The sum of outcomes for rolling an N-sided dice K times.

maths/sum_of_outcomes_for_rolling_n_sided_dice_k_time.py

@@ -0,0 +1,131 @@
+import numpy as np
+
+
+def outcome_of_rolling_n_sided_dice_k_time(n_side: int, k_time: int) -> list:
+    """
+    Sum of outcomes for rolling an N-sided dice K times.
+
+    This function returns a list. The first element is an array.
+    That array contains indexes.
+    The second element is another array.
+    That array contains probabilities for getting each index values
+    as the sum of obtained side values.
+
+     Algorithm Explanation:
+
+     1. Explanation of range:
+     When we are rolling a six-sided dice the range becomes
+     1 to 6.
+     While rolling 5 times range becomes 2 to 12.
+     The sum outcomes becomes 5 when all rolling finds 1.
+     30 happens when all rolling finds 6.
+     1 is the minimum and 6 is the maximum of side values
+     for a 6 sided dice. Therefore, the range is 5 to 30.
+     Therefore, the range is k to n*k.
+
+     2. Explanation of probability distribution:
+     Say we are rolling a six-sided dice 2 times.
+     for 0 roll, the outcome is 0 with probability 1.
+     For the first roll, the outcome is 1 to 6 equally distributed.
+
+     For the second roll, each previous outcome (1-6) will face
+     an addition from the second rolling (1-6).
+     If the first outcome is (known) 3, then the probability of
+     getting each of 4 to 9 will be 1/6.
+
+     The sum becomes 2 for two 1 outcomes. But the sum becomes
+     3 for two outcomes (1,2) and (2,1).
+
+     Link:
+     https://www.thoughtco.com/
+     probabilities-of-rolling-two-dice-3126559
+
+     That phenomenon is the same as the convolution. However, the
+     index position is different. Therefore, we adjust the index.
+
+
+     NB: a) We are assuming a fair dice
+     b) Bernoulli's theory works with getting the probability of
+     exactly 3 sixes while rolling 5 times. It does not work directly
+     with the sum. The same sum can come in many combinations.
+     Finding all of those combinations and applying Bernoulli
+     is more computationally extensive.
+     c) The algorithm can be used in playing games where the sum of
+     multiple dice throwing is needed.
+
+     I used that method in my paper to draw the distribution
+     Titled: Uncertainty-aware Decisions in Cloud Computing:
+     Foundations and Future Directions
+     Journal: ACM Computing Surveys (CSUR)
+     link: https://dl.acm.org/doi/abs/10.1145/3447583
+     The PDF version of the paper is available on Google Scholar.
+
+
+     >>> import numpy as np
+     >>> outcome_of_rolling_n_sided_dice_k_time(.2,.5)
+     Traceback (most recent call last):
+         ...
+     ValueError: The function only accepts integer values
+     >>> outcome_of_rolling_n_sided_dice_k_time(-1,5)
+     Traceback (most recent call last):
+         ...
+     ValueError: Side count should be more than 1
+     >>> outcome_of_rolling_n_sided_dice_k_time(3,-2)
+     Traceback (most recent call last):
+         ...
+     ValueError: Roll count should be more than 0
+
+     >>> outcome_of_rolling_n_sided_dice_k_time(4,2)
+     [[2, 3, 4, 5, 6, 7, 8],
+      array([0.0625, 0.125 , 0.1875, 0.25  , 0.1875, 0.125 , 0.0625])]
+     >>> outcome_of_rolling_n_sided_dice_k_time(6,2)
+     [[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12],
+      array([0.02777778, 0.05555556, 0.08333333, 0.11111111, 0.13888889,
+             0.16666667, 0.13888889, 0.11111111, 0.08333333, 0.05555556,
+             0.02777778])]
+     >>> outcome_of_rolling_n_sided_dice_k_time(6,3)
+     [[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18],
+      array([0.00462963, 0.01388889, 0.02777778, 0.0462963 , 0.06944444,
+             0.09722222, 0.11574074, 0.125     , 0.125     , 0.11574074,
+             0.09722222, 0.06944444, 0.0462963 , 0.02777778, 0.01388889,
+             0.00462963])]
+    """
+
+    if n_side != int(n_side) or k_time != int(k_time):
+        raise ValueError("The function only accepts integer values")
+    if n_side < 2:
+        raise ValueError("Side count should be more than 1")
+    if k_time < 1:
+        raise ValueError("Roll count should be more than 0")
+    if k_time > 100 or n_side > 100:
+        raise ValueError("Limited to 100 sides or rolling to avoid memory issues")
+
+    probability_distribution = 1
+    distribution_step = np.ones(n_side) / n_side
+
+    iter1 = 0
+    while iter1 < k_time:
+        probability_distribution = np.convolve(
+            probability_distribution, distribution_step
+        )
+        iter1 = iter1 + 1
+
+    output = []
+    index = list(range(k_time, k_time * n_side + 1))
+    output.append(index)
+    output.append(probability_distribution)
+    return output
+
+
+"""
+# Extra code for the verification
+
+index_dist = outcome_of_rolling_n_sided_dice_k_time(6, 5)
+
+print("Indexes:",index_dist[0])
+print("Distribution:",index_dist[1], "Their summation:",np.sum(index_dist[1]))
+
+import matplotlib.pyplot as plt
+plt.bar(index_dist[0], index_dist[1])
+
+"""