What prevents somebody from "re-mining" an already existing Bitcoin? This may reveal that I know nothing about math or cryptography.
Let's say that a given Bitcoin was &quot;lost&quot; in that it was once mined and assigned/given to a person who had it in their wallet.dat, but then they either lost the password to this file, or deleted it, or otherwise lost access to it forever (or without any realistic hope of ever retrieving it).
Now, that Bitcoin has been mined out of the 21,000,000 total ones ever, but it's &quot;lost&quot;.
But if a computer once guessed the right private key for that coin, why can't another computer do it again?
Basically, isn't it possible that all the Bitcoins that are currently mined and in possession of somebody, could be &quot;re-mined&quot;?
Is it just &quot;extremely unlikely&quot;? And you cannot &quot;target&quot; a specific existing private key/receive address/Bitcoin?
I'm always fearing that one day, somebody &quot;re-mines&quot; the Bitcoins that I have, thus seizing access to them, either by sheer accident or by somehow deducing the ones I own and specifically try to &quot;re-mine&quot; those in order to make me poor?
But if a computer once guessed the right private key for that coin, why can't another computer do it again?

It wasn't guessed the first time. It was generated randomly from a set of 115792089237316195423570985008687907852837564279074904382605163141518161494336 valid keys (that's the actual number).
Generating the same one a second time is so astronomically unlikely that it's not worth talking about.

Basically, isn't it possible that all the Bitcoins that are currently mined and in possession of somebody, could be &quot;re-mined&quot;?

What you're talking about is guessing, or otherwise retrieving, the private keys to existing public keys that hold coins. It has nothing to do with mining.

Is it just &quot;extremely unlikely&quot;?

Beyond extremely. Even if every computer on earth would generate 100 million keys per second, and there are 2 billion computers, they'd take several billion times the age of the universe before they have anything resembling a feasible change of randomly hitting any key that holds a balance currency.

And you cannot &quot;target&quot; a specific existing private key/receive address/Bitcoin?

You can. There are algorithms to find the private key for a given public key faster than just brute force, but the numbers are still astronomical, and believed infeasible with humanity's current technology.