#Aniket Shah
#Jan 30, 2-15, Assignment hw2.txt
#OK to Post

#1
seq(sin(i*pi/2),i=0..10);

y:=1,2,3;
y:=op([op([y])]);

S:= table([(a)=1,(x)=4,(3)=5]);

f:=(x,z) -> (x*z)^2;
F:=array(1..2,1..3,[[seq(diff(f(i,z),z),i=1..3)],[1,2,3]]);

#2
#MHg(q): simulates ONE instance of a variation on the Monty Hall 
#game show where there is a car behind one of the doors 
#(w.l.o.g door #1) and the two other doors have goats.
#implementing the switching strategy
MHg:=proc(q) local p, i, door1, UnpickedDoors, MontyDoors,MontyDoor,ND :

p:=convert(q,fraction):

i:=rand(1..denom(p))():

if i<=numer(p) then
  door1:=1:
  print(`You picked door `, door1 ):
else
  door1:=rand(2..3)():
  print(`You picked door `, door1 ):
fi; 

UnpickedDoors:={1,2,3} minus {door1}:

MontyDoors:=UnpickedDoors minus {1}:

MontyDoor:=MontyDoors[rand(1..nops(MontyDoors))()]:

print(`Monty shows you that door`, MontyDoor, `has a goat `):
print(`and he kindly (or "kindly") offers you to switch your`):
print(`choice, and you agree`):

ND:={1,2,3} minus {door1, MontyDoor}:
ND:=ND[1]:

print(`You decided to change your guessed door to door`, ND):

if ND=1 then
print(`Congratulations! You won a Porche! `):
else
print(`Monty tricked you, have fun with the goat.`):
fi:

end:

#3
#MHgs(q): same as MHg(p) but silent
MHgs:=proc(q) local p, i, door1, UnpickedDoors, MontyDoors,MontyDoor,ND :

p:=convert(q,fraction):

i:=rand(1..denom(p))():

if i<=numer(p) then
  door1:=1:
else
  door1:=rand(2..3)():
fi; 

UnpickedDoors:={1,2,3} minus {door1}:

MontyDoors:=UnpickedDoors minus {1}:

MontyDoor:=MontyDoors[rand(1..nops(MontyDoors))()]:

ND:={1,2,3} minus {door1, MontyDoor}:
ND:=ND[1]:

if ND=1 then
true:
else
false:
fi:

end:

#MHgsN(p,N): Runs MHgsN(p) N times, calculates the number of
#wins to losses
MHgsN:=proc(p,N) local i, numPorsche, propWins:

numPorsche:=0:

for i from 1 to N do
  if MHgs(p) then
    numPorsche:=numPorsche+1:
  fi:
od:

propWins:= evalf(numPorsche/N):

end:

#SDsim(N): simulates scenario where 1 person gets sick and another
#gets falsely diagnosed, out of N people
SDsim:=proc(N) local rs, fs, people, healthyPeople:

people:={seq(i,i=1..N)}:
rs:=rand(1..N)():

healthyPeople:=people minus {rs}:

fs:=healthyPeople[rand(1..(N-1))()]:

[rs,fs]:

end:

#6
#If k*(2*r+1)>N, then the probability of BDg(k,N,r) is 0, 
#basically by pigeonhole
#if k*(2*r+1)<=N, then the probability is binomial(N-2*r*k,k)*k!/N^k.