A Bitcoin address is the end of an assembly line. It starts with one random number, your private key, and runs through elliptic-curve multiplication, two hash functions, and a checksummed encoding. Every step is computed here, for real, by hand. Change one bit of the secret and the whole address avalanches.
The same secret in Wallet Import Format: version byte + key + checksum, Base58. This is what you back up.
You can run this line top to bottom in microseconds. Running it backwards, address to public key to private key, would mean breaking both the curve and the hashes.
Hit Flip one bit. A single changed bit in the secret produces a completely unrelated address (changed characters in red). No partial credit, and that's a good hash.
Those last 4 bytes are a hash of everything before them. Mistype an address and the checksum won't match, so your wallet refuses to send. Typos can't burn coins.