Similarly, the 27 bit rule for 100,000,000 people assumes that the bits have equal numbers of people who are yes and no on a question. In fact, some bits are more discriminating than others. “Have you ever been elected to an office that requires a statewide vote or been a Vice President?” (perhaps two bits of information), is going to eliminate 99.9999%+ of potential candidates for President, yet work nearly perfectly to dramatically narrow the field from the 100,000,000 eligible candidates. “Do you want to run for President?”, cuts another 90%+ of potential candidates.
It’s not an assumption, it’s a definition. Whatever is enough to cut your current set of candidates in half is “one bit”- the first bit will eliminate 50,000,000 people, the last bit will eliminate 1. An answer that reduces the set of candidates to .000001 times its original size contains 20 bits of information. (Notice that the question doesn’t have bits of information associated with it, since each possible answer reduces the candidate set by a different amount- if they said “no,” you acquired only a millionth of a bit of information.)
It’s not an assumption, it’s a definition. Whatever is enough to cut your current set of candidates in half is “one bit”- the first bit will eliminate 50,000,000 people, the last bit will eliminate 1. An answer that reduces the set of candidates to .000001 times its original size contains 20 bits of information. (Notice that the question doesn’t have bits of information associated with it, since each possible answer reduces the candidate set by a different amount- if they said “no,” you acquired only a millionth of a bit of information.)