“In raquetball, a player continues to serve as long as she is winning; a point is scored only when a player is serving and wins the volley. The first player to win 21 points wins the game. Assume that you serve first and have probability .6 of winning a volley when you serve and probability .5 when your opponent serves. Estimate, by simulation, …

prob.py

import numpy as np 

# Parameters - First game and winning prob.
initial = np.array([0.6,0.4]) #first game
prob_matrix = np.array([[0.6,0.4],[0.5,0.5]]) 


# Calculate the n product of a vector and prob_matrix
def multiple_dot(vector, n):
    if (n == 0):
        return vector
    else:
        vector = vector.dot(prob_matrix)
        return multiple_dot(vector, n-1)

# multiple_dot(initial,20) -> array([0.55555556, 0.44444444])

# Save vectos into list. 
def generate_21_list():
    prob_list = [] 
    prob_list.append(multiple_dot(initial,21)) # considering 21 wins in 21 games
    for i in range(1,21): # consideration winning 21 scores in 21+i games
        next_vector = prob_list[i-1].dot(prob_matrix)
        prob_list.append(next_vector)
    return prob_list
  
# Mean and get result
problist = generate_21_list()
(sum(problist) / len(problist))[0]