I don't think anyone has made this particular point here at all, so here goes.
My own view is that neither of the two views of probability has independant truth. Rather, I don't personally operate on the idea of Absolute Truth, I operate off of what is workable
. It doesn't make sense to operate any other way, at least not to me (solipsists may disagree).
That's moving slightly off topic, or at least it would be if I clarified it fully. The point is, which viewpoint, if any, of probability can actually be used to attain some result? Let's take gambling, because knowledge of probability has an observable relevance to the field of gambling. I like the example given above:
jjono wrote:Depends what you mean by "calculate the answer". Think of it this way - would you bet money that the ball was a particular colour? Imagine if I told you I have a ball in my hand that is either red, green or blue. Then I offered you odds of 5:1 to guess the colour right. Would you take it? What about if I offered you odds of 1:1? If you would take the first bet but not the second, doesn't that say that you think the probability is between 1/6 and 1/2 that the ball is your chosen colour? But wait...the ball is one colour and one colour only, right?
I doesn't matter that I know what colour the marble is, or that the marble is definitely either red, blue or green. To the best of your knowledge, each outcome has equal probability.
I like this example. Look at it this way: either I am telling the truth or I am not. If I am not, and the marble turns out to be purple, you would be justified in renegeing on your bet, as I'm sure anyone would agree. So let's assume I'm telling the truth. If the marble turns out to be either red, green or blue, some money will change hands if you accept the bet. If I give (to exaggerate and make the point easier to see) 1000:1 odds, then you are putting down one dollar to 1000 of mine on the probability that you can guess the color of the marble. I'd call that a good bet, no matter how you view probability.
IF, however, I give you the same odds, tell you that the marble is either red, green or blue, and offer you 1000:1 odds only if you want to bet that the marble is RED--you'd be a fool to take the bet, unless you knew from other means that I was very seriously trying to get rid of a lot of money. This is where motive comes into play, and when motive comes into play, "all bets are off" as the probabilities go to hell.
These are both fairly obvious in terms of the choices involved. The thing that may be missed is, probabilities in any real life circumstance do NOT necessarily sum up to one, nor are they absolute. We normally only look at a specialized cross section of probabilities. I like what an earlier poster said (that I am too lazy to look for and quote) in another forum: if you choose a point randomly on a line with even distribution, the probability of you choosing a point OTHER than the exact center is 1. "Or, to put it another way, events with probability 0 happen all the time."
Goplat wrote:There is no justification for assigning a uniform distribution on anything when the problem doesn't say so. Sometimes it'll give you something ridiculous: "Goplat's pet is either a cat or a rhinoceros. What is the probability that it's a rhino?" Uniform distribution says 1/2.
Sometimes it'll even give you different answers for equivalent phrasings of the same question! "I am holding either a shoe or a sock; what is P(sock)?" 1/2, right? "I am holding either a left shoe, a right shoe, or a sock; what is P(sock)?" Now the "Bayesian" answer is 1/3, despite the fact that the semantics have not changed.
Really, it's the "Bayesians" who have some 'splaining to do. I'm not the one who's pulling probabilities out of nowhere..
It will indeed give different answers for different phrasings of the same actual scenario. So what if I say, "It's a sock?" You can then know that it is a sock, with probability 1, according to your logic. What you're forgetting is that all of these situations only deal in the specialized cross section of situations wherein I am telling the truth
The simplistic view is, P(heads) + P(tails) = 1/2 + 1/2 = 1. In actual practice that works
, so we use it. But if you're going for Absolute Truth, you have to ask, what coin? Where? You wind up factoring in P(the coin spontaneously combusting) + P(the coin never landing due to gravity failure) + P(the coin landing on its edge) + P(the coin falling through a hitherto unnoticed crack in the floor and winding up unobservable in its final position) + ... + P(heads) + P(tails) = 1. This is all quibble, though, because P(heads) + P(tails) = 1/2 + 1/2 = 1 is a good, workable
truth, as far as it goes.
What are the odds of a quarter landing heads when I flip it? Most people would say 1/2. What about when I add the fact that we are on an orbiting space station?
There's no such thing as a funny sig.