🚀 go-pugleaf

In article <1993Jan26.143630.1@skaro.demon.co.uk> richard@skaro.demon.co.uk (Richard Develyn) writes:
>
>You and two other people (call them B and C) are locked, in separate 
>cells, in a prison.  You are told that two of you are to be executed. 
>So you calculate your chances of survival to be 1 in 3.
>
>The gaoler wanders round and you say to him that, since clearly at 
>least one of the two other chaps is going to die, he wouldn't be 
>giving anything away by telling you which one. He does - say B. Now, 
>amazingly, you calculate your chances of survival as 1 in 2 since it 
>is clear that either C or you will be the other victim.

Aaargh--it's Monty Hall all over again.

Cajoling the gaoler into giving you information about the other
prisoners doesn't tell you anything about yourself in this case.
Your odds of survival are still 1/3.  The odds of C's survival
(based on what you now know) have risen to 2/3, however.

So if the warden comes around and says he will not tell you who
is to be executed but he will let you change places with one of
the other prisoners if you like (and so to suffer whatever fate
had been decided for them), it is to your advantage to take
C's place.

-- David Karr (karr@cs.cornell.edu)

In article <1993Jan26.143630.1@skaro.demon.co.uk> richard@skaro.demon.co.uk (Richard Develyn) writes:
>Probability puzzle
>This puzzle sounds similar to the two children one discussed here 
>already but I've never heard a satisfactory explanation:
>
>You and two other people (call them B and C) are locked, in separate 
>cells, in a prison.  You are told that two of you are to be executed. 
>So you calculate your chances of survival to be 1 in 3.
>
>The gaoler wanders round and you say to him that, since clearly at 
>least one of the two other chaps is going to die, he wouldn't be 
>giving anything away by telling you which one. He does - say B. Now, 
>amazingly, you calculate your chances of survival as 1 in 2 since it 
>is clear that either C or you will be the other victim.
>
>So, you think to yourself, you fooled the gaoler. His information was 
>(somehow) able to improve your chances of survival. Then you wake up 
>and realise that you never had that conversation at all. However, you 
>pretend that you do have this conversation. Whichever other prisoner 
>he tells you is for the chop your chances are still 1 in 2. So in fact 
>your chances of survival were 1 in 2 all along ...
>
>Errrmmmm.
>

This question is in the rec.puzzles Archive.

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From the Archive, the answer is:
********
decision/prisoners.p
********
Three prisoners on death row are told that one of them has been chosen
at random for execution the next day, but the other two are to be
freed.  One privately begs the warden to at least tell him the name of
one other prisoner who will be freed.  The warden relents: 'Susie will
go free.'  Horrified, the first prisoner says that because he is now
one of only two remaining prisoners at risk, his chances of execution
have risen from one-third to one-half!  Should the warden have kept his
mouth shut?


********
decision/prisoners.s
********
Each prisoner had an equal chance of being the one chosen to be
executed.  So we have three cases:

Prisoner executed:         A    B    C
Probability of this case: 1/3  1/3  1/3

Now, if A is to be executed, the warden will randomly choose either B or C,
and tell A that name.  When B or C is the one to be executed, there is only
one prisoner other than A who will not be executed, and the warden will always
give that name.  So now we have:

Prisoner executed:  A    A    B    C
Name given to A:    B    C    C    B
Probability:       1/6  1/6  1/3  1/3

We can calculate all this without knowing the warden's answer.
When he tells us B will not be executed, we eliminate the middle two
choices above.  Now, among the two remaining cases, C is twice
as likely as A to be the one executed.  Thus, the probability that
A will be executed is still 1/3, and C's chances are 2/3.

you, eduardo, and annette are prisoners, and the warden says,
"two of you, chosen at random, are going to be executed tomorrow."
clearly, you have a 1/3 chance of living.  at least one of annette
and eduardo will die, so you ask the warden to tell you the name of
a person, besides yourself, who will be executed.  "annette's gonna
bite it," he says.

now, what's the chance of your survival? it is the chance that eduardo
will be executed, given that annette will be executed.  that is equal
to the chance that both eduardo and annette will be executed, divided
by the chance that annette will be executed.  that's 1/3 divided by
2/3.  that's 1/2.

now you wake up.  your conversation with the warden was a dream.  but,
you think, if i did ask the warden the question, and he did tell me
a name, no matter what name it is, i'd still have a 50/50 chance of
living - so i must have a 50/50 chance anyway!

well, sorry, the fact remains that you did NOT have a conversation, and
the warden did NOT tell you a name.  the only information that you know
is that at least one of eduardo and annette will die.  the chance that
you will live, given that at least one of e. and a. will die, is the
chance that you will live AND at least one of e. and a. will die, divided
by the chance that at least one of e. and a. will die.  that's one third
divided by one.  that's one third.

this is much like the children puzzle - the chance that the other kid is
a girl is different if you know that at least one is a girl than it is
if you know that the younger is a girl.

bob vesterman.

In article <93027.161640RVESTERM@vma.cc.nd.edu> <RVESTERM@vma.cc.nd.edu> writes:
>you, eduardo, and annette are prisoners, and the warden says,
>"two of you, chosen at random, are going to be executed tomorrow."
>clearly, you have a 1/3 chance of living.  at least one of annette
>and eduardo will die, so you ask the warden to tell you the name of
>a person, besides yourself, who will be executed.  "annette's gonna
>bite it," he says.
>
>now, what's the chance of your survival? it is the chance that eduardo
>will be executed, given that annette will be executed.  that is equal
>to the chance that both eduardo and annette will be executed, divided
>by the chance that annette will be executed.  that's 1/3 divided by
>2/3.  that's 1/2.
 ^^^

The 2/3 should be 1, since the warden has told you that Annette *will*
be executed.  So your odds of survival compute to 1/3.  No paradox.

The way you have set this up, it is *exactly* analogous to the Monty
Hall puzzle.

-- David Karr (karr@cs.cornell.edu)