You were right except for your last statement
The probability of two random bytes being the same is 1 in 256 (the first byte can be anything, but the second byte must be the same as the first). Another way of thinking about it - there are 256 pairs of identical bytes and 256*(1/65536) = 1/256.
You can extend the same to three bytes... When you look at any sequence of three random bytes and want to know the probability that they are all the same, you are really asking "what is the probability of the second and the third each being the same as the first"? And the answer is (1/256)*(1/256).
In reality bytes do not occur "randomly" - in uncompressed data in particular there are significant patterns so these probabilities do not translate into equivalent observed frequencies of pairs and triples for most file types.