Probability & Perspective

[timebomb]

I hope the moderators won't mind me opening another thread on the topic of probability but I don't want to clutter up the other mahjong thread. Please bear with me for a while as the puzzle that I'm going to describe can be very interesting for those who like logical thinking.

There are 3 prisoners, Tom, Dick and Harry. One day, the warden comes up to their cell and says, "One of you will be hanged tomorrow morning, the other 2 will be set free". Naturally, all 3 prisoners become very anxious.

Tom who's the most anxious of all, managed to pull the warden to one corner and asked him privately, "Please tell me if I'm the one who will be executed tomorrow?". The warden refused to tell him the answer. Tom, who does not give up easily, asked again, "If you won't tell me if I'm the one, then please at least give me the name of one of the other 2 who will be set free tomorrow?" The warden thought for a while and said, "Tomorrow, Dick's life will be spared".

"Alamak", cried Tom, "Now that I know it won't be Dick, it would mean it's either Harry or me who will hang tomorrow. My chances of living has gone down from 1/3 to 1/2".

Here's the question: Is Tom correct to say that? From his perspective, has his chances really gone down?

Second question: What about Harry who does not know that Dick isn't the one who's going to be hanged? From our perspective, has his chances change?

Loh K L

[hwchoy]

First of all there is a technical mistake, his chances of LIVING was 2/3 and it has gone down to 1/2, assuming a completely random distribution of probability.

This can be presented as a variant of the famous Schrödinger's Cat Paradox. Base on that conjecture, the probability of their living or dying would have been determined the moment someone made a decision (or "observation"). So even if Harry does not know, his chances is also determined when the "observation" was made.

[timebomb]

I've never heard of Cat's paradox so I'll tackle the second part of your post later, Choy. As for the technical mistake, you beat me to it. I logged in just to make the correction but I was too slow, I guess. Anyway, Tom's chances of living would have seem to have gone down from 66.6% to 50%. Is that true?

Loh K L

[Merviso]

Thanks for giving me the chance to use my much under used brain...

If I'm Tom, my initial chance of being set free is 2/3 or 66.66%. After knowing that Dick will be spared, my chances now becomes 1/2 or 50%. Of course to me, my chances of being set free is lower.

If I'm harry, my initial chance of being set free is also 2/3 or 66.66%. Since I do not know Dick will be spared, the chances to me is unchanged.

However, IMHO, the fact is that Harry's chances is now at 50%, even if he did not know about Dick's release. Being ignorance or lacking of relevant data will not give him more chances of survival, just lesser ability to make better judgment.

[hwchoy]

I've never heard of Cat's paradox

everything you wanted to know about the Cat but was afraid to ask

http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat

[hwchoy]

merviso, read about the Cat and think about it again. but it is a bit brain twisting if you are not used to this subject.

[Merviso]

wow... Uncle hwchoy, i'm now starting to think like the poor cat in the box

[timebomb]

Thanks for the link, Choy. I read the article and although I don't understand most of it, I'm quite certain it has little to do with the puzzle I posted. My puzzle is one on probabilities. It has got nothing to do with quantum mechanics as applied to atomic or subatomic systems. I also took a look at the definition of "Copenhagen interpretation" and this is what it says on wiki - "The Copenhagen interpretation is an interpretation of quantum mechanics, usually understood to state that every particle is described by its wavefunction, which dictates the probability for it to be found in any location following a measurement. Each measurement causes a change in the state of the particle, known as wavefunction collapse."

Truth be told, I haven't a clue what that statement means.

If you like to know however, the 3 prisoner puzzle is a variation of the 3 door puzzle I posted in killies.com several months ago. Take a look at this thread and you should be able to figure out the answers for the 3 prisoners puzzle.

Loh K L

[hwchoy]

exactly, quantum mechanics is probabilistic. it has to do with the philosophical question, when does a random probabilistic function becomes deterministic, as in when does it comes determined that tom, dick, or harry will be the one to die.

I have not read your 3 door puzzle, but I think there is a good chance that it is also a represenation of the Cat paradox.

[hwchoy]

If you are interested in quantum mechanics, there are quite a lot of good books out there that lays it on gently for the lay person. I read the subject many many years ago so cannot recommend any good books.

[fisherw]

I hope the moderators won't mind me opening another thread on the topic of probability but I don't want to clutter up the other mahjong thread. Please bear with me for a while as the puzzle that I'm going to describe can be very interesting for those who like logical thinking.

There are 3 prisoners, Tom, Dick and Harry. One day, the warden comes up to their cell and says, "One of you will be hanged tomorrow morning, the other 2 will be set free". Naturally, all 3 prisoners become very anxious.

Tom who's the most anxious of all, managed to pull the warden to one corner and asked him privately, "Please tell me if I'm the one who will be executed tomorrow?". The warden refused to tell him the answer. Tom, who does not give up easily, asked again, "If you won't tell me if I'm the one, then please at least give me the name of one of the other 2 who will be set free tomorrow?" The warden thought for a while and said, "Tomorrow, Dick's life will be spared".

"Alamak", cried Tom, "Now that I know it won't be Dick, it would mean it's either Harry or me who will hang tomorrow. My chances of living has gone down from 1/3 to 1/2".

Here's the question: Is Tom correct to say that? From his perspective, has his chances really gone down?

Second question: What about Harry who does not know that Dick isn't the one who's going to be hanged? From our perspective, has his chances change?

Loh K L

Firstly, Tom is an idiot. If only 1 is to die, the warden would definitely be able to come up with one name of a person to be set free. So asking that does not help in his knowledge (and therefore help his anxiety). Now, if there are 2 to be hanged, then that question makes sense but then his anxiety would be 100%.

[checkerboard]

This is a question that can perhaps be solved using Bayes Theorem:

In this case, using alternative representation of Bayes Theorem:

P(Harry Dying|Dick surviving) = P(Harry Dying)*P(Dick surviving|Harry Dying)/[P(Tom Dying)*P(Dick surviving|Harry Dying)+P(Harry not dying)*P(Dick Survivng|Harry not dying) =(1/3)*(1)/[(1/3)*(1)+(2/3)*(1/2)] = 1/2

[checkerboard]

Anyway, timebomb, the question you asked on the doors in killies in a famous problem, solved using bayes theorem:

==============================

Attached is the solution from Wikipedia

Example #4: The Monty Hall problem
We are presented with three doors - red, green, and blue - one of which has a prize. We choose the red door, which is not opened until the presenter performs an action. The presenter who knows what door the prize is behind, and who must open a door, but is not permitted to open the door we have picked or the door with the prize, opens the green door and reveals that there is no prize behind it and subsequently asks if we wish to change our mind about our initial selection of red. What is the probability that the prize is behind the blue and red doors?

Let us call the situation that the prize is behind a given door Ar, Ag, and Ab.

To start with, , and to make things simpler we shall assume that we have already picked the red door.

Let us call B "the presenter opens the blue door". Without any prior knowledge, we would assign this a probability of 50%

In the situation where the prize is behind the red door, the host is free to pick between the green or the blue door at random. Thus, P(B | Ar) = 1 / 2
In the situation where the prize is behind the green door, the host must pick the blue door. Thus, P(B | Ag) = 1
In the situation where the prize is behind the blue door, the host must pick the green door. Thus, P(B | Ab) = 0
Thus,


Note how this depends on the value of P(B).

[hwchoy]

wow I remember someone introduced me to Mr. Bayes about 25 years ago

[zyblack]

Seeing this thread just reminded me how dumb I am... Altogether different language....

[timebomb]

Seeing this thread just reminded me how dumb I am... Altogether different language....

Well, actually, you don't have to - think you're dumb, that is.

I've never heard of Bayes Theorem and I know nothing about The Cat's paradox and even less about Quantum Mechanics but I can understand the puzzle just as well. It's quite simple logic, actually. The key to understanding both the 3 door puzzle and the 3 prisoners one is to have an open mind. That's all to it, really.

Fisherw didn't actually come up with the answer but he gave a very good clue when he remarked that Tom asked a stupid question. Truth be told, Tom learnt nothing new when the warden said Dick's life would be spared. The warden gave him a name (Dick's) but he could have said Harry and it wouldn't make any difference to Tom. Because whichever name was given out, the warden could have definitely name one from the other 2 prisoners. So if he didn't learn anything new, how could Tom's probability of dying the next day change? In other words, from Tom's perspective as well as from ours, his probability of being executed remains the same, at 1/3.

But what about Harry? From his perspective, nothing has changed. He wasn't aware of the conversation between Tom and the warden so he did not know that Dick's life will be spared. But from our perspective, Harry's probability of being executed has now doubled from 1/3 to 2/3. Look at it this way - Divide the 3 prisoners into 2 groups. Group 1 consists only of Tom. He has a 1/3 chance of dying. Group 2 consists of Dick and Harry. Collectively, this second group has a 2/3 chance of dying. But now that we know Dick's probability of dying has gone down to zero, Harry would have to shoulder the whole thing by himself. In other words, he takes over Dick's probability as well.

It's almost the same as the 3 door puzzle but with a clever twist.

Loh K L

[hwchoy]

actually I would urge you to take a longer look at the Cat's Paradox. It is easy to brandish around terms like "from xxx's perspective" but if you look at Relativity, that is exactly what happens, different perspective produces different answers but has something fundamental changed about the probability of living for any of the three? what event triggered such a change and how do we reconcile the differences?

How many different ways can they each die?

[Ah_ZhaN]

Aiyah, from a simple logical perspective, both of them got the probability of 50/50 -- Either die or spare!


But mathematically, can also add on the probability of whether the Prison Warden is saying the truth to Tom - 50/50 truth/lie, on top of his probability to die/spare.


Morale of the story - Life is short, Keep more fishes now!

[bossteck]

I don't think who to die is a randomly distributed event right?