Let

MM

be an algorithm that solves a decision problem

$$

so that on every string

xx

$,$

if

xx

is a $$yes$$ instance of

$$

$,$ then

MM

returns the answer $$yes$$ with probability

$11$

;$$

if

xx

is a $$no$$ instance of

$$

$,$ then

M$$M$$

returns the answer $$no$$ with probability

$1/31/3$

$.$

Alice, Bob, and Carol want to amplify the success probability of

MM

from

$1/31/3$

to a higher value

pp

$.$ To this end. they suggest three different algorithms:$$

Alice suggests the following algorithm

AA

$.$ On a given input

xx

$,$ run

MM

five times. If at least one of the answers is $$yes$,$ return $$yes$.$ Otherwise, return $$no$.$

Bob suggests the following algorithm

BB

$.$ On a given input

xx

$,$ run

MM

five times. If at least one of the answers is $$no$,$ return $$no$.$ Otherwise, return $$yes$.$

Carol suggests the following algorithm

CC

$.$ On a given input

xx

$,$ run

MM

five times. If the majority of the answers is $$yes$,$ return $$yes$.$ Otherwise, return $$no$.$

Questions:$$

Which of

AA

$,$

BB

$,$ and

CC

indeed amplifies the probability of success?$$

What is the new success probability

pp

$?$