h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi again! Thanks for the beer hunt vid, that is a good practical demonstration of the nature of the problem. Florida Sailor re point 8: That is MY FORMULATION of the problem that I'm asking you to analyse ... in other words, analyse it like this for the sake of simplifying the problem. If this was a real game show things would be different, but it isn't a real game show, it is a thought experiment. Remember I'm the one asking, I'm asking that you imagine the situation exactly as I'm describing it - and the reason I'm asking you to imagine the situation this way is because it is the *simplest* possible situation to imagine and analyse mathematically. I am trying to get people to understand certain simple mathematical principles, and the thing is if people can't understand the simplest principles in the simplest situations, they can certainly not understand much more complicated situations. My goal here is to try and explain to as many people as possible a very simple situation ... which with this particular problem is hard enough! If the problem is COMPLICATED beyond what I'm describing here, then indeed people will have way too much difficulty understanding and will go away thinking 'this is way too complex to understand'. But then they'll have learnt nothing! I would like at least a person here and there to actually learn something about mathematics and probabilities which is why I formulate the problem with rigorous conditions - which indeed makes it as simple as possible. So that people have a chance of actually grasping certain principles and learning something.

So, if you want to set it up as an actual experiment, then set it up as closely as possible to the way I'm formulating it. It IS possible to set it up in a way where neither the host nor the contestant is trying to 'cheat' or influence the outcome. The beer hunt vid is an example: they were trying to stay as close as possible to a 'pure' outcome for the sake of actually testing the mathematics. If you set the experiment up you can even have both the host and the contestant simulated by a computer.

Formulated like this it is a problem that can be mathematically analysed. When you have this analysis, then you have a model predicting certain outcomes. So when you turn this into a practical game, then you can, using the mathematical outcomes predicted by the model, compare the workings of the actual game to that, and you can then based on that see if the real game show diverts from that, and if it does, you can better understand if non-random elements are factoring in.

But first you need to understand the 'pure' and simple situation. Every factor of non-randomness entering an actual game will make the situation more complex. But I'm asking that you just look at the most straightforward and simple situation which is the one I formulated in my eight points. The contestant chooses any door; the game show host opens a door showing a goat: he HAS TO open a door, and it HAS TO be one with a goat; he ALWAYS gives the contestant a chance to siwtch and he DOESN'T try to influence the contestant. Make him a robot or a computer if you're still worried BUT THIS IS JUST A THOUGHT EXPERIMENT! Can't you even just in your own imagination, for the sake of having a mental model, imagine a host that doesn't care either way and therefore does NOTHING to influence, confuse or clue in the contestant? This is what thought experiment means - it means setting up in your mind a situation that may not exist in real life, but it is for the sake of understanding a principle. But as the beer video shows you can even in real life set up a test of the principle and because it is not about who wins or loses, it's about testing the strategy and whether the statistics pan out in reality, you will have a situation that comes very close to what the mathematics predict.

So I'm asking as a matter of courtesy that you don't complicate the problem by introducing elements beyond what I've formulated unless we've managed to get everyone to understand what is involved in the simplest, 'purest' situation.

So just let me add: once we've looked at the situation where there's no bias at all, any way, THEN we can see what difference it would make if for example the contestant catches the host hesitating before opening a door with a goat, or if the host is biased towards opening one door rather than another - and the contestant notices that - or if the host acts somehow betrays knowledge of the car being behind any particular door, such as perhaps using a more persuasive tone of voice when asking if the contestant wants to switch if he knows the first chosen door was hiding the car, etc. etc.

All I can think of is what are the chances the contestant would rather have the goat than the car?

As far as preferring the goat to the car: someone who lived in a culture with no paved roads, and no gas, might prefer the goat.

However, I will bow out of this discussion now. I'm grateful to Florida Sailor and Willem for explaining my error.

I thought it was like the game show situation. And Ii'm with them and Tav on that interpretation.

If it's a thought experiment based on pure math, I haven't a clue. I can't eve follow notational logic.

I'll just sit this one out, and let you all explain it.

Please Dmitri, don't bow out! This is mathematics but it's really really simple mathematics ... the way I want to explain it, I want to try and make it as simple as possible. No notational logic ... no formulae, even. Just simple concepts needing to be grasped. I started this thread because I wanted to test whether I could effectively explain something regarding probabilities to people here. If you bow out you're not even giving me a chance to try and explain! And there's a lot riding on this. If I can successfully explain this it will mean a lot to my self-confidence. I have a chance of teaching maths to kids ... but I need to know if I can do it. I have a belief that just about anyone can learn mathematics ... if the student doesn't learn then the teacher has failed. But this is also an experiment in people's *willingness* to learn. Will people stay around and actually give me a chance? Will they 'get' why I'm setting the problem up like this? All of this is relevant to real-world teaching: how does a teacher get a concept across to his/her students? What is best: a teacher giving children a formula for solving a problem, or trying to teach children to understand a problem so as to be able to solve it? Saying 'my way or the highway'? Who or what counts as authority today? Could you actually find the answer to this problem on the internet? Suppose you found several websites and they say different things ... which one do you believe? Suppose you have two college professors, and the one says one way is right, and the other one says differently? Do you then give up and consider the problem to be insoluble? What if it isn't? What if it *still* is actually quite simple, and there is a clear solution but some people just can't see it due to how they're wired? If that is the case, how do you prove *that*? What does that imply for humans trying to prove things ... anything at all? This has implications all over the place. It's really not for nothing that I started this discussion.

So OK just to show how simple it is to grasp *a* principle of probability. Suppose you DO want the goat. What are your chances of getting it, if the game is set up as in my eight points?

Okay, now I get it, Willem. No problem. I just didn't want to hold the discussion back.

If I understand it correctly - and that's very iffy - you're setting up a thought problem that goes like this:

There are 3 doors.

1 door leads to a car.
1 door leads to a goat.
1 door leads to something else, neither good nor bad.

You choose one. That's a random probability of 1/3.

Now, someone opens another door, and it's not the goat - which is the one you want. Okay, now you have 2 doors to choose from.

The question is: Do you change your choice?

As I understand it, you now have a 50/50 chance of choosing the right door. I don't see any way around that.

I also don't see why eliminating the third door affects the choice you make now.

Remember: If you explain this to the simple, you need simple sentences.

(And remind me later to tell you the funny story about the German colonel and the math students. But I don't want to interrupt now.)

Hey Willem

When I read this thread I thought you were posing a question to us who read and participate in your journal as well those who may

It seems to have spiralled now and I am not keeping up at all

I thought your posts to Florida Sailor and Dmitri were difficult to keep up with and quite intense

Coming up with a lesson plan for the children would be interesting and challenging, perhaps we could start on that one

My first suggestion would be smarties and a cola bottle instead of a car and goats, because they will all want to take the goats home otherwise

HI again Dmitri and Peanut! Let me tell you a problem of this thread - and if you can help me then I'd be very happy. My goal is to explain something about probabilities. But I want to approach the problem NOT like a typical teacher/student situation, so I don't want to give the answer and then say 'OK everyone learn that, do it that way'! I want to hear people's ideas first. Now everyone here has his/her own idea about this. So we're left with a lot of ideas, now I have the huge task of looking at all these ideas and on my side seeing how other people are seeing things. And then after that I have to try and give an explanation that satisfies *everybody*. Now this is a pretty huge task I'm setting myself, so I get a bit panicky about it! In essence the task I'm setting myself is one of harmony: of trying to harmonize the different interpretations here to the maximum extent possible. That needs to be done before I can try actually explaining everything.

The reason I use the goat/car setup is because this is the way the problem has been setup by other people so *after* I've given my own explanation I can point people to certain other websites where the problem is also being discussed. But I don't want to do that yet, because I want to try my own explanations before pointing people to other people's explanations. The experiment can be done with smarties and cola, or with playing cards, or with thimbles and peas, provided it's set up in the same way, mathematically it's the same problem!

Anyways. I'm not anywhere near applying for a math teaching job yet, so this is just a test for now to see if I can explain a mathematical sort of situation. But this is about more than math, it's about understanding and about explanation in themselves which for me are pretty intense issues. I can *try* and be less intense! But you see it's also difficult to keep the conversation on topic, because lots of elements are being introduced to this conversation that are well outside the problem itself! I don't want to be a dictator and demand 'stick with the problem'!

Ok,

if the host knowingly chooses a door with goats that is a different question to the host making a random choice,

isn't it

Otherwise from the beginning isn't your chance of winning a car or a goat 50/50?

a *simpost* with Willem

No problem Peanut! But OK now for Dmitri's sake and for other folks too I'm going to dramatize it and introduce one more element that might just connect something for some of you! But the problem is mathematically still the same. SO:

"Who wants a try at this splendid game of chance? You, ma'am? Well then! Here you have three doors. One hides a Ferrari; two hides mangy goats. You don't want those, you want the Ferrari. Just choose right and you'll have it!

"Now be sure your choice is free and fair! Choose any of those doors. You can throw a dice if you want to - until it comes up either 1, 2 or 3, then take that door. I shall turn my back. May I be trampled by a herd of hippopotami if I try and influence your choice. There, I turn my back.

"Finished? All right you chose that door. Now ... you sure about your choice? Quite sure? You understand the odds: you have a one-in-three chance the Ferrari is behind that door.

"Say what. I'm feeling generous. How about ... instead of that door, the one you chose ... I offer you the chance to switch to *both* of the other doors! If the car is behind the one, OR behind the other, you win it! You can double your chances - from 1 in 3, to 2 in 3! Wanna go for it?

"No? You sure? Look, let me show you ... I am willing to open one of these other doors. I open this one ... see, there's a goat behind this one. Now the car might be behind this other one. Do you want to switch? The odds remain the same ... 2 in 3 ... but just 1 in three if you stick with your first choice."

"You choose to switch? Wise choice! See there what you've won!"

Yes, EXCEPT...

That doesn't mean the Ferrari is behind that door!

It's a *probability*.

I think I see what you mean now. You're talking about mathematical probabilities. That makes sense to me.

But it isn't real. It's math.

Over time, that will account for testing it 100 times, and getting 66 right answers.

But it won't tell the individual that 'Yes, that's the right choice, right now.'

Am I right?

You're right Dmitri! It doesn't give you a 100% certain answer but it gives you a 66%-or-thereabouts certain answer! But if you repeat it, say 100 times, and it's the same situation every time ... you will win about 66 cars if you switch, while if you don't switch, you'll only win about 33 cars.

I think you have your answer about how to explain it to kids.

This exercise WON'T help you decide which door to pick - if you only pick once.

It WILL help you decide - if you get to pick a whole lot of times.

Actually, that's what my teacher said in school, too.

Willam



I was not trying to change your formula, only to explain to Dmitri why your problem was different from the game show.

The only reason I mentioned the hesitation is because it effect the results of an experiment.

Your example of offering both doors without seeing the contents is a good way explaining your point.

In playing multiple games it is always best to switch as you can only be correct 1/3 of the time.

I will maintain that in a single game the odds drop to 'was I right or was I wrong?' a 50% - 50% chance. As the number of games increase so does the odds of your being wrong, but the odds are the same in any single game.

Your example was for a single game

I hope you don't think I am arguing with you, I am just presenting my point of view of your question.


F S

Hi Dmitri and Florida Sailor! Florida sailor, I just was getting a bit panicky because this thread is a bit more than I can handle with my social anxiety and all!

But anyways re what you and Dmitri say here about an event just happening *once*.

I studied physics, mathematics and philosophy at university so I get a lot of ideas from there. In reality: what is the significance of probability when it comes to a single, unrepeated event?

Suppose you have a six-sided die and you throw it only once. You have a one-in-six chance of throwing a six ... in fact you have a one-in-six chance of hitting *any* particular number. Suppose you throw it just once aiming for a six. You get a four.

Does it make sense to say you had a one-in-six chance of getting a six ... or a four?

In *retrospect* you had a 100% chance of getting a four - since that's what you got - and a 0% chance of getting any other number.

UNLESS ...

The 'many worlds' theory of quantum mechanics is true! This explanation is one that tries to make sense of quantum probabilities. Let us forget quantum mechanics and imagine it as just pertaining to ordinary real-world probabilities (again a thought experiment!) If the many-worlds theory was true for the throw of a dice then this would happen:

Each of the six possible outcomes turn into ACTUAL outcomes! So the universe splits into six different ones and in each universe one of the possibilities happen. There's a universe in which the die landed on one; another where it landed on two; another that landed on three and so on. THEREFORE to say that you had a one-in-six chance of getting a six means there is one universe out of six in which exactly that happens.

The same with the car/goat problem: if you stick with your first choice, there is one universe in which you win a car and two in which you win goats. If you switch, then there are two universes in which you win the car, and one in which you win a goat.

Remember this was actually a theory that was trying to make sense of quantum probabilities... but the principle actually also applies here because the throw of a die or any other even *always* incorporates lots of quantum probabilities the result of which also determine their outcome ... in practice though we're talking of millions or billions of quantum events, not just two or three or six or so!

That's true enough - though the 'many worlds' theory is still pretty hypothetical. It can't be demonstrated to exist, I don't think.

Now, mind you, I tend to read a lot of weird anecdotal evidence online.

One man insisted that when he was a kid, he got caught on a railroad trestle, just when a train went over. He said he felt an odd shift in his perception, and then found himself off the track. It was his assertion that he had experienced a universal shift, and that there was a version of reality in which he was hit by the train.

Then there was the strange tale of the man who appeared at Tokyo airport, many years ago. The man had a passport from a non-existent country, which he described as being somewhere in the area where Andorra is. He had odd currency on him, as well. The man insisted that he had travelled to Tokyo in the normal way, and didn't know what the fuss was about.

The authorities locked him in a hotel room, and guarded the door. The next day, he'd disappeared! Or so the story goes.

Now, whether these anecdotes have any bearing on the 'many-worlds' hypothesis,I'm not inclined to say. Is the existence of quantum moments evidence? I dunno. But this could account for the resistance of some people to games of chance. And Einstein's claim that 'Gott wuerfelt nicht'.

I am with Willem on this.

Let's imagine there were 9999 goats behind 9999 doors and 1 lamb-orghini behind 1 door.

You choose 1 door - you have a 1/10000 chance of finding the Lamborghini behind that door.

The host then says these 9999 doors that you haven't selected - they are all mine. Then the host opens
9998 of these doors to reveal 9998 goats. The host only has one more unopened door - but she offers
you the chance of making a swap - your selected door for her remaining unopened door - could one
conceivably imagine that there was only a 50-50 chance of finding a goat behind the hosts remaining
unopened door.

I meant only 50-50 chance of finding the Lamborghini behind the hosts remaining unopened door ... (but I haven't read that jstor paper so maybe they will pull out a plum)