Problem $7(3$ points$)$

Let M be an algorithm that solves a decision problem Pi so that on every string x$,$

if x is a "yes" instance of Pi$,$ then M returns the answer "yes" with probability $1$ ;

if x is a $"$no$"$ instance of Pi$,$ then M returns the answer $"$no$"$ with probability $1//3.$

Alice, Bob, and Carol want to amplify the success probability of M from $1//3$ to a higher value p$.$ To this

end. they suggest three different algorithms:

$$Alice suggests the following algorithm A$.$ On a given input x$,$ run M five times. If at least one of the answers is "yes", return "yes". Otherwise, return $"$no$".$

$$Bob suggests the following algorithm B$.$ On a given input x$,$ run M five times. If at least one of the answers is $"$no$",$ return $"$no$".$ Otherwise, return "yes".

$$Carol suggests the following algorithm C$.$ On a given input x$,$ run M five times. If the majority of the answers is "yes", return "yes". Otherwise, return $"$no$".$

Questions:

$$Which of A$,$ B$,$ and C indeed amplifies the probability of success?

$$What is the new success probability p$?$

please make sure you consider whole answer