import numpy as np

"""

A dynamic programming solution for the World Series problem.

Two players/teams play as many games as needed until one of them wins

n games. So 2n - 1 games at most are played.

"""

def world_series(n, i, j):

"""

The function takes as input a given state (i, j): i games have been won

so far by the first player and j games have been won by the other player.

The function returns the probability that the first player wins the

competition (reaches n games) given the current state (i, j).

:param n: int | number of games to win to be declared champion

:param i: int | number of games won so far by the first player

:param j: int | number of games won so far by the second player

:return: float | probability that the first player wins the competition

"""

# Sanity check

if i >= n or j >= n:

raise ValueError("i and j must be strictly smaller than n")

# Create matrix of size n^2

# M[i][j] represents the probability to be in the state (i, j), i.e. that

# i + j games have been played i of which were won by the first player

# and the other j the second player

M = [[1]*(n - j + 1) for _ in range(n - i + 1)]

# Fill first column of M

for i_ in range(1, n - i + 1):

M[i_][0] = M[i_ - 1][0] * 0.5

# Fill first row of M

for j_ in range(1, n - j + 1):

M[0][j_] = M[0][j_ - 1] * 0.5

# Fill other values of M by filling row by row

for i_ in range(1, n - i + 1):

for j_ in range(1, n - j + 1):

M[i_][j_] = 0.5 * (M[i_][j_ - 1] + M[i_ - 1][j_])

# Correct last value

M[n - i][n - j] = 0

# Cast M to numpy array: easier to slice

M = np.array(M)

# Returns the proba that first player wins

return np.sum(M[n - i, 1: -1])

if __name__ == "__main__":

print(world_series(4, 3, 1))

#print(world_series(5, 4, 2))

print(world_series(8, 4, 6))