Resource preservation exploit

[Dominae]

Talk of future and past events with regards to probability is misleading. It generates the assumption that past events are "certain". Certainly (!) in the eyes of an omniscient being all past events are either 1 or 0 (to use One_Brow's terminology). But I'm rather certain (as DaveMcW showed) that the results of placing a bet before or after a die is rolled and subsequently covered will be the same (in other words, you won't make more money in either case).

I think it's better to think of it as "availability of information". Knowledge of a fact yields 100% probability of that event (whatever that means). Any uncertain knowledge is granted some lesser probability (if possible). Thus the paradox I posted is definitely a paradox, even though the judge may have "rolled the die" (and therefore chosen which one was to guilty) many days before A even asked the guard for information.


Dominae

[punkbass2000]

Originally posted by Dominae


So, if I roll a die and cover it, you would argue that the probability of any given number was different from 1/6 (assuming a fair die)? I think we'll both agree that (othe than the realm of quantum mechanics) things either are or they aren't, but there is still something to say for probability, no?

I had completely forgotten about this poser that I posed, so sorry to anyone who was eagerly awaiting the answer (probably no one, but I feel bad not posting anyways).

Here we go:

The paradox can be resolved by noting that 'the guard said B is to be released' implies 'B is to be released', but not vice versa; if 'B is to be released' is true, then differing probabilities can be assigned to 'the guard said B is to be released'. Another way to look at it is that the fact that 'the guard said B is to be released' could indicate two very different state of affairs, namely, that C is to be killed (where the guard has no choice but to say 'B'), and that A is to be killed (where the guard does have a choice).

Here's the dirty math:

---

Let:

IB; 'B will be declared innocent'
GA; 'A will be declared guilty'

Then:

P(GA|IB) = P(IB|GA)*P(GA)/P(IB)
= P(GA)/P(IB)
= (1/3)/(2/3)
= 1/2

This is the straightforward analysis that leads to the paradox, since it's stipulated that all three prisoners have an equal chance of being declared guilty.

---

Let:

IB2: 'The guard said B will be declared innocent'

Then:

P(GA|IB2) = P(IB2|GA)*P(GA)/P(IB2)
= (1/2)*(1/3)/(1/2)
= 1/3

This resolves the paradox.


I'm now not speaking off-topic (on any thread) for another month.


Dominae

I thought you said the odds changed to 1/2.

[nbarclay]

The problem was misstated. It should have been written, "It looks like the odds have changed," Rather than saying that the odds actually have changed. As others have noted, the original list of possibilities was seriously flawed in that it listed only the who lives/dies aspect, not the who got the note aspect. Thus, the 1/2 figure based on that flawed list of possibilities is invalid.

Nathan

[One_Brow]

Originally posted by nbarclay


From the prisoners' perspective, who will live and who will die is certainly an unknown future event. Similarly, from the time a die is rolled or a coin is flipped until the time someone looks at it, what value will be seen when someone looks is an unknown future event.

However, if there is no decision to be made with regard to that event, then there is effectively no probability assigned to that decision.

/QUOTE]As any science fiction fan knows, human language is not always well suited to dealing with temporal (time-related) phenomena. In this case, terms such as "probability" are used in what might be called a retroactive tense: even though the event has already been decided, we treat it as undecided in our speech until we actually know the outcome. And my impression is that such usages are more than sufficiently common to be regarded as "correct" in terms of common usage of the language.

Nathan [/QUOTE]

In temporal terms, perhaps. However, when you remove the decision-making process as well as place the event in the past, you don't really have a probability.

[Dominae]

Originally posted by punkbass2000
I thought you said the odds changed to 1/2.

Yes, sorry to mislead you there, but that's the source of the paradox. It seems clear (at least to me) that the person shouldn't have a greater chance of dying, because it doens't appear that he's obtained any additional information. So the chances of him dying should be 1/3. Yet if B is to live, it's between him and C, so it now appears to be 1/2. I don't think it can be both. The analysis above shows that the right answer is 1/3.


Dominae

[Bambul]

I was going to make a post about resources but it seems inappropriate now. So I'll just tell an annacedote about my year 9 maths teacher.

We were doing probability and he was telling us that a long time ago people would gamble on tossing two coins and guessing whether they would come up HH, HT or TT. Mathematicians knew that the probabilities were 1/4, 1/2 and 1/4 (because it can be HT or TH). But everyone else thought it was 1/3 for each of them.

There was an unwriten rule between the mathematicians that no-one would reveal the true probability since they were making so much money out of gambling! Unfortunately someone did.