Probability of choosing the same number?

Hi all.

As good as I was at maths in high school, I'm afraid I never could quite grasp the Addmaths syllabus at school. Therefore, I'm not that great at probabilities :D

I have a slight dilemma. I'm going to try and explain this properly:

Let's assume I have two random numbers (each containing 10 digits). I now join those two 10-digit numbers to create one long 20-digit number.

Now, at random, I pick out 7 digits to create a new, smaller number. This process continues indefinitely, providing me with a long list of 7 digit numbers.

This is where my question comes in. What are the chances of picking out 7 digits that match, exactly, a previous 7 digits?

This puzzle has been worrying me for the entire morning, and like I said, I'm not that great at probabilities.

I'd really appreciate the help :)

Thanks,

Geoff