$2.(10$ points$)$ Blockchain and Zero Knowledge Proof

$($a$)(5$ points$)$ What is zero knowledge proof?

$($b$)(5$ points$)$ Robert has uncovered the secret code used to open a magic door in a cave. The cave is shaped like a ring, with the entrance on one side and the magic door blocking the opposite side. Mary wants to know whether Robert knows the secret code; but Robert, being a very private person, does not want to reveal his knowledge to Mary or to reveal the fact of his knowledge to the world in general. First, Mary waits outside the cave as Robert goes in$.$ Robert takes either path A or B; Mary is not allowed to see which path he takes. Then, Mary enters the cave and shouts the name of the path she wants him to use to return, either A or B$,$ chosen at random. How many time of successful checks could make Mary believe Robert know the code at $\backslash (99\backslash \%\backslash )$ confidence level.