What is the chance of someone (randomly) finding the private key for an existing Bitcoin address? Say I have created a paper wallet with one bitcoin address in a vault. What are chances that someone else will find the private key for this wallet?

Related: what are the chances of vanitygen address - private key pair collision?
Essentially zero. They would have to find a private key whose public key hash matched yours. The public key hash is 160 bits long. If they had a billion computers, each of which could try a billion keys a second, and they tried for a billion years, they'd have much, much less than a one in a billion chance of getting it.