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A bit in the data sense is just an element of the set of booleans. A bit in the entropy sense is the amount of information revealed by an observation with two equally probable outcomes. These are not the same thing because the amount of information contained in a bit is not always equal to one bit of entropy. For example, if a boolean is known to be 0, then the amount of information it contains is 0 bits. If it is known that the boolean is equally 0 or 1, then the information content is 1 bit. It depends on the prior probability distribution.
A bit in the data sense is just an element of the set of booleans. A bit in the entropy sense is the amount of information revealed by an observation with two equally probable outcomes. These are not the same thing because the amount of information contained in a bit is not always equal to one bit of entropy. For example, if a boolean is known to be 0, then the amount of information it contains is 0 bits. If it is known that the boolean is equally 0 or 1, then the information content is 1 bit. It depends on the prior probability distribution.