# The Prisoner Paradox: Examining the Relativity of Probability in Everyday Life

• I
• lukka98
In summary, if you live through the previous day, the probability of surviving the next day is always 0.5.
lukka98
For example, If I have a constant probability of 50% to death per day (sorry for the macabre example but I find it good);

Every day I wake up, I can say:"Today, at the end of the day, I have a probability of 0.5 to be live( or to be dead)!".

Now, one can calculate the probability that after some numbers of day I am death:
is equal to: 0.5 (prob. 1st day death) + 0.5(1st.live)*0.5(2nd death) + ... and so on , and it tends to 1.
For example after 2 days: P = 0,75 to be death.
If I calculate the probability to death exactly the 2nd day is: P = 0.5*0.5 = 25%, and it decrease with day number.
And the probability to be live at the "end of the n-day is" : (0.5)^n.Now there is my question:
If I wake up the second day, I can say:"Today I have a probability to live of 0.25 ( 0.5 1st day * 0.5 2nd day) "
But If I cannot say when I start to have "a life on probability" I can say:"Today I have 0.5 of probability to live!".

But this is wrong, because probability cannot change from different observer, so what I can say?
Thank you

PeroK
lukka98 said:
If I wake up the second day, I can say:"Today I have a probability to live of 0.25 ( 0.5 1st day * 0.5 2nd day) "
You can't say this. Each day is independent of they other, just like flipping a coin. I could flip a coin heads up 5 times in a row and on the sixth flip the probability of getting heads is still 50%.

lukka98
lukka98 said:
If I wake up the second day, I can say:"Today I have a probability to live of 0.25 ( 0.5 1st day * 0.5 2nd day) "
But If I cannot say when I start to have "a life on probability" I can say:"Today I have 0.5 of probability to live!".
No. You can say that "Today I HAD a probability to live of 0.25.". The events of yesterday have changed the probabilities.
lukka98 said:
But this is wrong, because probability cannot change from different observer, so what I can say?
Probability CAN change with new events. If you lived through yesterday, that changed today's probabilities.

PeroK and lukka98
Drakkith said:
You can't say this. Each day is independent of they other, just like flipping a coin. I could flip a coin heads up 5 times in a row and on the sixth flip the probability of getting heads is still 50%.
So what I calculate is the probability to be live "2 days consecutively?"

$$N(t)=N(0) 2^{-t/t_{1/2}}$$
where ##t_{1/2}## is named half life of nuclei.

If probability were relative as you say, the big radioactive waste problem should have been removed.

lukka98
FactChecker said:
No. You can say that "Today I HAD a probability to live of 0.25.". The events of yesterday have changed the probabilities.

Probability CAN change with new events. If you lived through yesterday, that changed today's probabilities.
Ok, but every day I have always a probability of 0.5 to survive, is correct?
and for example each 4 days i have always a probability of 0.0625 to survive, now or after 1 year?

lukka98 said:
Ok, but every day I have always a probability of 0.5 to survive, is correct?
and for example each 4 days i have always a probability of 0.0625 to survive, now or after 1 year?
GIVEN that you live to that day, you are correct. Of course, if you die before that day the probability changes.

lukka98
Of course, if you are considering the real-world situation of the actuary tables, the probabilities do change as you age.

lukka98
FactChecker said:
Of course, if you are considering the real-world situation of the actuary tables, the probabilities do change as you age.
My situation was ideal, just for this question.

there is a thing that I have difficult to accept:
If, instead of a person I take an atom ( only for more real example) with the same 0.5 prob. to decay per day.
After 1 million of year of him "creation" there is a probability of 10^-50 that is already live, I know every day it survive with 0.5 prob. (numbers are only for example).
But If I start to observe it tomorrow, the probability to survive is 0.5. is correct?

"there is a small probability that resist up to there , but once I have the probability is always the same 0.5."

lukka98 said:
But If I start to observe it tomorrow, the probability to survive is 0.5. is correct?
Sure (assuming it hasn't already decayed). Just like flipping a coin: The probability of getting 10 heads in a row is small, but if you've already gotten 9 heads then the probability is 0.5 that you'll get 10.

lukka98
fresh_42 said:
Here is a famous example which I stumbled upon in a book from Martin Gardner:
I thought the solution to this was simply that there's a self-reference in the assumptions. It boils down to the logically invalid formulation of an assumption:

Assumption 1: You cannot predict (i.e. logically deduce) using assumption 1 ...

In other words, this is an assumption about what you can deduce using the assumption being put forward.

That's self-reference (like "this statement is false"), which cannot legitimately be used in logical reasoning.

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lukka98 said:
there is a thing that I have difficult to accept:
This is what's called a mental block. It's become a bit of a problem for you. We can only repeat the same answer (which you always "like"), so it's up to you to try to free your mind.

pinball1970
lukka98 said:
For example, If I have a constant probability of 50% to death per day (sorry for the macabre example but I find it good);

Every day I wake up, I can say:"Today, at the end of the day, I have a probability of 0.5 to be live( or to be dead)!".
One final attempt to explain this. Let's stick with the macabre. You have 64 prisoners. Every day each prisoner has to toss a coin. If it's heads, then they get executed that day; and, if it's tails they stay alive for at least one more day.

On the first day 64 coins are tossed. Let's say 32 are heads - and 32 prisoners are taken away to be executed - and 32 are tails, hence 32 prisoners escape exceution.

On the second day, there are only 32 prisoners left and 32 coins tossed. Let's say there are 16 heads and hence 16 prisoners are taken away to be executed.

On the third day, there are only 16 prisoners, 8 of whom get heads and get executed and 8 of whom escape again.

As the days go by, there are half as many prisoners each day and half as many executions. Until eventually there is no one left.

PeroK said:
I thought the solution to this was simply that there's a self-reference in the assumptions. It boils down to the logically invalid formulation of an assumption:

Assumption 1: You cannot predict (i.e. logically deduce) using assumption 1 ...

In other words, this is an assumption about what you can deduce using the assumption being put forward.

That's self-reference (like "this statement is false"), which cannot legitimately be used in logical reasoning.
This is a thinker.

pinball1970 said:
This is a thinker.
A crude analogy would be to compare the following assumptions:

1) Assume there are only finitely many primes.

That is a logically valid assumption, which turns out to be false.

2) Assume we cannot prove there are infinitely many primes.

That is not a valid assumption, because it refers to what we can prove using the assumptions. This sort of self-reference leads to problems. Although, it is tricky to see precisely how and where the logic break down.

PeroK said:
A crude analogy would be to compare the following assumptions:

1) Assume there are only finitely many primes.

That is a logically valid assumption, which turns out to be false.

2) Assume we cannot prove there are infinitely many primes.

That is not a valid assumption, because it refers to what we can prove using the assumptions. This sort of self-reference leads to problems. Although, it is tricky to see precisely how and where the logic break down.
Sort of circular reasoning?

I read the discussion and it was a little technical

I tried working through it and twisted myself in knots.

The liar/Barber paradox seem more obvious in terms of a self reference, (not when I first read about them. )

lukka98 said:
there is a thing that I have difficult to accept:
If, instead of a person I take an atom ( only for more real example) with the same 0.5 prob. to decay per day.
After 1 million of year of him "creation" there is a probability of 10^-50 that is already live, I know every day it survive with 0.5 prob. (numbers are only for example).
But If I start to observe it tomorrow, the probability to survive is 0.5. is correct?

"there is a small probability that resist up to there , but once I have the probability is always the same 0.5."
For discussion among the two on probability, they should agree that which state has probability 1, 100%.
It may be "He is living this morning 100%." or "He was living yesterday 100%" or so. Probability is assumption based on our current knowledge. It depends on how much we know about it and is to be renewed by new knowledge.

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pinball1970 said:
Sort of circular reasoning?

I tried working through it and twisted myself in knots.
My point is that is unnecessary. It is perhaps interesting precisely where this particular example breaks down, but the whole concept of having an assumption that says what you can and can't prove is invalid. "Being surprised" is one way of describing what you can and cannot deduce from the given assumptions. So, "being suprised" cannot be part of an assumption.

For example, if you proposed an alternative Group Theory where we have the usual group axioms plus something like:

5) You cannot prove Lagrange's theorem.

Then that cannot be taken seriously as mathematics. What precisely goes wrong with this version of group theory may be very difficult to untangle. But, the basic concept of an axiom such as 5) is invalid from the outset.

I'm going to go out on a limb here, as I'm not quite sure I follow this thread and I'm probably misinterpreting something, so I'll simply phrase it from my own perspective. I could go assuming things and proposing some new axioms along with perhaps countable choice and a simple correction for the axiom schemas of specification and replacement, etc. (ZF'CC maybe, lol, I don't know), but I simply don't care to. Real mathematicians can handle that sort of thing. Same with the probability question originating in this thread which I may have an answer to, but only because I put the problem into a slightly different context that I won't be able to put my finger on in this thread so to speak (my apologies).

Anyways, I hope you find what you're looking for lukka98. PeroK too.

Edit: There is no formula that can be applied to a set so as to generate the class of ordinals, so I think the axiom schemas are still ok. Just countable choice would probably work just fine and we're back to good. Again, shooting from the hip here and not really caring (sorry), but hopefully that helps.

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Warning: off-topic.

PeroK said:
For example, if you proposed an alternative Group Theory where we have the usual group axioms plus something like:

5) You cannot prove Lagrange's theorem.

Then that cannot be taken seriously as mathematics. What precisely goes wrong with this version of group theory may be very difficult to untangle.
I propose:

5) You cannot prove the continuum hypothesis.

Do you think anything would go wrong?

pbuk said:
Warning: off-topic.I propose:

5) You cannot prove the continuum hypothesis.

Do you think anything would go wrong?
My point is that your axiom 5 is invalid. We don't need to get tangled up in what sort of system we have created.

The trap in the prisoner paradox is to dive into the logical system you have been given and try to figure out what goes wrong. That's the approach that leads to arguments, debates and confusion.

Instead, I propose that we step back and first examine the axioms we have implicitly been given. If we identify that those axioms are invalid or not well formed then we shouldn't attempt any logical reasoning on the basis of them.

Continuum hypothesis is solved already, sorry. It was independent from ZF because you need countable choice and it was independent from ZFC because ZFC is inconsistent. that's already been proven. Sorry...

pbuk and PeroK
AplanisTophet said:
Sorry...
Rather than post nonsense and apologise, it would be better not to post the nonsense at all.

PeroK
I understand skepticism. Countable choice is enough to show ##\omega_1## can't exist per the "Limits of Two Ordinal Sequences" thread in this forum. If ##\omega_1## can't exist, there are no uncountable ordinals. The axiom of choice is enough to prove the existence of uncountable ordinals, so that is why ZFC is inconsistent. ZF is not enough. ZFCC solves the continuum hypothesis.

I dropped out of college when I learned Cantor's Theorem and then went on to become a CPA instead. I've been auditing the foundations of Mathematics ever since and preparing taxes for a living. CPA stands for certified pain in the ass. I completely understand skepticism. This has been a 20 year journey for me with quite a few scars... I don't take it lightly.

I truly am sorry though... In many ways, this stinks.

PeroK
lukka98 said:
Now there is my question:
If I wake up the second day, I can say:"Today I have a probability to live of 0.25 ( 0.5 1st day * 0.5 2nd day) "
But If I cannot say when I start to have "a life on probability" I can say:"Today I have 0.5 of probability to live!".

But this is wrong, because probability cannot change from different observer, so what I can say?
Say professor and students are going to observe a Shroedinger cat in a box.
Professor says "I see living cat two days ago so she is living with probability of 0.25 today."
A student says " I see Box yesterday, at that time the cat was ..."
Professor says " Stop saying it. That would disturb my probability."
The probability for the professor and that of the student are different. Probability depends on "who assumes with how much knowledge" If you say "relative" in this sense I will say yes to the titled question.

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How did this thread go from flipping coins to axioms and logical systems??
The OP's question seemed easy enough to answer with a simple example of flipping coins.

pbuk
PeroK said:
My point is that is unnecessary. It is perhaps interesting precisely where this particular example breaks down, but the whole concept of having an assumption that says what you can and can't prove is invalid. "Being surprised" is one way of describing what you can and cannot deduce from the given assumptions. So, "being suprised" cannot be part of an assumption.

For example, if you proposed an alternative Group Theory where we have the usual group axioms plus something like:

5) You cannot prove Lagrange's theorem.

Then that cannot be taken seriously as mathematics. What precisely goes wrong with this version of group theory may be very difficult to untangle. But, the basic concept of an axiom such as 5) is invalid from the outset.
I posted a long break down of the paradox for my understanding.
I try to rephrase it so it made sense and there were no logical inconsistencies
It was deleted because it looked like I quoted you, I actually wanted you to give it the once over.
Anyway it was deleted, I had just deleted it myself from my mail so I cannot remember what I put!
In short, I think I get it. Thanks @PeroK

EDIT: I do have a copy but I am going off topic so ill leave it

PeroK

Update -- user @AplanisTophet has left us, so thread it reopened.

pbuk
berkeman said:
Update -- user @AplanisTophet has left us, so thread it reopened.
I was thinking of starting a new thread as the original OP was not he half as interesting as fresh_42 's prisoner link and Peroks explanations.

PeroK
pinball1970 said:
I was thinking of starting a new thread as the original OP was not he half as interesting as fresh_42 's prisoner link and Peroks explanations.
I'm on a train with no WiFi, but I can post my analysis tomorrow if you start a thread on the prisoner paradox.

pinball1970 and berkeman
PeroK said:
I'm on a train with no WiFi, but I can post my analysis tomorrow if you start a thread on the prisoner paradox.
Great will do.

berkeman

## 1. What is the Prisoner Paradox?

The Prisoner Paradox is a thought experiment that explores the concept of probability in everyday life. It presents a scenario where two prisoners are given the chance to reduce their sentence by cooperating with each other, but face a dilemma when it comes to trusting each other.

## 2. How does the Prisoner Paradox relate to probability?

The Prisoner Paradox highlights the relativity of probability in decision making. It demonstrates how different perspectives and information can impact the perceived likelihood of an outcome. In this case, the prisoners' trust in each other affects their decision and the probability of a favorable outcome.

## 3. What is the significance of the Prisoner Paradox?

The significance of the Prisoner Paradox lies in its ability to challenge our understanding of probability and decision making. It shows that probability is not always objective and can be influenced by various factors, such as personal beliefs and information. It also highlights the importance of critical thinking and considering multiple perspectives in decision making.

## 4. Can the Prisoner Paradox be applied to real-life situations?

Yes, the Prisoner Paradox can be applied to real-life situations where decision making involves uncertainty and trust. For example, it can be used to understand the dynamics of negotiations, relationships, and even financial investments. It can also be used as a tool to improve decision making by considering different perspectives and information.

## 5. How can we use the lessons from the Prisoner Paradox in our daily lives?

The lessons from the Prisoner Paradox can be applied in our daily lives by recognizing the relativity of probability and the impact of different perspectives on decision making. It encourages critical thinking and considering all available information before making a decision. It also highlights the importance of trust and communication in relationships and negotiations.

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