you don't get it by now, you're just being difficult.I'm still chewing on it though.

### Re: Follow the Maths

I think he has gone off on anther bender and is not thinking straight

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### Re: Follow the Maths

The probability is extremely low that I have a choice of being any other way!

Suffice it to say, I get what the math is saying, I just don't agree with it quite yet.

### Re: Follow the Maths

This assumes that I finished the first bender . . .

### Re: Follow the Maths

Oh, don't worry, I only buy winning lottery tickets.

### Re: Follow the Maths

Now who's being difficult, hmmmmm????

### Re: Follow the Maths

He started it! He started it!!

### Re: Follow the Maths

Here ya go spruce

"Actually Monty does you a confusing (to most people) disservice by throwing one away early. If he left the unpicked two doors closed and said you could open them simultaneously, and if you see a car … you win. Almost all would then trade for the two doors." - Don M from https://betterexplained.com/articles/un ... l-problem/

The key is knowing that your first choice is probably wrong. 2/3rds chance. By monty opening a door, the field is narrowed and the odds are increased that the door nobody chose is the car ONLY because the odds are your first choice is wrong.

With monty opening the inevitable goat, and factoring that your first choice is probably wrong, its more likely that its the unchosen door.

"Actually Monty does you a confusing (to most people) disservice by throwing one away early. If he left the unpicked two doors closed and said you could open them simultaneously, and if you see a car … you win. Almost all would then trade for the two doors." - Don M from https://betterexplained.com/articles/un ... l-problem/

The key is knowing that your first choice is probably wrong. 2/3rds chance. By monty opening a door, the field is narrowed and the odds are increased that the door nobody chose is the car ONLY because the odds are your first choice is wrong.

With monty opening the inevitable goat, and factoring that your first choice is probably wrong, its more likely that its the unchosen door.

### Re: Follow the Maths

Thanks for the input, DT, I'll check out the link.

### Re: Follow the Maths

Hit the nail on the head Texas To me, this makes it a no brainer..... I'm not sure if Sprucey will get it yet, thou

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### Re: Follow the Maths

Yes! That makes it VERY easy to understand.

Pick a door. Now you get a choice. Either stick with that door you picked, or open the other two doors sumultaneously. In either scenario if a car is revealed you win.

That whole "reveal and choose whether to switch" thing adds mystique and obfuscation to it. Funny how our minds work.

Pick a door. Now you get a choice. Either stick with that door you picked, or open the other two doors sumultaneously. In either scenario if a car is revealed you win.

That whole "reveal and choose whether to switch" thing adds mystique and obfuscation to it. Funny how our minds work.