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?

As a start, let's say it was 2 marbles instead of 100.  I interpreted "you have been told that all possible quantities of white marbles are equally probable" to be equivalent to the following situation:

There are 3 jars, one contains 2 black marbles, one contains 1 black and one white marble, and one contains 2 white marbles.  You pick one of them, and then draw 2 marbles from that jar, one at a time, with replacement. 

Label the marbles in each jar a) and b).

Jar 1:  a) black, b) black
Jar 2:  a) white, b) black
Jar 3:  a) white, b) white

After you've chosen your jar, each jar has 4 equally likely sequences of marbles: aa, ab, ba, bb, for a total of 12 possible sequences.  Given that you observed 2 white marbles, there are 5 possible situations:

1) jar 2, aa
2) jar 3, aa
3) jar 3, ab
4) jar 3, ba
5) jar 3, bb

4 out of these 5 involve choosing jar 3, so the answer to "what is the probability that the jar contains only white marbles?" would be 4/5.

Now, to extend this to the 100 marble case ...

Posted by iamkobe on 2006-05-18 12:06:39