A fascinating mathematical problem

Michael, it actually makes no difference what you choose to start with.

If doors A & B are 'goats' and C is 'the car', because the host is always going to eliminate A OR B, the choice is actually between (A OR B) and C ie. there are two choices NOT three.

So to 'correct' John's example

Stick | Switch

Goat | Goat Car )
----------------)->>> Switch wins
Goat | Car Goat )

Car | Goat Goat ->>> Stick wins

or if you prefer:

Stick | Switch
Goat | Goat Car ->>> Switch wins
Car | Goat Goat ->>> Stick wins
Car | Goat Goat ->>> Stick wins
Goat | Car Goat ->>> Switch wins

(because if you choose the car at first, the host can choose either of the other two to eliminate, thus giving four possible sequences and two outcomes).

quote:

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"If you are not convinced, try an experiment for yourself and see if you can 'win' the car 66.6% of the time rather than 50% of the time"

I bet you a million pounds it doesn't.

---

?????

Patrick - I initially agreed with you but now I realise the following with the two options below -

Car | Goat Goat ->>> Stick wins
Car | Goat Goat ->> Stick wins

If you are picking A, then the two scenarios above have a combined probability of 1/3, not 1/4 each.
The reason being is that 1/3 of the time the car is behind A then 1/2 the time the host reveals a goat behind B and 1/2 the time he reveals a goat behind C.
1/3 x 1/2 = 1/6

So each scenario happens 1/6 of the time.

Whereas for the other two scenarios -

Goat | Goat Car ->>> Switch wins
Goat | Car Goat ->>> Switch wins

The probabilities are -
1. Goat is behind A = 2/3
2. After that, the chance of a goat being behind B or C is a 1/2.
= 2/3 x 1/2 = 1/3
So each of the scenarios above have a 1/3 chance of happening. Since each of those scenarios result in you winning if you switch then the probability of you winning if you switch is 2/3.

Does that make sense? I am a bit vague on probability formula these days, but I think it goes something like that.

Charlotte

OK, I give in! It's not analagous to the heads/tails example because the doors were set-up when there were 3 possibilities, and they are not reset once there are only two.

So how can I use this 'amazing new discovery' to make me rich?

Gosh, now I'm really confused! Simon are you 'Charlotte' just at weekends?

quote:

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(I only ever scored 100% in one degree exam and it was Probability Theory)

---

Yes, you're right - it was a fluke.

Erm no, but my girlfriend seems to have discovered the Naim message board!

Patrick,

The Monty Hall Show ran for quite a while. So most, if not all the contestants knew the process. It was always the same.

Contestant picks a door
Host opens one of the OTHER doors and ALWAYS shows a goat
Contestant gets to change his mind or stick

For those who suggested an "experimental" run, well, that what my computer simulation did.....and sure enough, STICK and you win 0.33 of the games. CHANGE you mind and you win 0.67 of the games.

NOW....if part way through the game, just after Monty has opened the door to reveal a goat, an alien drops in, unaware of the rules and what has just happened.....well, he has two doors to choose from, one with a goat and one with a car. So HIS chances of choosing the car at this point ARE 50/50

Cheers

Don

But Minky, what did you think of "The curious incident of the dog in the night-time"?

I read it a few weeks ago, just interested what you though..... Apart from the intriguing mathematical bit

Computers are supposed to work on 1's and 0's - in other words "Yes" or "No" - why does mine frequently say "Maybe"?......

Don

"NOW....if part way through the game, just after Monty has opened the door to reveal a goat, an alien drops in, unaware of the rules and what has just happened.....well, he has two doors to choose from, one with a goat and one with a car. So HIS chances of choosing the car at this point ARE 50/50"

NO! It can never be 50/50. The process of choosing left or right, 1 or 2, can only be 50/50 if no process of elimination occurs.

OK - It's getting late and I am totally baffled by this problem.

3 doors - you've a 1 in 3 chance of winning if you stick. Whatever happens to any other door is totally unimportant.

Host eliminates one door and then offers a switch. The host MUST know where the car is to open a losing door in order to offer the switch. If he opened the door to reveal the car then game over and your chance was only ever 1 in 3.

Surely your chance is now 1 in 2 regardless if you switch or not as you are not getting the option of chhosing the opened door.

The opened door and the offering of a new question must surely negate the original question.

It's a 50:50 chance - win or lose - car or goat. It's surely that simple.

Blythe,

It was very thought provoking. I must admit that I read it with a sense of impending doom because at any moment it could easily have drifted into "about a boy" style mawkishness. Thank god that Haddon never allowed this to happen (although the last page jarred a bit in this respect).

It's interesting that Oliver Sacks was quoted on the front cover because I got some of the same visceral response reading this book as his "Awakenings". Behind things that most of us dismiss as "easy" lie extremely delicate and complicated mechanisms. I'm not sure that dwelling on this in real-time is constructive but it could help with things like tolerance and compassion.

Disturbingly, I felt myself identifying with Christopher on some level. I thought his analysis of R&R was spot on.

What did you think ?

By an odd coincidence I was reading some stuff this last week which uses basic probabilities and simple statistics not a million miles away from those at work in this thread to work out the rather elegant solution to a seemingly intractable problem. I should warn you this is a pretty tough problem, but if you fancy yourself a bit handy at this sort fo thing then read on.

Mulder and Scully are each sent hundreds of thousands of uniquely numbered steel boxes. Each box has three doors numbered 1, 2 and 3.

With the boxes comes a letter which explains that the boxes contain a machine which, when you open one of the doors, will randomly flash a light either red or blue. Once you have opened the box and the flash has been emitted the box will be useless and no longer function. There is no way to determine before opening the door which colour the box will flash and the box bursts into flames after the flash which prevents any post-test investigation into how it works. Finally, the letter states, the boxes come in pairs, and if Mulder and Scully both open the same door on boxes with the same number then they will both see the same colour.

Intrigued by this Mulder phones Scully and they start opening boxes. Sure enough each time they open the same door on boxes with the same numbers the machine inside flashes the same colour for both Mulder and Scully.

"This is amazing", says Mulder. "I'm here in New York and you're in LA and yet these boxes are communicating with each other using an advanced alien technology".

"Pah! Why would you say that?" asks Scully.

"Becuase the flash colour is randomly determined when you open the door and yet our boxes always flash the same colour when we open the same door on boxes with the same number. So when I open my box it must somehow tell your box what colour to flash for the door I just opened so that it gets the same colour as mine".

"Don't be stupid, " says Scully, "obviously the flashes are not random but pre-determined. Each pair of boxes is just programmed to flash a certain colour when a certain door is opened."

"So inside my number 2,754 box there might be a program that says "1=red, 2=red, 3=blue" and your box 2,754 just has the same instructions. Give each pair of boxes a random 'program' and it looks like they are randomly flashing red or blue but still agree."

"Yes but...", starts Mulder.

"No buts", says Scully, "given that the boxes self destruct after we open the door so we cannot work out how they work, there is no way you can prove its truly random and not pre-programmed. And since them being programmed is a much more likely explanation than some bizarre, unknown form of instantaneous communication over 1000s of miles I think we have to believe the former".

Mulder is initially stumped and concedes defeat. But after thinking about it for a while he phones Scully back and announces that he can prove that the boxes are not pre-programmed.

"Ok," says Scully, "I'm listening".

"What we need to do is each open a lot of boxes but each of us randomly determine which door to open. We record the flashes for the door we opened on each box and compare our results".

So they each open their half of (say) 10,000 pairs of boxes randomly and independently deciding which door to open. Then Scully crunches the combined numbers in her iMac and analyses the results -- and as it turns out that Mulder and Scully saw the same colour in 50% of cases.

So who is right, Mulder or Scully?

Matthew

quote:

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"NOW....if part way through the game, just after Monty has opened the door to reveal a goat, an alien drops in, unaware of the rules and what has just happened.....well, he has two doors to choose from, one with a goat and one with a car. So HIS chances of choosing the car at this point ARE 50/50"

NO! It can never be 50/50. The process of choosing left or right, 1 or 2, can only be 50/50 if no process of elimination occurs.

---

For the alien no process of elimination has occured - unless the alien is then told which door had been originally chosen by the contestant prior to the host having opened one of the other two doors.

Thus for an uninformed alien, the odds are at 50/50; for the contestant who switches, the odds are 2/3 in his favour.



Regards,

Steve.

Three guys are in a prison cell condemned to death at dawn. However, one of them is to be spared, and the decision as to which one it will be has already been taken by the guard. The three prisoners are aware that one of them is to be spared but they don't know which.

One of the three prisoners is awake while the other two sleep. He asks the guard to whisper the name one of the prisoners who will die at dawn other than himself. (The two sleeping prisoners obviously, for whatever reason don't care whether they live or die, which is the only explanation I can give for their being able to sleep through their few remaining hours before their fate becomes known to them.)

The guard duly obliges and the anxious prisoner awake looks upon the sleeping inmate to be condemned to die with certainty and mistakenly thinks that his own chances of survival have just risen from 1/3 to 50/50.

Meanwhile another guard overhears the name of one of the three condemned prisoners mentioned. He asks the first guard why he mentioned the name of one of the prisoners, and receives the answer that one of the prisoners asked who would die other than himself, but he doesn't ask which one of the three condemed uttered the question.

With the knowledge of what the question was but not which of the three prisoners asked it, he has a 50/50 chance of of guessing the survivor.

The prisoner with the knowledge of the question asked and that it was himself who asked the question, he still only has a 1/3 chance of survival.

To me, this isn't mathematics or probability, but pure logic.

Regards,

Steve.

[This message was edited by Steven Toy on Mon 19 April 2004 at 3:41.]

quote:

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Originally posted by Matthew Robinson:
So who is right, Mulder or Scully?

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I'll have a stab.

If it were truly random, each door would have a 1:2 chance of yielding a red. The chance of two people getting the same colour simultaneously is 1:4. This, together with the fact that the same doors on the same numbered boxes always yeild the same colour tends to say that the boxes ARE individually programmed.

If the boxes are individually programmed there would be 9 possible combinations of doors that could be opened :

Scully/Mulder

1/1
1/2
1/3
2/1
2/2
2/3
3/1
3/2
3/3

Three of these combinations (1/1,2/2,3/3) give a 1:1 chance of getting the same colour. The other 6 give a 1:4 probability, so the probability of getting the same colour at random from pre-programmed boxes would be (6/9*1/4) + (3/9*1) = 18/36 or 1:2.

So I'll put me money on Scully. Now PLEASE tell me I'm wrong, and why.

quote:

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Originally posted by Minky:
Blythe,

It was very thought provoking. I must admit that I read it with a sense of impending doom because at any moment it could easily have drifted into "about a boy" style mawkishness. Thank god that Haddon never allowed this to happen (although the last page jarred a bit in this respect).

It's interesting that Oliver Sacks was quoted on the front cover because I got some of the same visceral response reading this book as his "Awakenings". Behind things that most of us dismiss as "easy" lie extremely delicate and complicated mechanisms. I'm not sure that dwelling on this in real-time is constructive but it could help with things like tolerance and compassion.

Disturbingly, I felt myself identifying with Christopher on some level. I thought his analysis of R&R was spot on.

What did you think ?

---

Like you, I found it very thought provoking.
At times I felt slightly uncomfortable when I found myself laughing; was I laughing at someone's misfortune? I also found myself smiling at some of the apparent logic and of course the times such as "the policeman put his hand on my shoulder, I don't like anyone touching me, so I hit him" I couldn't help myself and howled with laughter, having read his feelings earlier and got to know the character to some extent.
I felt angry when I learned that the father had lied to Christopher, but laughed at the logic of Christopher and the way he handled it.
Certainly a book I'll never forget.

Computers are supposed to work on 1's and 0's - in other words "Yes" or "No" - why does mine frequently say "Maybe"?......

Minky,

"So I'll put me money on Scully. Now PLEASE tell me I'm wrong, and why"

You are most of the way there in that you are trying to solve the problem in the correct way: but you were 10 yards out on the wing with only a 10st flat footed Englishman to beat you then stepped out of bounds!

There are nine cases 1,1; 1,2; 1,3, etc. Thinking about what happens in the case of a box pair with a single "program" (say, 1=blue, 2=blue, 3=red) would be a good place to start.

BTW Einstein couldn't work this out (or at least failed to realise it) and the bloke who did was eventually awarded umpteen Physics medals and a Nobel nomination.

Matthew

OK, maybe I was wrong to mix programmed with random.

I'm not sure if every box has to have the ability to flash either red OR blue depending on which door is opened.

If every box does have the ability to flash either red or blue (no programmed BBB or RRR) we can cross the door matrix with the colour matrix. Assume that every program has one of one colour and two of the other (RRB, RBB, BBR, RBR, BRB BRR). Take RRB :

1(R)/1(R)
2(R)/1(R)
3(B)/1(R)
1(R)/2(R)
2(R)/2(R)
3(B)/2(R)
1(R)/3(B)
2(R)/3(B)
3(B)/3(B)

Gives 5/9 - pretty close to 50% - and all colour combinations should give the same result.

Bugger. I've just remembered that there were 10,000 iterations

[This message was edited by Minky on Mon 19 April 2004 at 10:55.]

Steven

"Thus for an uninformed alien, the odds are at 50/50; for the contestant who switches, the odds are 2/3 in his favour."

My meaning of elimination runs deeper than that. There is no such thing as a 50/50 choice. It's simply a comfortable, solid state universal equilibrium that is false when considering space-time, and its impact on 50/50 choice.

The big bang is considered to have 50% anti-matter, and 50% matter. Now, in zero space/singularity, they would have totally eliminated each other, and left nothing, but nothing. The fact we are here, shows that some matter escaped this paradox of total elimination. Probably due to space-time alterations or influences during the big bang sequence which allowed matter to escape freely into space.

Point being, it is impossible to have total randomness (computers give a calculative impression) and its reverse in creating a perfect solid state model. Therefore the fact we use 50/50 is false in its own making, and it can only be use in a purely mathematical / singularity state, in which nothing can exist to participate in choosing a door.

As all life is part of an imperfect equation (matter), we cannot make a perfect 50/50 choice, but can only choose based on a process of elimination while using a solid state hypothesis. Matthew is interested in your approach theory quite simply, but all is in vain as we are imperfect.

Minky,

You were *so* close.

"Bugger. I've just remembered that there were 10,000 iterations"

Exactly. If you do a lot of iterations 5/9 becomes significiant and the fact that Scully found she agreed with Mulder 50% of the time means that the machines cannot be programmed and must be truly random.

I.e. For a given program there are generally 5 of the nine cases where Mulder and Scully would see the same colours. e.g. for BBR we have 1,1; 2,2; 3,3; 1,2 and 2,1. In addition there are programs such as RRR and BBB which always give the same colour and so increase the discrepency.

This idea comes from Quantum Physics and is a statistical test called Spin Inequalities discovered by John Bell. It's a technique that allows us to prove that certain complementary particles can "communicate" instantaneously over arbitary distances -- a process Einstien refused to beleive and called "spooky action at a distance".

Matthew

Matthew

"This idea comes from Quantum Physics and is a statistical test called Spin Inequalities discovered by John Bell. It's a technique that allows us to prove that certain complementary particles can "communicate" instantaneously over arbitary distances -- a process Einstien refused to beleive and called "spooky action at a distance"."

... and maybe this takes us on to string theory or even magic?

Magic works, because it can predict the properties of other people, based on one person, or particle. Scientists tend to be the easiest fooled as they love to quantify all the data, in order to reach a conclusion for the trick. Rather than view the trick laterally for clues and other interpretations, like how the magician is focusing and moving, rather than on logic. Look for sign posts, not answers!

The trick allows for your reinterpretation and understanding if repeated, but magicians don't like to show you the trick twice, unless it's heavily based in mathematics. For example, the 21 card trick with 3 roles of 7, where the 13th card is always the one you chose, is far too difficult to figure out, even after repeated attempts for average Joe. Though the trick is easy to perform, and the mathematics’ are simple, it is such a sublime sequence of fact, that Joe will remain confused until he performs the trick himself.

Choosing the right door or box is a problem of time, rather than choice. If you were a time traveller, you could cheap. But in fact, if you could replay a thousand times the hosts action, your actions, audience auctions, and you could probably gather enough information to choice the correct door.

So here is my theory, if you have zero time to choose the door, it would be 50/50. The fact we have time, means it can never be.

And then you have String theory!


Mathematically, who has the best chance of getting the job, the person who went to the first interview or the person who went to the last?

[This message was edited by PR on Mon 19 April 2004 at 14:23.]

PR,

You appear to be talking bollocks.

Matthew

Matthew

So what part don't you understand, all of it?

Applying time value to decision making is the key. Ask you friend in the financial markets whether his choice in similar investments is based on mathematical logic (50/50) without time value being a component?

PR,

I don't understand any of it. It reads like utter bollocks to me. I assume its some kind of subtle joke but I must confess it has completely passed me by.

"Ask you friend in the financial markets whether his choice in similar investments is based on mathematical logic (50/50) without time value being a component?"

He would think I have gone mad and revise his previous impression of me as a moderately intelligent person.

Matthew