#Nathan Fox
#Homework 8
#I give permission for this work to be posted online

#Read packages
with(`combinat`):

#Read procedures from class
read(`C9.txt`):

#Help procedure
Help:=proc():
    print(` ExpectedDurationE(N, p) , GRsimWL(N, a, p, K) `):
end:

##PROBLEM 1##

#Conjecture: f(N, a) = a/N
#This checks out with the system, and the boundary conditions work,
#so it's right

##PROBLEM 2##

#ExpectedDurationE(N, p): inputs N (the exit capital if you are a
#winner), and p (the probability of winning a dollar at each
#round), and outputs the list of length N-1 whose a-th entry is
#the expected duration of the game (exiting either a winner
#or loser).
ExpectedDurationE:=proc(N, p) local E, eq, var, i, sol:
    var:={seq(E[i],i=0..N)}:
    eq:={E[0]=0, E[N]=0, seq(E[i]=1+p*E[i+1]+(1-p)*E[i-1], i=1..N-1)}:
    sol:=solve(eq, var):
    return subs(sol, [seq(E[i],i=1..N-1)]):
end:

##PROBLEM 3##

#Conjecture: g(a, N)=a*(N-a)
#This checks out with the system, and the boundary conditions work,
#so it's right

##PROBLEM 4##

#It checks out

##PROBLEM 5##

#For N=50, PrE(N,9/19)[N/2] is 0.06698122976
#For N=80, PrE(N,9/19)[N/2] is 0.01456559065
#For N=100, PrE(N,9/19)[N/2] is 0.005127349998
#These are probabilities of winning, and they are all really small,
#so gambling is bad

##PROBLEM 6##

#GRsimWL(N, a, p, K): the gambler goes to the casino K
#times and does GR(N, a, p)
#Returns a triple of numbers. The first number is as before, the
#ratio of wins to K, the second number is the average duration of
#the games where he exited a winner, and the third number is the
#average duration of the games where he exited a loser.
GRsimWL:=proc(N, a, p, K) local w, d, i, g:
    w:=0:
    d:=[0$2]:
    for i from 1 to K do
        g:=GR(N, a, p):
        w:=w+g[1]:
        d[g[1]+1]:=d[g[1]+1]+g[2]:
    od:
    return evalf(w/K), evalf(d[2]/w), evalf(d[1]/(K-w)):
end: