deeper inquiry about the transaction process.

[kthejung]

How do the miners know that a private key is the correct one while not knowing what it is?

[Foxpup]

Read up on public-key cryptography.

[DannyHamilton]

http://en.wikipedia.org/wiki/Public-key_cryptography

There is a mathematical relationship between the private key and the bitcoin address.

This mathematical relationship allows the transaction creator to use the private key to calcualte a digital signature.  This digital signature is provided with the transaction.  Because of the mathematical relationship, the solo-miners (or mining pool operators) and all full nodes can use the public key to verify that the digital signature is valid without ever knowing what the private key was.

[kthejung]

Read up on public-key cryptography.

okay, thanks.