If you pick 24 words randomly, the probability of having a seed phrase which passes the checksum would be 1 in 256. For the last word 8 out of the 2048 words would produce a valid seed phrase which passes the checksum.
I know the last word is a checksum generated from the first 23 words,
The checksum isn't the last word. In a 24 word seed phrase, the checksum is the last 8 bits.