A bag contains 10 marbles. 3 are drawn at random, all 3 are blue.

"Urn Problems"

>>9065176
Ah, right, but what about this more specific case where we don't know what's in the bag? The other marbles could be red, green, black, etc...

Is this problem even solvable, as stated, or is there not enough information?

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>>9065184
It is solvable in the sense that you can come up with solutions to it. But because the problem is not well defined, there exists multiple solutions to it. What's more, the solutions are likely to be as unrigorous as the problem itself. Mathematics thrives from unambiguousness; the problem you stated is more of a philosophical type.

In the universe of your problem, there exists only blue marbles since no other colors were ever specified. Thus all the marbles are blue with probability of 1.

>>9065165
Classically: no. There isn't a definite solution because not enough is known about the marbles in the bag. We don't know what colors are possible and therefore can't assign a probability to any one marble being blue.

We could, however, assume that each marble can exist as any ROYGBIV color. Meaning the probability of all 10 being blue is (1/7)^10. But since we know for certain that 3 marbles are blue, the probability that the entire bag is blue can reduce to (1/7)^7.

>>9065252
>>9065253
We know that there are 10 marbles in the bag (or 7 after the first 3 are removed).

Can't we assume that they are either "blue" or "not blue?" Then we can list all the possible states of the bag...

3 blue + 7 not blue
4 blue + 6 not blue
5 blue + 5 not blue
... etc

>>9065165
The odds would be 1 - the probability that they are not all blue. So you would have to come up with every variant where there were 10 marbles, and at least 3 of them were blue, to determine the answer.

>>9065271
is there equal probability of blue marbles and non-blue marbles in the urn?

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>>9065271
Let me construct two bags, A and B.

- Bag A: 10 marbles. I favored blue marbles so that for each marble I inserted there was 90% chance that it was a blue one.
- Bag B: 10 marbles. I favored non-blue marbles so that for each marble I inserted there was 90% chance that it was a non-blue one.

It is clear that the solution to your problem varies whether we are talking about bag A or bag B. But we cannot know whether the bag even is either A or B; it could be some mystery bag C with its own probability distribution.

Thus, still not enough information.

I think you can solve it if you define the possible colors that could be in the bag. Let's say it is red, blue, and green marbles added randomly to the bag. There is a small chance that it ended up with all 10 being blue. Getting your first 3 in a row as blue gives you more information and would increase the odds that all 10 are blue.

>>9065317
Good point. But the fact that we drew 3 blue marbles tells us that Bag B is much less likely to be what we're dealing with.

So we do have enough information to make some useful statements about this particular bag. If we drew all 10 marbles, we could define the bag completely. This is the heart of the problem.

>>9065271
This is clever, but without knowing the probability that any one marble is blue we can't know the probability that every marble in the bag is blue. Your assumption is one that essentially assigns a 1/2 probability of a marble being blue, which can't rigorously be concluded from the information given to us in the problem.

>>9065343
>>9065349
Hmm... So we probably can't come up with a numeric probability as an answer. But we could just label the probability of a marble being blue as P(blue) and find an answer in terms of that value, right?

>>9065362
Yes. It would be (P(blue))^7 (based on the information given that 3 marbles are known already to be blue).

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>>9065343
You cannot assume that past probabilities would affect future probabilities.

For example, if you play roulette and "red" wins 10 times in a row, it doesn't imply that "red" is more probable than "black". Neither does it imply that "black" is now more probable because "it hasn't come for a long time". These are assumptions that seem intuitive but are actually fallacies.

>>9065381
It does if you don't already know the layout of the wheel.

>>9065165
50/50

Reality trumps all math.

>>9065165
Without proper information on what the other marbles could be, there's no way to know. You seem to think that the fact that you drew 3 blue marbles has any effect on what the other marbles are, which is not necissarily true. if we had more information, we might be able to answer the question.