This $is$ for a Math Cryptology class:

An unenlightened professor asks his students to memorize the first $1000$ digits of $$ for the exam. To grade the exam, he uses a $100-$digit cryptographic hash function H$.$ Instead of carefully reading the students$$ 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.

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

$($b$)$ Suppose each student gets every digit of $$ wrong $($a very unlikely occurrence!$),$ and they all have different answers. Approximately what should the average on the exam be$?$