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Both are the same. Since it is a probability measure, it will be true only if a large number of bits are considered.
If you analytically arrive at a prob of error = 0.001, then when you actually build the system and observe a very large number of bits (say 1 million), then the number of bits in error should be nearly 1 error per 1000 bits recieved. (ie; ~1000 bit errors in the 1 milliom bits recieved).