Quick Probability Question

[Borachio]

Answer: two half logs and some sawdust. And a quizzical look on my face. (Why didn't I use a chopper and split it longitudinally?)

 

[metatron]

You say "without assuming error" as if you're not adding any assumptions into your interpretation, but you are making a number of implicit, unstated assumptions in interpreting it as a maths question. For a start, you're assuming that it is a maths/probability question, and not any other type of question, like a trick question or a management test question. If a person actually did tell you all that information, and you actually did have 2 bags in front of you with a different number of balls in it, you most certainly wouldn't expect the 2nd bag to have 60 balls in it: the guy could be completely crazy, he could be a magician, he could be lying, .... etc.

There is only one interpretation of the question if you interpret the question as a probability question. Now, I realise that the thread title is "quick probability question", but it's not like that isn't open to interpretation either. I asked a simple maths question above, but it is in fact not simple, not maths, and not even a question... It was intended to demonstrate that the (now explicit) assumption that all the information you are given is true is not always a rational one.

Another simple maths question: if you take one log and saw it in half, how many logs do you have?

2, right?

But that's clearly not the right answer to that question. If you interpret it as a maths question, then obviously, you start with 1 log, split it in half, and now you have 2 logs. But if you interpret it as not a maths question, then there's a bloody good chance that the answer is not 2. Why would you only have 1 log to start with? This is the implicit assumption when interpreting it as a maths question: that the question presents all information necessary to answer the question definitively, and all the information is correct. So you assume that you only have 1 log to start with. But who has just 1 log? What kind of person has a saw, has the need to saw a log in two, the skill and equipment to saw it in two, but doesn't have more than 1 log somewhere else in his house/workshop/logging camp/factory? I can't think of a plausible reason why someone would only have 1 log to start with, and no more than that.

This is the problem with reducing data to merely a simple maths problem. It's why, if somebody actually did tell you about the 2 bags, in real life, you would -- should, at any rate -- have serious doubts about whether the person telling you the information is acting in good faith, giving you all the information, and that all the information you are given is correct. It's also why the US mortgage market collapsed in 2007/2008.

We would have to get into semantics regarding the term Trurorian used: "correct"

In my view "correct" is not necessarily the same as "helpful in avoiding being tricked out of your money by some prankster or street magician".

The fundamental problem here still is that in your original post (#32) you are making an aweful lot of assumptions for which you have no supporting evidence. Your conclusion ("40-20") is virtually unsupported as a result.

Your issue with the "you are told" part can indeed be problematic in real life in the way you described. It's just that we don't know anything about the person doing the telling.
So we might as well, for arguments sake, asume the person is someone who wants to test our grasp on statistics, like a teacher or an obnoxious nerd at some party.
At least in my experience those are the most common situations in which people present urn problems to me.
So far anyway.
Until i am tied to a chair in a basement surrounded by four guys with Russian accents and suspicious smirks on their faces presenting an urn problem to me i might as well rely on that experience.

Anyway. The correct answer would remain correct no matter if it was the one said person (doing said telling) expected or hoped for or whatever.

 

[Mise]

I don't think 40-20 entirely unsupported: from experience, I can imagine that someone might well use "twice as likely to get a black ball as before" to mean "there are twice as many black balls as before". On the other hand, I can't imagine why they would use "twice as likely to get a black ball as before" when they could have just said "100% certain to get a black ball". They could, of course, mean a whole host of things, so in real life I would probably ask clarifying questions. And it's true that the only evidence I have is just a vague intuition or experience with people making these sorts of statements. Your intuition/experience tells you something else, which is fair enough -- it's certainly open to interpretation.

I bet if you did some clever data mining analysis using google or facebook or newspaper archives or something, you'd find that P("they meant 100% certain" | "they said twice as likely") < P("they meant twice as frequent" | "they said twice as likely"). It's possible though that P("they meant that the probability is doubled" | "they said twice as likely") is greater than both those other two statements, but then my judgement says that when "they meant the probability is doubled" is the same as "they meant 100% certain", that the probability of this final statement falls dramatically.

 

[Borachio]

But you're missing the "as what" from the "twice as likely".

"Twice as likely" on its own means nothing.

 

[Mise]

Well, yes, "twice as likely to draw a black ball compared to the 1st bag" could easily mean "you will draw a black ball twice as often/twice as frequently", rather than "the probability of drawing a black ball is doubled". People use the concept of "likelihood" quite vaguely; it does not always refer to probability.

 

[Borachio]

Mr M, sir. You really are clutching at straws with this one.

 

[Mise]

I really don't think I am. Just read any BBC News article on, for example, the latest findings from a university psychology study to see how people actually use phrases like "twice as likely". "Twice as likely" is very often used synonymously with "twice as frequently" in everyday speech, because "twice as frequently" often does imply "twice as likely" or "twice the probability". If you don't eat vegetables, you will get constipated twice as frequently as if you do eat vegetables <---> if you don't eat vegetables, you will be twice as likely to get constipated as if you do eat vegetables. The two statements have the same meaning, so "likelihood" and "frequency" are often used interchangeably in everyday speech. It's quite plausible that the person saying "twice as likely" means "twice as frequently".

It's not often, however, that "twice as likely" is used to mean "with 100% certainty"...

 

[Borachio]

Yes, but see, in this particular case that's the result you get from doing a rational calculation.

That's what people find so fascinating about maths, and science, in general: that the results are often counter-intuitive.

 

[Mise]

No it's not, it's the result you get from assuming that "twice as likely" means "the probability is doubled", and a whole host of other assumptions that you make when interpreting the text.

 

[Borachio]

OK. So how likely are you to draw a black from the first bag?

 

[Mise]

Depends on (a) what you mean when you use the term "likely", (b) the context of the question, (c) the factual accuracy of the information, and (d) that there is no other pertinent information that could influence the answer.

Assuming (based on your previous posts) that by "likely", you mean "what is the probability", that (b) the context (again, based on your previous posts) is either (i) a maths question, or (ii) intended to get me to admit that the answer is 50%, that (c) the information presented by the OP is factually accurate, and (d) that the bag is "fair" (i.e. there isn't a hidden compartment or a bag-within-a-bag that hides all the black balls, or that it isn't sealed shut), the black balls aren't all glued to the bottom or sewn into the bag, that the balls aren't positioned so that I'm more likely to pick one colour over the other, that there is no physical impediment to my ability to retrieve any balls at all from the bag (such my hands being tied behind my back), that the bag doesn't contain also 100,000 cubes or 20-sided dices, in addition to the 60 balls, that the bag isn't see-through and/or that I can't simply look inside the bag and pick whatever colour I want, etc etc (use your imagination!), then:

"I am highly unlikely to draw a black ball, or any ball at all, because I'm being deliberately annoying and want to ruin your thought experiment".

 

[Borachio]

Let's call that zero, then; for the sake of argument. So twice zero is how many?

(edit: I freely admit that natural language is ambiguous, btw.)

 

[Mise]

Well I think that illustrates the point: If the probability was "zero" in both instances, I wouldn't say that I was "twice as likely" to draw a black from the 2nd bag as the first, if actually what I meant was "I'm just as un/likely".

If I told you that "I'm twice as likely to draw a black from the 2nd than the 1st bag", it wouldn't be natural to assume that I wasn't going to draw a black from either bag, would it? (Unless you were using this conversation as a guide, that is )

 

[Borachio]

You're not really a mathematically inclined person, are you?

edit: there's really no reason why you should be.

 

[warpus]

Erm, how would you rephrase the problem to make it more explicit and unambiguous?

I would word it like this:

"The probability of pulling out a black marble out of the 2nd bag is twice that of the probability of pulling out a black marble out of the 1st bag. How many black and white marbles are in the 2nd bag?"

But then you might say "But then it's easy!"

That's the issue I have with this question - the "hard" part of the question is figuring out what the question is actually asking. Once you have that and you sit down to do probability.. that part is easy.

The question doesn't rely on your knowledge and expertise with probabilities - but rather your ability to figure out what the question is asking. Which is easy enough, but still. Probability questions should test your knowledge of statistics and probability foremost.

 

[Verarde]

Yes, the phrase "twice as likely" certainly is open to interpretation. I think that might be why there are varying answers, hmm? And I agree with warpus, a wording such as "twice that of the probability" as he puts it would be much easier to understand.

 

[Truronian]

Well, yes, "twice as likely to draw a black ball compared to the 1st bag" could easily mean "you will draw a black ball twice as often/twice as frequently", rather than "the probability of drawing a black ball is doubled". People use the concept of "likelihood" quite vaguely; it does not always refer to probability.

But then why even mention Bag 1 if it has nothing to do with the solution? Do you consider it to be a poorly worded trick question?

 

[Antilogic]

Do you consider it to be a poorly worded trick question?

Yes.

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