% Matlab program generating the Markov chain for sunny and cloudy days  %
% It also computes the answer to the problem           %

P = [.7 .3
     .5 .5];    % Transition probability matrix        %

P3 = P^3;       % Three-step transition probability matrix

P3(1,1)         % Answer to the exercise               %

% Generate this Markov chain                           %
% X=0 is a cloudy day, X=1 is a sunny day              %
% Let's generate 100 days starting with a cloudy day   %

n=1; X(1) = 0;
for day=2:100;
  U=rand;
  if X(n-1)==1; X(n)=(U < .7)*1 + (U >= .7)*0;
  else          X(n)=(U < .5)*1 + (U >= .5)*0;
  end;
end;

X
plot(X);