Infinite supply of marbles probability problem

[Roni1985]

Homework Statement

A has an infinite supply of marbles numbered 1,2,3,... A places the marbles, two at a time and in numerical order, into an urn (i.e. first 1 and 2, then 3 and 4, etc.). Each time A puts two marbles, B reaches in and pulls on out. This process goes on forever.

c) If B always removes marbles randomly (each marble in the urn equally likely to e removed), with what probability will marble number 1 remain in the urn forever?

Homework Equations

The Attempt at a Solution

Here is what I did:
1,2 =>prob that 1 stays is 1/2

1,3,4 => prob that 1 stays is 2/3

1,3,4,6 =>prob that 1 stays is 3/4

until (n-1)/n

so we need to multiply them all and we are left with 1/n
so as n->inf, the probability that it happens is zero ?

Is my answer correct ?

Thanks,
Roni.

 

[Dick]

Seems fine to me.

 

[Roni1985]

Seems fine to me.

I see, thanks very much :)