#Aniket Shah
#01/27/14
#HW1
#Ok to post

#A few examples from Garvan’s Maple book

#1
p:= x^9-1:
factor(p);

simplify(3*4^(1/2)+5);
simplify((x^2)^(1/2));

rationalize((1-sqrt(2))/(1+sqrt(2)));

#2
#CheckPie2(A,B)(): inputs two sets of integers and 
#checks the inclusion-exclusion principle for two
#sets: |A union B|=|A|+|B|-|A intersect B|
CheckPie2:=proc(A,B) local a,b, aub, anb:

  a:=nops(A):
  b:=nops(B):

  aub:=nops(A union B):
  anb:=nops(A intersect B):

  if aub=a+b-anb then
    print(`good`):
  else
    print(`bad`):
  fi;

end:

#3
#DoNs()(): modified DoN(N) which tosses a coin with 
#statements and only outputs the number of coin 
#tosses that it took to exit with one dollar
DoNs:=proc(N) local coin, c, ST, account,i,j:

  coin:=rand(0..1):
  ST:=1:
  account:=0:
  j:=0:

  for i from 1 to N do

    c:=coin():
    j:=j+1:

    if c=1 then
      RETURN(j):
    else
      account:=account-ST:
      ST:=2*ST:
    fi;

  od:
  RETURN(N);

end:

#4
#AveDuration(N,K)(): runs DoNs(N) K times and 
#computes the average duration
AveDuration:=proc(N,K) local i, average, totalLength:

  average:=0:
  totalLength:=0:

  for i from 1 to K do
    totalLength:=totalLength+DoNs(N):	
  od:
  
  RETURN(totalLength/K):

end:

#I got 1027/500 for AveDuration(100,1000)

#5
#limit of the expected value of AveDuration(N,K) 
#as N,K->infinity will be the sum of n/2^n  as n 
#goes to infinity, which is 2. If the probability 
#is p, then the expected value is 1/p.