# Probability: Infinite marbles placed in, and selected from an urn

1. Sep 12, 2010

### IniquiTrance

1. The problem statement, all variables and given/known data

I have a countably infinite set of marbles numbered; 1, 2, 3,..., n.

I also have an urn that can hold an infinite amount of marbles.

I then place marbles 1 and 2 into the urn, and remove one of them with the following probabilities:

The probability of removing a marble is proportional to its number.

So, the probability that I remove marble 1 is $$\stackrel{1}{3}$$, and that I remove marble 2 is $$\stackrel{2}{3}$$.

Once a marble is removed, I then place marbles 3 and 4 into the urn. I

Now if marble 2 was removed earlier, then marbles 1,3,4 are in the urn. The probability of removing any of them are now respectively, $$\stackrel{1}{8}, \stackrel{3}{8}, and \stackrel{4}{8}$$

I keep adding and removing marbles as above, in order of their number.

I am asked to show, that there is a positive probability that marble 1 remains in the urn forever.

2. Relevant equations

3. The attempt at a solution

Not quite sure how to pin this down. Any help is much appreciated!

2. Sep 12, 2010

### Dick

Ok, so the probability marble 1 survives the first pick is 2/3. The probability that it survives the second is 7/8. At this point you have a choice which one to pick which is not 1. Pick the one which has the least probability to be picked which is not 1. That would be a lower bound for the probability that 1 will never be picked, right? I haven't tried to show the resulting infinite product is positive. Can you? That should get you started.

3. Sep 12, 2010

### IniquiTrance

$$\stackrel{2}{3}*\stackrel{7}{8}$$, no?
$$\prod_{n}^{}\mathbb{P}\left\{A_{n}|A_{n-1}\right\}$$