1. with(statistics)
with(linear algebra)


EstGr= proc(p,i,N,K)
Local c, prob, r, cle:
r=0
c=0
prob=0
While r<K do
cle=GRt(p, i, N)
prob= c+cle[1]
c=c+cle[2]
r=r+1
end do
prob=prob/K
c=c/K
print(‘ProbabilityofExitingAWinner’, prob); 
print(‘AverageDurationOfGame’, c); RETURN([prob,c]);
end proc
EstGR (1, 5, 5, 100)


EstGR= proc(p, i, N, K)
Local c, prob, r, cle
r=0
c=0
prob=0
While r<K do
        Cle GRt( p, i, N): prob= prob +cle[1]: c=c+cle[2]: r=r+1
End do 
prob=prob/K
c=c/K
print(‘ProbabilityofExitingAWinner’, prob); 
print(‘AverageDurationOfGame’, c); 
RETURN([prob, c])
End proc


2.  
1. It is true due to the possibility of losing and winning being equal
2.  yN(i)=i/N satisfies the recurrence. 
3.  
4. Recurrence is equal to EN(i)=i(N-i)




3. 
ExactFairGR=proc(i,N)
Local x, c , XN, d;
x=i
c=0
While 0<x and x<N do
XN= RandomVariable(Bernoulli(i/N))
c=x+1
d= trunc(Sample(XN,)[1])
If d=0 then x=x-1
Else x=x+!
End if
End do 
RETURN([i/N,c])
End proc


4. 
p=RandSM(10)
StSa(p, 400)
evalf(StSp(p, 400))
evalf(StS(p))