How many white balls are there in the box most probably?

There are n balls in a box. Some of them are white. A ball drawn from the box turns out to be white. How many white balls are there in the box most probably?

Alright, well i know there are originally n white balls and if I'm grabbing a white ball then there will now be n-1 balls left.

P(grabbing a white ball)= I'm not sure at all, for all we know that was the only white ball. Or there could be 100 balls and 99 are white. I really don't know how to approach this. I feel like we are given very minimal information. If someone could provide me with a solution that would be awesome! Thanks guys! :)

• The fixed data are that there are $n$ balls in the box, that you draw a white ball, and that there was known to be at least one white ball. There are therefore $100$ possible states of the box, one for each possible number of white balls. If we knew the probabilities of these states, we could work backwards to determine which was most probable given that we drew a white ball when drawing at random. But we don’t have those probabilities, and I don’t see any reason to prefer one assumed distribution of them over another. Commented Oct 6, 2013 at 2:16
• This is classical Bayes statistics. Was a prior distribution provided? Otherwise, this is NARQ.
– Did
Commented Dec 11, 2013 at 7:07
• possible duplicate of Probability that the bag contains all balls white given that two balls are white
– MJD
Commented Sep 6, 2014 at 14:05

The question is a bit strange, admittedly. This is how I would interpret it (SEE COMMENTS): Given the knowledge that there is at least one white ball in the box, and that the probability of selecting any ball is equal, then what is the number of white balls that will maximize the chance that a randomly pulled ball is white?

In this case, the answer is of course $n$.

There is another, perhaps more sensible option. We could also consider that the box was at one point filled, presumably by another person who did not care how many balls were white. Maybe she chose from a large box that had white and blue balls in equal proportion. In that case $n$ white balls would make it very likely that you get a white ball, but this is not very likely to happen.

Of course, there is no reason why there should be equal probability of white and blue, or even that there should be only two choices. Perhaps an even more sensible answer is $1$, because there is a staggering array of possible choices for balls to put in the box, and most of them would result in one white marble (well, most of them would result in none, actually, but we know there's at least one).

Pick your poison, really, because I would bet money that this problem was chosen to make a point: incomplete information is not an excuse to throw up your hands and say "It can't be done!" Rather, it's a time to choose some reasonable assumptions, state them carefully, and derive results from them rigorously.

• See this question is driving me nuts. It is a question on my assignment that I just don't know how to approach? Is there a very general way of saying that there are (n-1)! ways of doing this? Or is this actually just a question to think about and there is no correct way of getting a 'solution'
– dee
Commented Oct 6, 2013 at 2:11
• I don’t think that your first interpretation is consistent with the wording of the question: it neither answers the question nor explains why the question can’t be answered. Commented Oct 6, 2013 at 2:14
• @BrianM.Scott: I agree it's not a good interpretation, but I don't see how it is inconsistent with the question: "How many white balls are there in the box most probably?" could mean no more than "Extrapolate". Commented Oct 6, 2013 at 2:18
• @monmer: As Brian Scott said in the comments, there really is no meaningful answer to this question. Commented Oct 6, 2013 at 2:21
• @Eric: The answer must be either: a function of $n$; an explanation of why the question is not well-posed, preferably with an indication of what additional information would make it well-posed; or, if one can make a case for assuming a particular distribution of probabilities for the different numbers of white balls, that case together with the relevant calculation of an answer in terms of $n$. I see now that you probably intended my last alternative; I initially interpreted what you wrote a little differently. Commented Oct 6, 2013 at 2:21

A white ball is most likely to be chosen if the all the balls in the box are white. I'm not sure if this answers the question though.