There is a bag of unspecified size, in which some unspecified number of red marbles and some unspecified number of blue marbles has been placed. Someone now adds three red marbles and two blue marbles. You are to draw a marble out of the bag at random. What is the probability that the marble will be red?
It may seem like there’s not enough information to answer, and that may be right, but I’m not sure if it is right, so I’m looking for the straight dope.
What makes me think there may be an answer is:
If we had stated that two red and two blue marbles had been added, I just feel intuitively the answer is now 1/2, and it doesn’t seem to me as though this version of the problem gives any more information than the problem as stated. But maybe I’m wrong to think the answer 1/2 is correct in this case?
If I had specified an upper limit to how many blue and red marbles were originally in the bag, I would be able to answer the question. So it seems like some sort of averaging or limiting process could get me the answer in the case where the number of marbles previously has no specified upper limit. But maybe that intuition is incorrect?
You can’t get an answer to this problem as stated.
Let’s suppose the bag originally had one red and one blue marble in it. You add three reds and two blues so you’ve got a total of four reds and three blues.
If you pull one out at random the probability that it is red is 4/7.
Now suppose you’d started with 1 red and 20 blue. You add 3 red and 2 blue for a total of 4 red and 22 blue. Now the chance of pulling a red at random is only 4/26.
The answer depends on how many of each color you start with in the bag.
If the problem stated that you had an unspecified BUT EQUAL number of reds and blues in the bag to begin with, then your first thought (50-50 chance of pulling red assuming that you add 2 reds and 2 blues since there will still be an equal number of reds and blues) would be correct but that’s an awful lot of changes to the original problem.
Basically I think you guys are saying you need to know the original number of marbles to be able to calculate the probability. But this is not right–if I just said I had started out with, for example, “up to ten” of each color, I can still calculate an exact probability w/out specifying how many marbles started out in the bag.
Say I said I had started w/ “up to one” of each color, for example. Then there’s a 1/4 chance there were no marbles, 1/4 that there’s one red, 1/4 that there’s one blue, and 1/4 that there’s one red and one blue. So after adding the 3 red and 2 blue, the chance of drawing a red marble is, I think,
Whatever that comes out to… I guess approximately 14/24.
So anyway, a similar method can be used to find the answer for any unspecified but limited number of original red and blue marbles.
But what about the case where the number of red and blue marbles is unspecified and unlimited? Is there no answer in this case? Why is there no answer in this case though there is for all unspecified-but-limited original amounts?
But now you are adding more information to your original problem.
Now the problem becomes:
r + 3 red marbles are added to a bag, b + 2 blue marbles are added to the bag.
r and b are random numbers, with upper limit L.
What is the probability P of drawing a red marble?
It looks to me like the limit of P as L -> infinity is 1/2.
The problem you describe (as quoted below) is exactly the problem I described in my original post, except I didn’t say anything about an “upper limit L”. The problem I am concerned with is when there is no upper limit. I mentioned a different but similar problem in a subsequent post which included upper limits, as part of an explanation as to why I wasn’t sure there isn’t enough info in the original problem.
Now, you agree with me that when there is an upper limit, you can get an answer. Then you say that as the limit approaches infinity, the answer approaches 1/2. This is the kind of answer I am looking for–note what you might be construed as saying, roughly, is that removing the limit makes the answer 1/2, which is exactly what I was asking about: what is the probability when you have no limit?
But is it safe for me to say that the probability is 1/2 if I don’t know the original numbers of marbles? Or do I just have to say that the probability approaches 1/2 as we raise the upper limit of how many marbles there may be, but that when we have no idea how many marbles there may be, we simply have no idea what the probability is?
In other words, is it safe to say the probability (when I don’t know the number of marbles) equals what the probabilities approach as I imagine different possible limits to the numbers of marbles and raise that limit towards infinity, or is this not a legitimate move?
That’s exactly correct. If you don’t know what the initial numbers of red and blue marbles are beforehand, then you don’t know how many there are afterwards (except that there are at least three red and two blue.) The only way to sensibly answer this question is to average over some set of “beforehand” numbers, and to calculate this average in a mathematically well-defined way, this set has to be finite. Once you’ve calculated this for some finite number L, then you can see if the answer still makes sense if L is arbitrarily large, but not before.
I’ll also add that there’s another implicit assumption in this approach, namely that all “beforehand numbers” are equally likely, i.e. a bag with x red marbles and y blue marbles is just as likely as that for one with x’ red and y’ blue for any values of x, x’, y, and y’. If not, then the problem becomes more difficult.
Yeah, I intended the implicit assumptions you mentioned to be taken as true.
You said “that’s exactly correct” but what exactly is correct? What you were responding to was an “or” statement…
So to clarify, is it correct that “It is safe for me to say that the probability is 1/2 if I don’t know the original numbers of marbles”?
Or rather, is it correct that “I just have to say that the probability approaches 1/2 as we raise the upper limit of how many marbles there may be, but that when we have no idea how many marbles there may be, we simply have no idea what the probability is”?
I think from your subsequent comments you were talking about the second option–meaning the answer to my original question is “the problem does not admit of an answer.” But I wasn’t a hundred percent sure, so please clarify this point for me.
You don’t have to know how many original marbles there are, but rather how many there can be.
If you know that the original number of marbles can be arbitrarily large, then I think it would be safe to say the probability is 1/2.
However, if you don’t know how many marbles there can be, then you cannot figure out the probability.
Clothahump, it’s sort of right in a way, but I was asking for an answer which does not involve the variables r and b–I was asking whether a numeric answer can be arrived at in absence of any knowledge of the values of those two variables.
You CAN NOT get an answer for this problem as it is stated. You can come up with a general formula a la Nighttime. If you know one of the following things then you can get an answer:
Exactly how many red and blue marbles you started with.
The probability distribution function (PDF) for how many red and how many blue marbles are in the bag to start.
That the number of red and blue marbles is “really large”.
Let me 'splain:
If you know exactly how many reds and blues you’ve got in total, it’s simple.
If you don’t know how many reds and blues you’ve got but you know the distribution function for them then you can figure out the expected value of reds and blues and then calculate (for example, suppose that both reds and blues have the following PDF - p(1)=0.1, p(2)=0.1, p(3)=0.2, p(4)=0.13 and p(5)=0.47). Without knowing the PDF you can’t necessarily assume that there is an upper or lower bound on how many reds and blues there are, and you can’t assume that the probability of there being, say, 10 reds is the same as the probability of there being 1 red or 74 reds or 10 jillion reds.
Special case. If the number of reds and blues are approximately equal then the answer will converge on 0.5 as those numbers increase. If you start with 100 blues and 102 reds you have a 105/207 chance (0.5072). If it was 1000 blues and 1002 reds it’s 1005/2007 (.5007).
But the problem as stated does not state exactly how many reds and blues you start with, it doesn’t give a distribution function for reds and blues and it doesn’t say “for the case where there are about the same number of reds and blues and there’s a very large number of each”.
Sorry, in my third case I should have explicity stated that not only are the number of reds and blues really large but those two values are pretty close.
If you don’t know how many marbles there can be, then there may still be a limit on how many there can be - you just don’t know what it is.
For example, the number may be limited by how many marbles exist; you don’t know how many marbles exist, so you don’t know how many marbles there can be, but that doesn’t change the fact that there is a limit on how many marbles there can be. You just don’t know what the limit is.
But if you know that the number of marbles can be arbitrarily large, then you know there cannot be a limit on how many there can be.
There’s something a bit troubling about this problem as it has developed in this thread.
The OP specified that there is “some unspecified number” of blue and red marbles in the bag. As I see it, this makes it clear that we are dealing with one specific (and unspecified) case. The correct response to this is clearly that there is no way, after adding 3 red and 2 blue marbles, to state what is the probability of drawing a red marble, except to say that the probability is less than 100% and greater than zero.
But the discussion seems to have transmogrified into consideration of all possible cases - a sort of meta-probability. This is fine, so long as we understand that this was not part of what was being considered in the OP. If (as later mentioned but in no way touched on by the OP) all possible original distributions of marbles are equally likely, then it is possible to say that the most likely probability is close to 50%. But that is very far from showing that under the OP’s specifications, the probability is 50%.
xeema, isn’t it the case that when calculating probabilities involving some unspecified quantity, we usually do something like averaging up all the possible values that unspecified quantity could have?
As in this simplified example: there is either one red marble, or no red marble. In other words, there is some unspecified number of marbles in the bag, either zero or one.
Now what is the probability that I will find one marble in the bag? Well… we take the unspecified amount, average it’s two possible values (zero and one) and we get our answer: 1/2.
So I’m not sure I’m following your objection. You said we started by talking about “some unspecified number” but ended up by talking about looking through all the possible amounts that unspecified number might be, given our epistemic position towards the bag. As I have just argued, these two lines of discussion you mention are not only compatible, but the second is naturally to be thought of as a discussion that follows from the first.
