Please Help: A Probability Math Question

##############################################
# Biased/unbiased  recursion of heads OR tails
##############################################

N = 9     # number of flips
m = 3      # length of run (must be  > 1 and <= N/2)
p = 0.5   # P(heads)

prob = rep(0,N)
h = rep(0,N)
t = rep(0,N)

h[m] = p^m
t[m] = (1-p)^m
prob[m] = h[m] + t[m]

for (n in (m+1):(2*m-1)) {
  h[n] = (1-p)*p^m
  t[n] = p*(1-p)^m
  prob[n] = prob[n-1] + h[n] + t[n]
}

for (n in (2*m):N) {
  h[n] = ((1-p) - t[n-m] - prob[n-m-1]*(1-p))*p^m
  t[n] = (p - h[n-m] - prob[n-m-1]*p)*(1-p)^m
  prob[n] = prob[n-1] + h[n] + t[n]
}

# m heads OR m tails
prob[N]

##################################
# Biased/unbiased  simulation of
# Heads OR Tails
# Heads AND Tails
# Neither heads NOR tails
# Code by BruceZ
##################################

N = 9     # number of flips
m = 3      # length of run
p = 0.50    # P(heads)

sims = 0
count1 = 0
count2 = 0

while (1) {
  flips = ifelse(runif(N,0,1) < p, "H", "T")
  runs = rle(flips)
  heads = runs$lengths[runs$values == "H"] >= m
  tails = runs$lengths[runs$values == "T"] >= m
  if ( sum(heads) | sum(tails) ) {count1 = count1 + 1}
  if ( sum(heads) & sum(tails) ) {count2 = count2 + 1}
  sims = sims + 1
}

p1 = count1/sims
error1 = 3.29*sqrt(p1*(1-p1)/sims)
p2 = count2/sims
error2 = 3.29*sqrt(p2*(1-p2)/sims)

# m heads OR m tails
p1
p1 - error1
p1 + error1
# m heads AND m tails
p2
p2 - error2
p2 + error2
# neither m heads NOR m tails
1 - p1
1 - p1 - error1
1 - p1 + error1
sims