Why do you need to specify that a coin is fair? This sounds like the kind of etherial question that generally gets dropped from stack exchange sites, but I don't know of a better venue to ask so I'm hoping this question will help other folks with a similar dilemma.
I recently posted this question: Probability of selecting a combination of two variables.

I have a bag of toys. 10% of the toys are balls. 10% of the toys are
  blue.
If I draw one toy at random, what're the odds I'll draw a blue ball?

One person provided an answer immediately and others suggested that more details were required before an answer could even be considered.  But, there was a reason I asked this question the way that I did.
I was thinking about probabilities and I was coming up with a way to ask a more complicated question on math.stackexchange.com.  I needed a basic example so I came up with the toys problem I posted here.
I wanted to run it by a friend of mine and I started by asking the above question the same way.  When I thought of the problem, it seemed very clear to me that the question was "what is $\mathbb{P}(blue \cap ball)$."  I thought the calculation was generally accepted to be $$\mathbb{P}(blue \cap ball) = \mathbb{P}(blue) \cdot \mathbb{P}(ball)$$
When I asked my friend, he said, "it's impossible to know without more information."  I was baffled because I thought this is what one would call "a priori probability."
I remember taking statistics tests in high school with questions like "if you roll two dice, what're the odds of rolling a 7," "what is the probability of flipping a coin 3 times and getting three heads," or "if you discard one card from the top of the deck, what is the probability that the next card is an ace?"
Then, I met math.stackexchange.com and found that people tend to talk about "fair dice," "fair coins," and "standard decks."  I always thought that was pedantic so I tested my theory with the question above and it appears you really need to specify that "the toys are randomly painted blue."
It's clear now that I don't know how to ask a question about probability.  


*

*Why do you need to specify that a coin is fair? 

*Why would a problem like this be "unsolvable?"

*If this isn't an example of a priori probability, can you give one or explain why?

*Why doesn't the Principle of Indifference allow you to assume that the toys were randomly painted blue?

*Why is it that on math tests, you don't have to specify that the coin is fair or ideal but in real life you do?

*Why doesn't anybody at the craps table ask, "are these dice fair?"

*If this were a casino game that paid out 100 to 1, would you play?



This comment has continued being relevant so I'll put it in the post:

Here's a probability question I found online on a math education site:
  "A city survey found that 47% of teenagers have a part time job. The
  same survey found that 78% plan to attend college. If a teenager is
  chosen at random, what is the probability that the teenager has a part
  time job and plans to attend college?" If that was on your test, would
  you answer "none of the above" because you know the coincident rate
  between part time job holders and kids with college aspirations is
  probably not negligible or would you answer, "about 37%?"

 A: Being precise and specifying the conditions under which you are asking for a probability avoids ambiguity, and impacts the result!
In mathematics, it is important to avoid ambiguity in general, when that is possible, if and when you want an answer to the question you "meant" to ask.
Your blue ball question asked: 

I have a bag of toys.
  10% of the toys are balls.
  10% of the toys are blue.
If I draw one toy at random, what're the odds I'll draw a blue ball?

I commented, asking for clarification, and posted my qualified answer, part of which stated:
"Note: we are assuming that "blueness" is uniformly distributed over all the toys. Otherwise, it may be the case that 10% of the toys are red balls, and 10% of the toys are blue blocks, in which case you have 0% probability that you'll draw a blue ball."
Note also that the answers two your question, in order to provide an answer, follow from the assumption that the qualities of "blueness" and of "ball" are independent of one another. This is rarely the case.

Without being precise, there are many possible answers, depending on the conditions for which an event is being "probabilized". Stipulating "fair die", "standard deck", and "uniformly distributed" all rule out the 

"but what if....?

questions.
A: I think you're missing the point here. If a maths problem doesn't specify one way or the other whether a coin is fair, then it's OK to assume that it's a fair coin, because that's the way things usually are. Coins are fair, by and large.
But it certainly is not the case that the colour of a toy is unrelated to what kind of toy it is. I would say that the probability of a ball being blue is much greater than the probability of a toy in general being blue.
So the question is fundamentally flawed.
A: the principle of indifference is required to compensate for the lack of precision but it might lead to some paradoxes.
the principle of indifference does not imply any independence it only assigns uniform probabilities to disjoint events:
if we are to apply the principle of indifference to those three sets:
{it will rain in Rome tomorrow it will not}
{it will rain in Paris tomorrow,it will not}
{It will rain both in Paris and Rome tomorrow,it will not}
we will end up having a clear paradox because applying the principle of indifference + independence to the first two sets will lead to a probability of 1/4 of it raining in both cities tomorrow instead of 1/2 deduced from mere application of the indifference principle.
A: $1$-You need to specify that a coin is fair so that you have a reasonable model, namely a completely specified probability density function.
$2$- such problems are even solvable in a well defined sense, for example
Minimax. However you still need to know the range of fairness of the coin, i.e., how much unfair or the bias should be known.
$3$- aprioi probability is the probability of occurrence of your hypothesis. One of the best examples can be the binary communication where the transmitter sends either $0$ or $1$ from a random source. This means the apriori probability of sending $0$ and sending $1$ at a channel use is roughly $0.5$. However the channel introduces noise and the probability that a $0$ sent is received as $0$ can be different from a $1$ sent is received as $1$. The a-posteriori probability is the multiplication of the a priori probability with the conditional probabilities such as mentioned "probability of a $0$ sent is received as $0$". Clearly if the a priori probability of sending $0$ is $0$ then all the time $1$ will be sent and whatever the channel condition is you will get a perfect decision at the receiver. But of course the entropy of this source is $0$ and such a communication doesn't make sense
$4$- if you see in a math test that the fairness is not mentioned it is their assumption or in a wide sense incompleteness.
$5$- the fairness of a dice or correlations among some events is not of only mathematical importance. They all appear in the nature and there has been alot of research going on in this matter. Just a little example: if you have two sensors sensing the room temperature and if you situate them very closely, they will observe similar temperature values as they will have correlated observations. Now how is this linked to a fair dice? If you throw an unfair dice some numbers will occur more than a probability $1/6$ and some with less that that of. Then one can simply deduce that the number $5$ comes very frequently and play to that number. The relation to temperature sensors is generally related in time, namely to the correlations in time. If at time $t$ you observe the number $5$ then for example at time $t+1$ you observe $4$ with a probability say $90\%$ the same story that a sensor at time $t$ would observe almost the same thing in time $t+1$ or the second sensor which is close to the first one will have similar observations. Note that if the observations have correlations, then the entropy will be much less and the fairness will be reduced and the diversity will be also much less. As a result according to Shannon, the information with respect to that source will be less.
A: Probability theory is the study of theoretical random processes that have no essential unknowns, i.e., all aspects are determined. The "unknowns" of problems are merely aspects that must be determined from others that are given. This is precisely why a coin or die etc. is said to be fair. If it is NOT fair, sufficient other characteristics MUST be given to DETERMINE it's probabilities; or what is asked for must NOT depend on its probabilities - which is unlikely.
This is in stark contrast to statistics which is the study of random processes that HAVE essential unknowns. So statistics is essentially the study of estimation. There is no such concept in classical probability theory.
Modern theoreticians have attempted to blur (they call it "generalize") probability theory to include what really is part of statistics; but it waddles like statistics, quacks like statistics, looks like statistics, and IS statistics.
Probability theory has no concept of "prior" or "posterior" probabilities or distributions. These are strictly terms from statistics.
Finally, if the labels of the balls (or whatever) are not MUTUALLY EXCLUSIVE, the probability of drawing a specified one depends not only only on how many there are of each but also how many are in common. This is the idea of what is called "independence." The simple product formula for the union is ONLY valid for independence. 
