

The Prisoners’ Paradox - TriinT
http://www.futilitycloset.com/2009/07/03/the-prisoners-paradox/

======
knowtheory
Oh dear. So this is an implementation of the Monty Hall problem but now is
crossing over into Metaphysics.

There are several different ways to consider the ontological reality of
probabilities (i guess wikipedia only has two listed, Frequentists and
Bayesians <http://en.wikipedia.org/wiki/Probability_interpretations> ).

The thought experiment in question probably should be answered with a Bayesian
world view, and does not appear to be.

Prisoner A's chances of being executed is _not_ materially effected when he
learns new information, because the decision as to who has been pardoned _has
already been made_.

Finding out new information, does change a probability, but not the
probability that he'll be executed. It only changes the likelihood of whether
or not the assertion that he will be executed is true. The nature of the
universe has not actually changed, merely the state of belief of Prisoner A.

The point being, he's got no cause to celebrate, upon merely discovering
information, because learning more information does not actually change the
state of the world, in this case.

~~~
huherto
But, did he discovered new information? What if the guard had said "Prisioner
C is sure to die"?

~~~
gambling8nt
He did discover new information. The probabilities of survival switched from:
A: 1/3, B: 1/3, C:1/3 to A: 1/3, B:0, C: 2/3

He just didn't discover any information to change the probability of his own
survival.

~~~
huherto
nice. I think this sums it all.

------
jonsen
A's chances has not changed.

But the chance of C being the pardoned has doubled to 2/3.

------
pushingbits
As non-intuitive puzzles go I think the hat guessing game
(<http://www.relisoft.com/science/hats.html>) takes the cake. I still remember
a dailywtf thread where 19 pages worth of people claim that the correct
solution is completely bogus (<http://thedailywtf.com/Comments/Riddle-Me-An-
Interview.aspx>).

------
gort
Before he asks the question, there are 4 possibilities:

    
    
      1. A lives, A gets told B dies
      2. A lives, A gets told C dies
      3. B lives, A gets told C dies
      4. C lives, A gets told B dies
    

Since in fact A got told that B dies, we are either in situation 1 or 4.
However, these did not start off at equal probability. Lets define P(x) as the
probability that x is the actual situation. It is obvious that, before the
question is asked:

    
    
      P(1 or 2) == P(3) == P(4) == 1/3
    

Thus P(1) < P(4). This remains true once situations 2 and 3 have been ruled
out.

~~~
jonsen
Interesting setup. So

    
    
      P(1) == P(2) == 1/6
    

When A is told that B dies, it's down to 1. or 4.

    
    
      P(1 or 4) == 1/6 + 1/3 == 1/2
    

So chance A lives is now

    
    
      P(1) of P(1 or 4) == 1/6 of 1/2 == 1/3
    

And the chance C lives is now

    
    
      P(4) of P(1 or 4) == 1/3 of 1/2 == 2/3

------
mhb
<http://en.wikipedia.org/wiki/Monty_Hall_problem>

~~~
mannicken
Yeah, except it's pretty obvious why there is a switch to 1/2 from 1/3 in
Prisoner's Paradox and it's not obvious in MH.

What I understand happened in Prisoner's Paradox is that the very field of
probabilistic distribution of pardons has shrinked from 3 to 2. It's like
binary search. You cheated your way by knowing where _not to look_.

Another simple analogy: you can increase the value of rational number both by
increasing the numerator and decreasing denominator. In this case you
decreased denominator.

Monty Hall requires a slightly bigger leap of logic.

Actually, here's a more directly linked problem:
<http://en.wikipedia.org/wiki/Three_Prisoners_problem>

~~~
mattmaroon
You mean it's pretty obvious why there is no switch to 1/2 from 1/3? As the
Wikipedia article you linked points out: "The answer is he didn't gain
information about his own fate, but he should switch with C if he can."

This isn't the same as Monty Hall. Monty Hall only increases the odds after
switching doors. This actually seems like a trick meant to fool people who
know about the Monty Hall problem but don't really understand it.

~~~
mannicken
It's strange why my blatantly stupid post was upvoted.

I re-read all three problems after you replied and this is what I understood.
Talking about Three Prisoners' problem since the article's wording doesn't
include the very crucial ingredient for easier understanding: coin flip on
warden's side.

For A, it doesn't really matter if B or C were executed. In fact, he knew that
either B or C are going to be executed for sure. He knew that at least one
person who is not him will be executed.

Warden just named the sure-to-be-executed-guy. Didn't change what he
previously knew about his chances, since he still doesn't know for sure
whether the-other-not-named-guy is going to be executed.

However, the distribution changed for C. Now A still knows that there 1/3
chance of him living but since we know for sure that the-guy-who-warden-named
(B or C) will be killed the-guy-who-warden-didn't-name-and-who-isn't-A (C or
B) will have 1 - 1/3 = 2/3 probability of living.

~~~
mattmaroon
Correct.

------
mhb
The clearest explanation I've seen is this. Suppose there are 1,000 prisoners
one of whom is to live. You ask the guard to tell you which 998 out of the 999
other prisoners will be killed. So there's one he didn't tell you about. Who
do you think has a better chance of living - you or the prisoner in the other
group whose fate the guard didn't reveal?

------
devin
Forgive me, but doesn't it seem odd that the problem defines Prisoner A from
one perspective, and Prisoner B from another?

Prisoner B could just as well be Prisoner A since A was defined by the
narrator, and Prisoner B by the guard. I know we're descending into the
theoretical stuff, probabilities and so on, but might there not even be enough
information to go down any path at all given this dilemma?

~~~
jonsen
Prisoner A is the one that asks. That sets the one an only perspective for the
guards answer.

~~~
devin
Really? From whose perspective is Prisoner A described as asking the question?
The narrator's. Please correct me if I'm wrong, but there is an implicit third
person in my view.

