Gambling, R - betting analysis

Some of the gambling games are really coin flips, with 50/50 chances of success. Along those lines we have coding from http://forumserver.twoplustwo.com/25/probability/flipping-coins-getting-3-row-1233506/ that determines the probability of a series of heads or tails in a coin flip, with a trigger that can be used if you know the coin/game is biased towards one result or the other.

We have the following script:

##############################################
# Biased/unbiased  recursion of heads OR tails
##############################################
import numpy as np
import math

N = 14     # number of flips
m = 3      # length of run (must be  > 1 and <= N/2)
p = 0.5   # P(heads)

prob = np.repeat(0.0,N)
h = np.repeat(0.0,N)
t = np.repeat(0.0,N)

h[m] = math.pow(p,m)
t[m] = math.pow(1-p,m)
prob[m] = h[m] + t[m]

for n in range(m+1,2*m):
  h[n] = (1-p)*math.pow(p,m)
  t[n] = p*math.pow(1-p,m)
  prob[n] = prob[n-1] + h[n] + t[n]


for n in range(2*m,N):
  h[n] = ((1-p) - t[n-m] - prob[n-m-1]*(1-p))*math.pow(p,m)
  t[n] = (p - h[n-m] - prob[n-m-1]*p)*math.pow(1-p,m)
  prob[n] = prob[n-1] + h[n] + t[n]

prob[N-1]  

The preceding code produces the following output in Jupyter:

We end up with the probability of getting three heads in a row with an unbiased game. In this case, there is a 92% chance (within the range of tests we have run 14 flips).