An unenlightened professor asks his students to memorize the first $1000$ digits of

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for the exam. To grade the exam, he uses a $100-$digit cryptographic hash function

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$.$ Instead of carefully reading the students$$ answers, he hashes each of them individually to obtain binary strings of length $100.$ Your score on the exam is the number of bits of the hash of your answer that agree with the corresponding bits of the hash of the correct answer.

If someone gets $100\%$ on the exam, why is the professor confident that the student$$s answer is correct?

Suppose each student gets every digit of

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wrong $($a very unlikely occurrence!$),$ and they all have different answers.