Add Project Euler problem 205 solution 1 (#5781)

* updating DIRECTORY.md

* Add solution

* updating DIRECTORY.md

* Fix

* Fix
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DIRECTORY.md

@@ -865,6 +865,8 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_191/sol1.py)
   * Problem 203
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_203/sol1.py)
+  * Problem 205
+    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_205/sol1.py)
   * Problem 206
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_206/sol1.py)
   * Problem 207

project_euler/problem_205/sol1.py

+"""
+Project Euler Problem 205: https://projecteuler.net/problem=205
+
+Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4.
+Colin has six six-sided (cubic) dice, each with faces numbered 1, 2, 3, 4, 5, 6.
+
+Peter and Colin roll their dice and compare totals: the highest total wins.
+The result is a draw if the totals are equal.
+
+What is the probability that Pyramidal Peter beats Cubic Colin?
+Give your answer rounded to seven decimal places in the form 0.abcdefg
+"""
+
+from itertools import product
+
+
+def total_frequency_distribution(sides_number: int, dice_number: int) -> list[int]:
+    """
+    Returns frequency distribution of total
+
+    >>> total_frequency_distribution(sides_number=6, dice_number=1)
+    [0, 1, 1, 1, 1, 1, 1]
+
+    >>> total_frequency_distribution(sides_number=4, dice_number=2)
+    [0, 0, 1, 2, 3, 4, 3, 2, 1]
+    """
+
+    max_face_number = sides_number
+    max_total = max_face_number * dice_number
+    totals_frequencies = [0] * (max_total + 1)
+
+    min_face_number = 1
+    faces_numbers = range(min_face_number, max_face_number + 1)
+    for dice_numbers in product(faces_numbers, repeat=dice_number):
+        total = sum(dice_numbers)
+        totals_frequencies[total] += 1
+
+    return totals_frequencies
+
+
+def solution() -> float:
+    """
+    Returns probability that Pyramidal Peter beats Cubic Colin
+    rounded to seven decimal places in the form 0.abcdefg
+
+    >>> solution()
+    0.5731441
+    """
+
+    peter_totals_frequencies = total_frequency_distribution(
+        sides_number=4, dice_number=9
+    )
+    colin_totals_frequencies = total_frequency_distribution(
+        sides_number=6, dice_number=6
+    )
+
+    peter_wins_count = 0
+    min_peter_total = 9
+    max_peter_total = 4 * 9
+    min_colin_total = 6
+    for peter_total in range(min_peter_total, max_peter_total + 1):
+        peter_wins_count += peter_totals_frequencies[peter_total] * sum(
+            colin_totals_frequencies[min_colin_total:peter_total]
+        )
+
+    total_games_number = (4 ** 9) * (6 ** 6)
+    peter_win_probability = peter_wins_count / total_games_number
+
+    rounded_peter_win_probability = round(peter_win_probability, ndigits=7)
+
+    return rounded_peter_win_probability
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")