Which Twin has the Tony?

[TomCollins]

After reading this thread, it is quite clear that your understanding of the term probability is far worse than the people you describe, especially DS.

[PairTheBoard]

[ QUOTE ]
How is this not like saying: "I've flipped a coin, but I haven't looked down at it yet; the odds of it being heads is not 50:50, instead it's either 100% or 0% heads, I just don't know which". Because such a statement is, at best, a semantic argument, at worst a huge waste of our collective time.

[/ QUOTE ]

It depends on what the repeated experiment is. If the coin gets flipped one time and is now to be revealed repeatedly to various observers then with respect to that experiment the probabilty that it is heads is either 100% or 0%, we just don't know which.

If the coin is reflipped before each time it's revealed then with respect to that experiment the probability that it is heads is 50%.

This is not a trivial exercise in semantics. In the 2 Envelope problem where the amounts in the envelopes are fixed, having been chosen by some unknown and irrelevant means, this is exactly the point on which the so called paradox twists. It's known that one envelope has twice the amount of money in it than the other. Before opening any envelope the probabilty is 50% that the Second Envelope has twice the amount as the first and 50% that it has half the amount of the first. But after opening the First envelope and seeing it has a certain amount in it, say X dollars, it is no longer true that the probabilty is 50% that the second envelope has twice the known amount X and 50% probabilty that it has half the known amount X in the first envelope. This even though we have no idea whether X is the larger or smaller amount.

If the probabilities were still 50-50 then we could calculate the expected value of the second envelope to be 1.25X which would imply that it's always better to take the second envelope after seeing what's in the first - clearly a mistaken conclusion. If the amounts in the envelopes are A and 2A then the proper probabilty statements upon seeing X in the first envelope are: The conditional probabilty the Second Envelope has 2X given X=A is 100%. The conditional probabilty the Second Envelope has .5X given X=A is 0%. Similarly for the case X=2A. These proper conditional probabilties allow us to make the correct expected value calculation for the second envelope to be equal to X as it should be. Notice these conditional probablities are working exactly like the Twin with the Tony.

PairTheBoard

[MikeL05]

[ QUOTE ]
I'm going to be really disappointed if the answer is shown to be pointless and a huge waste of my time

[/ QUOTE ]

FMP

[inlemur]

The reason this issue is cloudy is due to different interpretations of probability. DS takes a Bayesian approach to probability, and clearly PTB does not.
http://en.wikipedia.org/wiki/Bayesian_probability

[Stephen H]

It seems your point is "probability only applies to future events; events that have already occurred (such as a girl getting a haircut) cannot have probability assigned to them." This is perhaps semantically true, but useless. My contention is that we can equally use probability to apply to events that have happened in the past but that we do not know the outcome of, and that this is just as valid as using probability on future events.

Case in point:
You have the A[img]/images/graemlins/spade.gif[/img]K[img]/images/graemlins/spade.gif[/img]. The board comes Q[img]/images/graemlins/spade.gif[/img]J[img]/images/graemlins/spade.gif[/img]2[img]/images/graemlins/heart.gif[/img]. What is the probability your flush will hit?
We use probabilty in poker all the time. But the deck is already shuffled; if you have perfect information about the state of the deck, you *know* that the flush either will come (100%) or won't come (0%). The randomizing event has happened. But we don't know the outcome yet; it's hidden. Therefore we use probability to analyze the problem. I can say "the repeatable part of the experiment is flipping the cards over; I'm not going to reorganize the deck" all I want - the use of probability is still valid.

Clearly you disagree with this, but you aren't convincing me that your point is valid very easily.

[icepoker]

To PairTheBoard: I read the whole thread, and I've come to the conclusion that you are a moron. Please never again use the word probability, you have no idea what it means.

[probman]

I have to agree with icepoker. Form your posts in this thread, it is quite clear that you have fundamental flaws in your understanding of probability. What makes it worse is your biting attempts to lecture others on their misuse. While the former is acceptable, when combined with the latter, they make you out to be a complete idiot.

[Stephen H]

I disagree with the analogy to the 2 envelope problem. In that case, the apparent paradox, as you've said in a past thread, relates to an incorrect assumption on the probability distribution:
[ QUOTE ]

This is the heart of the problem - the distribution from which the envelope amounts are chosen. Even advanced probabilty students miss the underlying bogus assumption that all envelope amounts are equally likely. ie. that the envelope amounts were chosen from a uniform distribution over the positive real numbers. Of course there is no such probabilty distribution.


[/ QUOTE ]
However, in this case, the probability distribution of haircuts onto twins is a lot simpler than money in envelopes. We're talking discretes and Booleans, not analysis across the real number line. I'm still unconvinced that we cannot model it as an equal probability of each twin having the haircut and get useful results.

[AaronBrown]

I'll stay out of the meaning of probability discussion, but I have to defend the envelope paradox. It is considerably deeper than suggested here. Like all good paradoxes, the point is not to explain it, but to peel away the layers.

You do not need to assume that the amount in the envelope is equally likely to be any real number. It is true that from a Bayesian perspective if seeing the amount of money in one envelope does not change your estimate of the probability of it being the larger amount, then your prior probability distribution must have an infinite expectation. If you expect infinity, any finite number is a disappointment, so you should always switch.

That's not a logical inconsistency, although it's hard to think of an example that doesn't involve God or the Devil.

But in the first place, that argument only makes sense to Bayesians. In the second place, you can make the paradox work with finite expectations as well.

One example is an American bets a European that the US dollar will buy less that 0.80 Euros in one year (or set any rate such that the probability is 50%). If the American wins, he gets 800 Euros. If he loses, he pays US $1,000.

The American will either lose $1,000 or win Euros he can exchange for more than $1,000. The European will either win 800 Euros or dollars he can exchange for more than 800 dollars. So both parties have positive expectations. In fact, the American can convert his bet to a 50/50 proposition for $1,000 and get paid $50, by selling an option in the financial markets. The European can get a 50/50 800 Euro bet and be paid 40 Euros.

[PairTheBoard]

[ QUOTE ]
The reason this issue is cloudy is due to different interpretations of probability. DS takes a Bayesian approach to probability, and clearly PTB does not.
http://en.wikipedia.org/wiki/Bayesian_probability

[/ QUOTE ]

Thanks for that link inlemur. I did not realize there was such a "Baysian" camp out there. The term "Baysian" is a little misleading I think. I believe all Mathematical Probabilty is being done by "Relative Frequentists" and they all make extensive use of Bayes Theorem. According to the link, "Baysian Probabilty" refers to a Philosophy. The philosophy evidently asserts that it is meaningful to use the term probabilty in ways that don't have the precise mathematical meanings that are used in Mathematical Probabilty - ie. Relative Frequency Probabilty. It's not suprising that as the link points out, Mathematical probablists don't attend the Conferences of Baysian Probablists. They are basically speaking a different language.

But you are correct in pointing out that David has the right to go with the Baysian Crowd in his use of language if he so desires. I don't see the need for creating the confusion though myself. You can use "degree of certainty" language just as easily without constantly making us wonder whether the "probabilty" you refer to carries the precise meaning of mathematical probabilists or the Philisophically Debatable meaning of the "Baysians".

PairTheBoard