longrun_function(n,k,pb)
{
# n:length of the whole binary sequence (independent) (n>2) #
# pb : probibility of sucess for each trial#
# k:length of the longest run L#
# outcome : P(L>=k)#
  a_matrix(1,k,1)
  b_diag(a,k)
  c_matrix(0,1,k)
  p0_cbind(1,c)
  U_rbind(a,0)
  c_cbind(c,1)
  M_cbind((1-pb)*a, pb*b)
  M_rbind(M,c)
  Mn_M%*%M
  for (i in 3:n) {  Mn_Mn%*%M }
  1-p0%*%Mn%*%U
}


Elongrun_function(n,pb)
{
# outcome : expectation of L #
  s_0
  for (k in 1:n) {  s_s+longrun(n,k,pb)  }
  s
}