This is famous game.

1> He doesn't switch
if he choose prize door
wi= win in ith choose
p(w1) = p(w2)=1/3

if he choose wrong door(no prize)
p(w1) = 0

2>He switch to the other remaining door than shown by host

if he was in prize door and he switch
p(w2|w1) = 0
if he was not in prize door ,host shows non prize door and u switch prize door u win

p(w2|not w1) = 2/3

p(w2) = p(w2|w1)p(w1)+p(w2|not w1)p(not w1)
= 0*1/3+1*2/3 = 2/3

3> U switch randomly

p(w2) = p(w2/s)p(s)+p(w2/not s)p(not s)
s= switch door randomly
= 2/3*1/2+1/3*1/2 =1/2

clearly switch is good .5>.33 :)