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 +1,116 @@
+import numpy as np
+
+def outcome_of_rolling_n_sided_dice_k_time(n_side: int, k_time: int):
+    """
+   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(2,2)
+    ([2, 3, 4], array([0.25, 0.5 , 0.25]))
+
+    >>> 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]))
+    """
+
+    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
+
+    index = list(range(k_time,k_time*n_side+1))
+    return index, probability_distribution
+
+
+'''
+# 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])
+
+'''