It's possible to compress (some) random data, based on just luck. It's not generally possible to compress any random data.
If I generate a thousand random 1024 byte sequences, I'm sure I could pick a few of them that I can compress, so long as I don't have to compress all of them.
It's also possible that the total of my compressed subset and the uncompressed rest, is smaller than the sum of the sequences! This still doesn't mean I could compress any random sequence, just specific ones. Ones that had lower entropy, by pure luck.
To translate to your analogy: I can make a compressor that compresses 2 bits into 1 BUT it works only works for 50% of all two bit sequences...
If I generate a thousand random 1024 byte sequences, I'm sure I could pick a few of them that I can compress, so long as I don't have to compress all of them.
It's also possible that the total of my compressed subset and the uncompressed rest, is smaller than the sum of the sequences! This still doesn't mean I could compress any random sequence, just specific ones. Ones that had lower entropy, by pure luck.
To translate to your analogy: I can make a compressor that compresses 2 bits into 1 BUT it works only works for 50% of all two bit sequences...