You have a jar that is filled with a hundred marbles, each of them either black or white, but you have no idea how many of each color there are. However, you have been told that all possible quantities of white marbles (from 0 to 100, both inclusive) are equally probable.

You randomly select 100 marbles from the jar one at a time, with replacement, and they are all white. What is the probability that the jar contains only white marbles?

(In reply to A solution by Dej Mar)

There are two types of probability we're talking about here: The a priori probability of selecting a particular jar, and the probability of selecting a white marble from any given jar.

The sampling is with replacement, so all drawings from a particular jar have the same probability: i/100, where i is the number of white marbles in the jar, so for any given initial jar contents, that set of contents stays the same, so the overall probability of all white marbles from that jar is (i/100)^100.

On the other hand, the choice of jar is made only once, and you have 1/101 chance of choosing a particular jar (that is, of having a jar that has a given number of white marbles out of the hundred).

As you draw out marbles, some probabilities do in fact change. Most notably, as I think you have in mind, as soon as you draw the first white marble, you know that the jar can't be one with all black marbles.  But this is automatically taken care of in the Baye's rule analysis where the denominator is the sum of all the ways of choosing all white marbles, regardless of the initial composition of the jar: Sigma{i=0 to 100} (i/100)^100 / 101.

The 101 is the a priori probability that that jar would have been chosen.  While there are 101 items being summed, note that when i = 0, nothing is contributed toward the total, as (0/100)^100 is zero.  That's the contribution of the all-black-marble jar.

So your concern is addressed in the solution given.

Posted by Charlie on 2006-05-19 09:23:08