Ah, your name is of course Xema. Not Xeema. Won’t happen again.
My point exactly. Yet for your friend it is 100/0. So probability does depend on point of view. In one universe your friend looked at the marble and found a blue one and in another he found a red. You don’t know which universe you’re in, and both are equally likely.
No, you cannot. It is not clear if you’re in the ‘red’ or ‘blue’ universe (jumping on the parallel universe analogy here 
)
From the point of view of someone who knows that, it’s 90/10, but for you it might as well be the other way around. Remember, nothing outside your head is interesting in these calculations. Everything you don’t know doesn’t exist.
In your universe the outcome is 100 to zero. In my universe it’s 50/50. Unless I know the teams, in which case I’m in yet another universe with say 60/40. Of course I won’t take your bet. I am geographically disadvantaged.
The chances depend entirely on your state of mind. But I’m banging on, and saying the same stuff over and over. The parallel universe analogy became a bit muddled too. I’ll get back to it if I find a new and illuminating way of stating my case. It seems the thread has been abandoned by everyone else.
This is absolutely not true. One distribution (the true distribution) has a probability of 1, and all others have a probability of 0.
One other point: the possible distributions form a countable set (pairs of natural numbers). It’s well-known by probability theorists that a uniform distribution over a countable set is not possible. So some distributions must be more likely than others, or there must be an upper limit on the number of marbles.
I didn’t phrase this very well, I admit. When you don’t know the distribution, it could be any of them. The identity of the true distribution is itself a random variable to you. Saying this necessarily leads to a flat distribution may have been a bit much. Maybe you can answer this question: For the purposes of probability calculations is there any difference between lack-of-information randomness and “real” randomness?
I agree with this too (and so was wrong–or over-simplifying–in that earlier statement). But the distributions favouring red over blue are no more prevalent than those favouring blue over red, averaging them out.
Aargh! No sorry. I don’t completely agree with the last part. I was myself confusing the different meanings of the word ‘distribution.’ Not all percentages of red marbles are possible, but all possible percentages could be equally likely. That’s what I meant by uniform. You can have uniform (or flat) discrete distributions too, as I’m sure you will agree. Thus it makes more sense to speak of a ‘symmetric’ distribution, where any percentage X of red marbles is as likely as 100-X.
The answer to this question gets into the issue of frequentist versus subjective theories of probability, and there are whole books written about that, so I’m not going to try to cover it here. For the purposes of this problem, I don’t see a difference.
While it’s true that there’s a bijection between red-favoring and blue-favoring distributions, that doesn’t mean that both are equally likely.
See my comments above.
I see now that the last part of this thread boils down to a frequentist vs Bayesian argument, with me leaning toward Bayes. In short, a philosophical question.
This is a frequentist view (right?) Then it’s difficult for us to reach an agreement. But allow me to force the issue: If persuaded at gunpoint, what ratio of red to blue marbles would you predict to come out of a bag with an unknown configuration of marbles?
It’s not a frequentist view. There are distributions of distributions that favor red-favoring distributions, as well as those that favor blue. They exist as much as any mathematical object exists, and everyone* can agree to that.
As for the gunpoint issue, there’s no meaningful frequentist answer. Either I pull something out of my ass, or I get shot.
*Everyone except the intuitionists, but screw them.
You’re right: It doesn’t matter. Any guess is as good as the other. :smack: Eh… I’m getting tired. Not quite on board with your claim that bijection doesn’t imply equal probability, though. If there exist as many distributions favouring red as favouring blue you’d be equally likely to get the one as the other.
By the way, I never said you’d get shot if you guessed wrong, only if you didn’t guess at all
Only if your pdf satisfies sum(p(r, b), r < b) = sum(p(r, b), r > b). There are plenty of distributions that don’t, but it’s difficult to actually construct one.
But for every pdf that doesn’t satisfy this, the same one, only with r and b, interchanged also exists.
Yes, but they’re still not equally likely. Take it out as many levels as you’d like–distributions of distributions of distributions of distributions–and the fact remains that sets of equal size need not have equal measure.
I don’t know much about math, and I won’t be able to type out any fancy formulas, but I have noticed that this problem seems to come down to misunderstandings or mischaracterizataions of the original problem.
Some people seem to see the problem as an ongoing (infinite?) number of single marble draws from a(n) large (infinite?) number of bags. The assumption seems to go further: that each bag is filled with a truly random number of red versus blue marbles (plus the known 3 reds and 2 blues). If this were the case, then I think we could say that after a million or billion draws (or whatever) the overall tallies would be quite close to 50% red draws and 50% blue draws.
HOWEVER-- this is not what I understand the original problem to be. I read it as being a ONE-TIME draw from ONE bag with an unknown ratio (and unknown total-- remember this is a hypothetical exercise so you don’t get to feel the weight of the bag and think “hmmm feels like there’s about 15 marbles in here therefore…”). If this is indeed the problem then there is not enough information to be able to make any prediction. The information you DO have (the added 3 reds and the added 2 blues) is meaningless because you don’t know how many other marbles there are.
One more thing… what color is the bag? 
‘Size’ is not a ‘measure’? Then what do you mean by ‘size’ (I’m superficially familiar with measure theory)? r and b are completely interchangeable, no? I’m not able to imagine how one can have a set of all possible distributions of r and b that isn’t symmetric with respect to these two, or how that can mean they’re not equally likely.
I don’t think that there’s a misunderstanding about that. This is how I see the problem too. The disagreement seems to be over what this means. I’m thinking that the distinction between an infinite number of draws and one draw, and between ‘real’ randomness and apparent randomness is unimportant. The probability of flipping heads or tails in a one-time coin toss is half of each, even though you can’t throw a ‘hails.’
I’m somehow picturing it as brown. Don’t know if that means anything, though.
I think you may have missed my stipulation that the marble I chose was red.
I think we have here a definite point, with which I can emphatically disagree. In my view, the chances depend entirely on the distribution of marbles in the bag and have nothing whatever to do with my state of mind.
I will certainly agree that, assuming the goal is to chose a marble of a certain color, my state of knowledge could affect my strategy: if, for example, I was given a choice between a bag that was 50/50 and one that was 90/10 (favorable), the right bag from which to choose would be obvious. But I hope that is not what is meant by “state of mind.”
Here’s a story that might illustrate where I think you may be going wrong: Suppose a friend takes me to an empty football stadium; as we look over the vacant field, he says, “Observe the distribution of players on the field - in particular, note how exactly the same number are standing south of the 50-yard line as are north of it.”
“But … there are no players on the field.”
“Right - they won’t be here for several hours. But nonetheless you must admit that the current distribution is symmetrical around the 50-yard line.”
The “symmetry” of no data supports no inferences. It’s just as accurate to say that the distribution of players on the field is symmetrical around the 2-yard line. So I remain convinced that the OP in no way supports the conclusion that the chance of each color is 50% - or close to that, or indeed close to any number you choose to name.
When someone tells me “one of these three cards is the Ace of Spades, the other two are Jacks,” and then I pick one w/out looking at it, and I say “The probability that this is the ace of spades is 1/3”, and then he turns it over and it’s a jack, and then I point to another card, then I naturally say “The probability that this card is the ace of spades is 1/2.” Unless you think I’ve made a mistake here, then it seems clear to me that the probability changed–precisely because my state of knowledge changed.
Just my two cents…
-FrL-
Um, you know that’s the most famous example of an error in probabilistic thinking, right?