Can You Solve This Hilarious Limit Problem Involving Sine and Infinity?

[uart]

Originally Posted by Galileo
Not so much a joke as a brainteaser.
Three prisoners, strangers to each other, were suspects of a murder case. One day they came to hear that a sentence has been drawn. Two of them have been found guilty and will be executed, but they don't know which of the two . One guy, a statistician, figures his chances for survival are 1/3, so he goes to the bars of his cell and hails the guard: "Hey psst, do you know which of us has been sentenced?".
"Eh, yes.", says the guard, "But I'm not allowed to tell you.".
"Tell you what", says the guy, "I already know that 2 of us will executed, that means at least one of the other guys will be. I don't know them or anything, surely you can point to one which is guilty?". The guard sees no harm in that and points one of the prisoners, "He is guilty".
"Thanks!", proclaims the statistician, "my chances have just increased to 1/2".

Alright, this is killing me! Can someone please explain it!

His chances have not changed as no new information (that would change the statics) was really added by the guards revelation.

One thing I do find weird about this problem is how two people whom have never meet can both be found guilty of the same murder. That's some pretty whacky justice system.

 

[GregA]

I don't really have much knowledge of stats and so I apologise if this seems like a stupid reply but I just have to chip in...

the three prisoners each have a 1/3 chance of survival...the guard however has just informed one prisoner that a different prisoner's chance of survival just dropped from 1/3 to 0/3...already the system has been changed no?
Also from the perspective of the statistician, before he found out who was doomed, the dice was weighted equally for all of them but now he can look at one prisoner and be absolubtely sure that his presence can have no effect on the outcome. (if we call the three prisoners A,B and C.., from C's perspective before speaking to the guard there are three people in the game. 'A' might die, but then again there is a chance that his luck is in and it will be B and C that die...knowing that 'A' is dead though, means there are only two people left in the game) thus his chances should be 1/2 no? (unless the guard is lying)
As I said earlier...my apologies If I am talking garbage

edit: had this discussion with a wiser person than myself and now know why I am wrong :)

 

[Denryoku]

Alright, this is killing me! Can someone please explain it!

Hi, this is my first post

I'm not really sure about this, but I agree with uart that no new significant information was given by the guard. I think the problem is in the way the prisoner simply eliminated the guy pointed by the guard from his reasoning.

Since we have no information about who commited the crime (or how the justice system works), we assume each one to have an equal chance of survival. At the beginning its clear that any given prisoner has a 1/3 chance of being declared innocent.

Later, one prisoner (say, prisoner A) asks the guard to point one of the other guys who is guilty. So he points one of the two suspects that will executed (lets say, prisoner B). Now A can figure his chances of survival using Bayes' theorem as follows:

P(A is innocent | B was pointed) = P(B was pointed | A is innocent) * P(A is innocent) / [ P(B was pointed | A is innocent) * P(A is innocent) + P(B was pointed | A is guilty) * P(A is guilty) ]

-The probability of B being pointed by the guard given that A is innocent is 1/2 (if A is innocent then the other two are guilty, but the guard had to point only one of them)
-The chances of A being innocent are still 1/3
-The probability of B being pointed by the guard given that A is guilty is, again 1/2 (since A is guilty, only one of the other two prisoners is guilty and thus was signaled by the guard; but we have no information to determine which one)
-The probability of A being guilty is 1 - 1/3 = 2/3

So the chances of A being innocent given that B was signaled by the guard are: (1/2 * 1/3) / (1/2 * 1/3 + 1/2 * 2/3) = 1/3 (not 1/2)


Well, back to the main topic... taken from http://www.xs4all.nl/~jcdverha/scijokes/1_6.html" :

A bunch of Polish scientists decided to flee their repressive
government by hijacking an airliner and forcing the pilot to fly them
to a western country. They drove to the airport, forced their way on
board a large passenger jet, and found there was no pilot on board.
Terrified, they listened as the sirens got louder. Finally, one of
the scientists suggested that since he was an experimentalist, he
would try to fly the aircraft.

He sat down at the controls and tried to figure them out. The sirens
got louder and louder. Armed men surrounded the jet. The would be
pilot's friends cried out, "Please, please take off now! Hurry!"

The experimentalist calmly replied, "Have patience. I'm just a simple
pole in a complex plane."

 

[murshid_islam]

don't know whether this one's posted already:

a mathematician is one who cannot tell a teapot from a doughnut.

 

[fourier jr]

don't know whether this one's posted already:
a mathematician is one who cannot tell a teapot from a doughnut.

i think you mean coffee cup and a doughnut, since a teapot has 2 handles while a doughnut has only 1

 

[murshid_islam]

i think you mean coffee cup and a doughnut, since a teapot has 2 handles while a doughnut has only 1

well, the teapots i have seen have only one handle. anyway, the point is that both a doughnut and a teapot with one handle (or a coffee cup) has one hole in it. in the doughnut, the hole is in the middle, whereas in the coffee cup, it is in the handle.

 

[benorin]

don't know whether this one's posted already:
a mathematician is one who cannot tell a teapot from a doughnut.

That sounds more like a topologist.

 

[NateTG]

well, the teapots i have seen have only one handle. anyway, the point is that both a doughnut and a teapot with one handle (or a coffee cup) has one hole in it. in the doughnut, the hole is in the middle, whereas in the coffee cup, it is in the handle.

I think we have a new candidate -- "A topologist is someone who thinks that most teapots have two handles."

Most teapots I am familiar with have two holes (or handles) in them. One that your fingers go through, and the other that tea goes through.

 

[hypermorphism]

The spout for the tea is not equivalent to a handle (a torus with a disc removed). If anything, the teapot (without the lid) would be equivalent to a (circular) cylinder with a handle attached. The equivalence of the coffee cup and the donut has less ambiguity. :)

 

[shmoe]

The spout for the tea is not equivalent to a handle (a torus with a disc removed). If anything, the teapot (without the lid) would be equivalent to a (circular) cylinder with a handle attached. The equivalence of the coffee cup and the donut has less ambiguity. :)

A teapot is a two handled coffee cup. If your cylinder+handle hopes to be a teapot then the cylinder part has thickness, i.e. a 3 dimensional volume (like a coffee cup with a hole in the bottom). This can be deformed into a two handled coffee cup for safety purposes or into a two holed donut for delicousness.

Think of the classification theorem of compact connected orientable 2-manifolds without boundary (did I get everything?). The surface of the teapot is one of these, hence it's either a sphere or torus with some number of handles.

 

[podboy6]

Worst math joke ever:

"Okay, when you rotate a conic section, you slap on -iod to the end of its name, i.e, paraboloid, hyperboloid, ellipsiod, ect...

Okay, what do you get when you rotate a human?

A humaniod."

 

[uart]

Hi, this is my first post
Since we have no information about who commited the crime (or how the justice system works), we assume each one to have an equal chance of survival. At the beginning its clear that any given prisoner has a 1/3 chance of being declared innocent.
Later, one prisoner (say, prisoner A) asks the guard to point one of the other guys who is guilty. So he points one of the two suspects that will executed (lets say, prisoner B). Now A can figure his chances of survival using Bayes' theorem as follows:
P(A is innocent | B was pointed) = P(B was pointed | A is innocent) * P(A is innocent) / [ P(B was pointed | A is innocent) * P(A is innocent) + P(B was pointed | A is guilty) * P(A is guilty) ]
-The probability of B being pointed by the guard given that A is innocent is 1/2 (if A is innocent then the other two are guilty, but the guard had to point only one of them)
-The chances of A being innocent are still 1/3
-The probability of B being pointed by the guard given that A is guilty is, again 1/2 (since A is guilty, only one of the other two prisoners is guilty and thus was signaled by the guard; but we have no information to determine which one)
-The probability of A being guilty is 1 - 1/3 = 2/3
So the chances of A being innocent given that B was signaled by the guard are: (1/2 * 1/3) / (1/2 * 1/3 + 1/2 * 2/3) = 1/3 (not 1/2)
:

Ouch Denryoko, I really think it can be put a lot more simply than that :). You're right to use Bayes thm with this type of problem however I'd prefer to tackle it as follows,

Define the events as follows,
Event A : Person asking guard is innocent.
Event B : Person picked out by guard is guilty.

The reason why this one is so simple is that P(B)=1 because it is given as data!

Now since P(B)=1, it is trivial that P(A and B) = P(A) hence Bayes theorem gives,

P(A | B) = P(A and B) / P(B) = P(A)

That is, the probability of event A given that event B has occurred is identical with the original probability of event A, so it remains at 1/3. This is really a trivial application of Bayes in this case.

---------------------------------

It's interesting to note that if you change the problem so that the guard does not have to pick guilty but instead selects one of the other two at random to reveal that persons fate (say by withdrawing a name from a hat or whatever) then you do end up with a non-trivial application of Bayes thm. In this case then the probability of the original guy's innocence does indeed rise from 1/3 up to 1/2 if the guard reveals one of the other two to be guilty. Using the same event definitions as above you now get the following,

P(A and B) = P(A), because if the original person is innocent then both the other two are guilty so the name withdrawn from the hat (or whatever) must be guilty.

This time however P(B) is not equal to unity but is instead P(B)=2/3.

So Bayes thm results in,

P(A | B) = P(A and B) / P(B) = 1/3 divide 2/3 = 1/2

I hope that helps anyone who was still unsure about that one.

 

[biggins]

another joke

alcohol and calculus don't mix, don't drink and derive.

 

[Lisa!]

A real mathematician is someone who writes 'A', then read it 'B', but actually he means 'c'.

 

[benorin]

I'm supposed to prove the freshman’s dream identity,

$$\left( a + b\right) ^p \equiv a^p + b^p \left( \mbox{mod }p\right)$$

for p prime.

Can't I just use the binomial theorem? j/k

 

[benorin]

How they prove that all odd integers higher than 2 are prime?

How they prove that all odd integers higher than 2 are prime?

Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, and by induction - every odd integer higher than 2 is a prime.

Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student.

Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime,...

Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime, 11 is a prime,...

Programmer: 3 is a prime, 5 is a prime, 7 is a prime, 7 is a prime, 7 is a prime,...

Salesperson: 3 is a prime, 5 is a prime, 7 is a prime, 9 -- we'll do for you the best we can,...

Computer Software Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release,...

Biologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 -- results have not arrived yet,...

Advertiser: 3 is a prime, 5 is a prime, 7 is a prime, 11 is a prime,...

Lawyer: 3 is a prime, 5 is a prime, 7 is a prime, 9 -- there is not enough evidence to prove that it is not a prime,...

Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime, deducing 10% tax and 5% other obligations.

Statistician: Let's try several randomly chosen numbers: 17 is a prime, 23 is a prime, 11 is a prime...

Computational linguist: 3 is an odd prime, 5 is an odd prime, 7 is an odd prime, 9 is a very odd prime,...

Psychologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime but tries to suppress it,...

I coppied this from here.

 

[Galileo]

A real mathematician is someone who writes 'A', then read it 'B', but actually he means 'c'.

Sounds like my professor. He writes A, says B, means C, while the answer is D.

 

[hypermorphism]

...
Think of the classification theorem of compact connected orientable 2-manifolds without boundary (did I get everything?). The surface of the teapot is one of these,...

Ah, then you're including the fact that the teapot contains a non-zero 3-dimensional volume, or the surface of a 3-dimensional manifold analogous to a teapot. I was thinking of a 2-dimensional manifold analogous to a teapot, in that the handle was attached to a sphere with two discs removed. In that case, my teapot does have a boundary, but yours doesn't.

 

[uart]

prove the identity,
$$\left( a + b\right) ^p \equiv a^p + b^p \left( \mbox{mod }p\right)$$
for p prime.
Can't I just use the binomial theorem? j/k

What's wrong with using the binomal theorem there ?

 

[Lisa!]

https://www.physicsforums.com/showpost.php?p=805754&postcount=268"

Sounds like my professor. He writes A, says B, means C, while the answer is D.

But he must be a bad mathematician since he doesn't get the answer!

 

[AlphaNumeric]

My first post here :)

Q : Why didn't didn't Cauchy like taking his dog for a walk?

A : It kept leaving residues at all the poles.

http://movies.collegehumor.com/items/2005/05/collegehumor.149448.wmv song cracks me up. I expected it to be cheesy as hell, but the amount of puns they fit in is very impressive.

 

[Cexy]

What's purple and commutes?

An abelian grape.

What's purple, and worshipped by only a limited number of people?

A finitely venerated abelian grape.

 

[Alkatran]

For this statistical problem, why are we going all the way into Bayes theorem? We can just distinguish between the three available cases, being:

A goes free
B goes free
C goes free

Each is just as likely until the guard points out that B isn't going free, so then you have:

A goes free
C goes free

being equally likely.

But let's assume the probability is still a third. Given that we know that B is not going free and that the sum of all the probabilities should equal to 1...

P(A) + P(B) + P(C) = 1
1/3 + 1/3 + 0 = 1
2/3 = 1
contradiction

This sounds a lot like that case with the chance of the 'other child being a boy' is really non-intuitive.

 

[alfredblase]

Not so much a joke as a brainteaser.
Three prisoners, strangers to each other, were suspects of a murder case. One day they came to hear that a sentence has been drawn. Two of them have been found guilty and will be executed, but they don't know which of the two . One guy, a statistician, figures his chances for survival are 1/3, so he goes to the bars of his cell and hails the guard: "Hey psst, do you know which of us has been sentenced?".
"Eh, yes.", says the guard, "But I'm not allowed to tell you.".
"Tell you what", says the guy, "I already know that 2 of us will executed, that means at least one of the other guys will be. I don't know them or anything, surely you can point to one which is guilty?". The guard sees no harm in that and points one of the prisoners, "He is guilty".
"Thanks!", proclaims the statistician, "my chances have just increased to 1/2".

No no no. Think of it like a bag of balls, hahaha :P. What I mean is there are three balls in his bag...odd..., the stats guy is cupping one ball in his hand... Since balls are indistinguishable he removes one of the others, ouch, and then he has two balls so he's happy! see?... oh let's face it I only put this post up for the crummy balls gag...

But seriously I'm still confused by this, emm its a 0.5 probability all along isn't it? He knows one of the others is guilty without asking the guard. The probability that either he or one of the other two prisoners is going to be executed is equal to one. Since this system has only two balls, about and both are equally likely to be innocent then the chances of innocence are 0.5 for each. He wrongly calculated his chances of survival in the first place, and asking the guard further compunded his stupidity.

 

[amcavoy]

My first post here :)
Q : Why didn't didn't Cauchy like taking his dog for a walk?
A : It kept leaving residues at all the poles.
http://movies.collegehumor.com/items/2005/05/collegehumor.149448.wmv song cracks me up. I expected it to be cheesy as hell, but the amount of puns they fit in is very impressive.

Nice video lol.

 

[Joffe]

No no no. Think of it like a bag of balls, hahaha :P. What I mean is there are three balls in his bag...odd..., the stats guy is cupping one ball in his hand... Since balls are indistinguishable he removes one of the others, ouch, and then he has two balls so he's happy! see?... oh let's face it I only put this post up for the crummy balls gag...
But seriously I'm still confused by this, emm its a 0.5 probability all along isn't it? He knows one of the others is guilty without asking the guard. The probability that either he or one of the other two prisoners is going to be executed is equal to one. Since this system has only two balls, about and both are equally likely to be innocent then the chances of innocence are 0.5 for each. He wrongly calculated his chances of survival in the first place, and asking the guard further compunded his stupidity.

Actually posting that just compounds your stupidity (joking). For anybody who cannot see, the main man still has a 1/3 chance of survival, the known guilty guy has a 0 chance and the other a 2/3 chance. It has already been explained so I won't bother.

 

[Alkatran]

Actually posting that just compounds your stupidity (joking). For anybody who cannot see, the main man still has a 1/3 chance of survival, the known guilty guy has a 0 chance and the other a 2/3 chance. It has already been explained so I won't bother.

Hold on, how is the other guy any different than the current one? In the context of the problem the only difference is who asked the question!

 

[alfredblase]

exactly (all I wanted to say was exactly)

 

[Joffe]

Hold on, how is the other guy any different than the current one? In the context of the problem the only difference is who asked the question!

The guy who asked the question differs from the other because he was not involved in the selection. Imagine it this way, there are a thousand people and all but one are guilty, I could ask the guard which 998 of the others are guilty and it would narrow it down to me and one other guy, obviously the other guy has a much greater chance of running free.

 

[alfredblase]

no, your argument is lame

 

[Cexy]

no, your argument is lame

Only if you 'lame' you mean 'correct'

This is a version of the Monty Hall problem, explained here:

http://www.comedia.com/hot/monty.html

 

[alfredblase]

bah not convinced, I trust my argument and the judgment of Alkatran over yours, Joffe's and that random webpage. I sahll repeat my argument for the readers benefit.

Before he asks the guard he knows one of the other prisoners is guilty. He can discard that prisoner from the system. He is left with himself and one of the other prisoners; 1 prisoner + 1 prisoner = 2 prisoners. Since the chances of survival are equal for both, and since one of them is going to be executed, then the probability of survival = 1/2.

He could have calculated all of this BEFORE asking the guard. Now when he asks the guard all he is doing is discarding one of the prisoners, something he could have done before asking him. So the calculation of his chances of survival remains the same, i.e he has a fifty percent chance of survival. His chances of survival were 0.5 all throughout the story. Nuff said.

 

[Alkatran]

The guy who asked the question differs from the other because he was not involved in the selection.

I understand now. A has more information, but it doesn't help his odds, it helps the other inmate's odds. To make sure of this, I wrote a simple little program:

int main() {
  int a, b, n;
  int data[3];
  int asum = 0, osum = 0;
#define size 1000
  for (b = 0; b < size; b++)  {
    for (a = 0; a < 3; a++)
      data[a] = 0;
    for (a = 0; a < 2; a++)
      do {
        n = (int)(rand() * 2);
      } while (data[n] = 1)
    
    osum += (data[2] == 1) ? data[1] : data[2];
    aSum += data[0]
  }
  printf("Asker chance: %f, other chance: %f\n", asum, osum)
  return 0;
}

The output makes it pretty clear: the asker dies 2/3 of the time and the one who isn't pointed to dies 1/3 of the time.

 

[Martin Rattigan]

When I first read the explanation I thought it meant C+log cabin i.e. a place where Dutchmen live.

 

[alfredblase]

OMG this is a great "brain teaser" and I think it contains something very deep about statistics. Like a good little physicist after my ally alkatran left me all alone I decided to experiment. The results are conclusive. I describe the experiment in the following paragraph. Please repeat the experiment yourself.

I made three squares and labelled two with a G for guilty, and one with an I for innocent, then folded them, then put them in a hat. There were three very little squares in the hat. So if I were to pick one of the squares and assign that to the main man, and I repeated the experiment 100 times, roughly 1/3 times the main man would be innocent.

Now I take one of the squares out and assign it to the main man. I don't look at the paper. Now I play the part of the guard. So I look at both papers remaining and remove one with G on it from the hat. Now I put the main mans paper back in the hat along with the other paper the guard didnt remove. What am I left with? One paper with I and one paper with G. This would be the outcome every time no matter how many times I repeat the excersise. If I were to then remove one paper and assign it to the main man, and repeated the experiment 100 times, roughly half the time the main would be guilty and roughly half the time innocent. There's no argument against this experiment is there??

So wow, statistics is odd.