You need every bit string to be maximally unpredictable a priori. EG, every bit string of length n is equally likely. This is not the same as having exactly e

You need every bit string to be maximally unpredictable a priori. EG, every bit string of length n is equally likely. This is not the same as having exactly equal numbers of 0s and 1s. That doesn't really even make sense, because with the constraint as mentioned, you can't ever transmit an odd length string. Also, if you are transmitting a bit string of length 2, knowing the first bit guarantees you know the second, which is low entropy (you only have one bit of entropy in a 2 bit message). On the other hand, it is the case that most long bit strings sampled from a maximally entropic source have roughly the same number of 0s and 1s.