Step by Step solution about How to resolve the algorithm Dice game probabilities step by step in the Python programming language

The provided code is a Python program that calculates the probability of winning in a dice game. It defines three different functions to calculate the probability of winning, each with a different approach. The first function, gen_dict, generates a dictionary that counts the number of ways to get each possible sum when rolling multiple dice. It uses the itertools.product function to generate all possible combinations of dice rolls and then increments the count for each sum. The second function, beating_probability, uses the dictionary generated by gen_dict to calculate the probability of one player beating another player in a dice game. It considers all possible combinations of dice rolls for both players and calculates the probability of each player winning. The third function, winning, uses the combos function to generate a list of the number of ways to get each possible sum when rolling multiple dice. It then uses this list to calculate the probability of one player beating another player in a dice game. The code includes examples of how to use each function to calculate the probability of winning in a dice game. The code is well-written and easy to understand. It uses Python's built-in functions and modules effectively to solve the problem. Here is a more detailed explanation of the code: The gen_dict function takes two arguments: n_faces, the number of faces on each die, and n_dice, the number of dice being rolled. It returns a dictionary that counts the number of ways to get each possible sum when rolling the dice. The beating_probability function takes four arguments: n_sides1, the number of sides on the dice for player 1, n_dice1, the number of dice being rolled by player 1, n_sides2, the number of sides on the dice for player 2, and n_dice2, the number of dice being rolled by player 2. It returns the probability of player 1 beating player 2 in the dice game. The winning function takes four arguments: sides1, a list of the numbers on the sides of the dice for player 1, n1, the number of dice being rolled by player 1, sides2, a list of the numbers on the sides of the dice for player 2, and n2, the number of dice being rolled by player 2. It returns the probability of player 1 beating player 2 in the dice game. The code includes examples of how to use each function to calculate the probability of winning in a dice game. The code is well-written and easy to understand. It uses Python's built-in functions and modules effectively to solve the problem.

from itertools import product

def gen_dict(n_faces, n_dice):
    counts = [0] * ((n_faces + 1) * n_dice)
    for t in product(range(1, n_faces + 1), repeat=n_dice):
        counts[sum(t)] += 1
    return counts, n_faces ** n_dice

def beating_probability(n_sides1, n_dice1, n_sides2, n_dice2):
    c1, p1 = gen_dict(n_sides1, n_dice1)
    c2, p2 = gen_dict(n_sides2, n_dice2)
    p12 = float(p1 * p2)

    return sum(p[1] * q[1] / p12
               for p, q in product(enumerate(c1), enumerate(c2))
               if p[0] > q[0])

print beating_probability(4, 9, 6, 6)
print beating_probability(10, 5, 7, 6)


from __future__ import print_function, division

def combos(sides, n):
    if not n: return [1]
    ret = [0] * (max(sides)*n + 1)
    for i,v in enumerate(combos(sides, n - 1)):
        if not v: continue
        for s in sides: ret[i + s] += v
    return ret

def winning(sides1, n1, sides2, n2):
    p1, p2 = combos(sides1, n1), combos(sides2, n2)
    win,loss,tie = 0,0,0 # 'win' is 1 beating 2
    for i,x1 in enumerate(p1):
        # using accumulated sum on p2 could save some time
        win += x1*sum(p2[:i])
        tie += x1*sum(p2[i:i+1])
        loss+= x1*sum(p2[i+1:])
    s = sum(p1)*sum(p2)
    return win/s, tie/s, loss/s

print(winning(range(1,5), 9, range(1,7), 6))
print(winning(range(1,11), 5, range(1,8), 6)) # this seem hardly fair

# mountains of dice test case
# print(winning((1, 2, 3, 5, 9), 700, (1, 2, 3, 4, 5, 6), 800))


from __future__ import division, print_function
from itertools import accumulate # Python3 only

def combos(sides, n):
    ret = [1] + [0]*(n + 1)*sides # extra length for negative indices
    for p in range(1, n + 1):
        rolling_sum = 0
        for i in range(p*sides, p - 1, -1):
            rolling_sum += ret[i - sides] - ret[i]
            ret[i] = rolling_sum
        ret[p - 1] = 0
    return ret

def winning(d1, n1, d2, n2):
    c1, c2 = combos(d1, n1), combos(d2, n2)
    ac = list(accumulate(c2 + [0]*(len(c1) - len(c2))))

    return sum(v*a for  v,a in zip(c1[1:], ac)) / (ac[-1]*sum(c1))


print(winning(4, 9, 6, 6))
print(winning(5, 10, 6, 7))

#print(winning(6, 700, 8, 540))