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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%?"

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    At a Texas toy store, $10\%$ of the toys are guns, and $10\%$ of the toys are pink. What is the probability that if we draw a toy at random, we will draw a pink gun?2012-11-28
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    That's my point. I'd say that without further information you have a 1 in 100 chance of running across a pink gun: http://is.gd/wFED462012-11-28
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    Is this really a question? For example, in a math test "coin toss" always means fair unless otherwise stated, because it is tiresome to keep stating that the coin is fair throughout the section on probability. If you are trying to "make a point," as you say, this is not a discussion site, but a Q&A site.2012-11-28
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    Heh, yeah, I'm serious. I have a daughter whom I'll eventually need to give sample statistics questions to to help with her homework. In her math book, it'll say, "you flip a coin." On stack exchange, folks want to know, "is the coin fair?" I want to know why. I seriously don't understand how my original question is any different than "You have two 10 sided dice. One die is for shape and the other for color. There is one ball shape and one blue color. What're the odds of rolling a blue ball?"2012-11-28
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    When we are trying to solve a "real" problem using tools from probability theory, the first step is to produce a mathematical *model* of the situation. The assumption of independence is physically reasonable when we model successive tosses of a coin. But assuming independence is not *always* reasonable.2012-11-28
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    In Prob(blue $\cap$ black) = P(blue).P(Black), you assume independence. It's not always true.2012-11-28
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    @D.Patrick Your question is not the same as two die rolls, unless it is unknown whether the die are somehow linked magnetically. If you asked: There are 100 toys in a bag, and 10 of them are blue and ten of them are balls, how many are blue balls? Would it be possible to answer definitively, or do you not have enough knowledge to know? But the probability of picking a blue ball is just that number divided by $100$.2012-11-28
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    Also, you subject should probably match the introduction to your question. The question about "fair coins" only recurs much later deep in the body of your question, which makes it very hard to figure out what your actual goal is. In particular, it would have been very nice to know up front that you were concerned about it essentially as a potential teacher - that it is a pedagogical question.2012-11-28
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    @ThomasAndrews, I think you're getting at my confusion. I don't understand why you wouldn't just assume that the toys were randomly assigned a color. It strikes me as odd how fundamentally different people think the question is as written vs. "10% are randomly blue." If you were writing a math test for 9th grade math students, how would you write it? What about for college seniors? Grad students? At what age would you be annoyed that they kept asking, "are the toys randomly painted or did you paint them with some strategy?"2012-11-28
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    You wouldn't assume it because, unlike die rolls, which are independent by all measures that we have experience with, the toys in the bag are a fixed finite set of objects, and the conditions that you have placed on the problem allow the number of blue balls to be anywhere from $0$ to $10$. We can easily imagine examples where there are no blue balls in the bag. We can't easily imagine examples where the die rolls are similarly dependent - I suppose magnetically entangled dice might be an example. The contents of the bag are not "random." The random process is just picking a toy the bag.2012-11-28
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    I edited my original post appending a real life sample question that's similar to mine that I found on a math education website quiz. Perhaps it's the way we think that changes so dramatically as we get older?2012-11-28
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    By the way, I think it's worth noting that almost universally, everybody hates my original question but nobody has upvoted a single answer to this question. In fact, the only upvotes are one on the question itself and 4 for the comment, "At a Texas toy store, 10% of the toys are guns, and 10% of the toys are pink. What is the probability that if we draw a toy at random, we will draw a pink gun?" Of course, almost certainly this comment will change all of that, but it was true at the time. :)2012-11-28
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    The survey question is simply a bad question. Change *None of the above* to *Insufficient information*, and it becomes an adequate question, though not a particularly good one. Change it from multiple choice to free answer, and it becomes a perfectly reasonable question.2012-11-28
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    @AndréNicolas, +1 for the Texas store example... Yet a similar fraction of theoretical statistics results remain valid if independence or i.i.d assumption are dropped.2012-11-30

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