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

DIRECTORY.md

@@ -942,6 +942,7 @@
   * [Sum Of Digits](maths/sum_of_digits.py)
   * [Sum Of Geometric Progression](maths/sum_of_geometric_progression.py)
   * [Sum Of Harmonic Series](maths/sum_of_harmonic_series.py)
+  * [Sum Of Outcomes For Rolling N Sided Dice K Time](maths/sum_of_outcomes_for_rolling_n_sided_dice_k_time.py)
   * [Sumset](maths/sumset.py)
   * [Sylvester Sequence](maths/sylvester_sequence.py)
   * [Tanh](maths/tanh.py)

maths/sum_of_outcomes_for_rolling_n_sided_dice_k_time.py

@@ -0,0 +1,126 @@
+import numpy as np
+
+
+def outcome_of_rolling_n_sided_dice_k_time(n_side: int, k_time: int) -> float:
+    """
+    The sum of outcomes for rolling an N-sided dice K times.
+
+    This function returns a list. The last two elements are the
+    range of probability distribution.
+    The range is: 'k_time' to 'k_time*n_side'
+
+    Other elements contain probabilities for getting a summation
+    from 'k_time' to 'k_time*n_side'.
+
+    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 5 to 30.
+    The sum outcomes become 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.
+
+    While rolling 2 dice simultaneously,
+    the sum becomes 2 for two 1 outcomes. But the sum becomes
+    3 for two different outcome combinations (1,2) and (2,1).
+    The probability of getting 2 is 1/6.
+    The probability of getting 3 is 2/6
+
+    Link to rolling two 6-sided dice combinations:
+    https://www.thoughtco.com/
+    probabilities-of-rolling-two-dice-3126559
+    That phenomenon is the same as the convolution.
+
+    The algorithm can be used in playing games or solving
+    problems where the sum of multiple dice throwing is needed.
+
+
+    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.
+
+    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)
+    array([0.25, 0.5 , 0.25, 2.  , 4.  ])
+    >>> outcome_of_rolling_n_sided_dice_k_time(2,4)
+    array([0.0625, 0.25  , 0.375 , 0.25  , 0.0625, 4.    , 8.    ])
+
+    """
+
+    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")
+
+    prob_dist = 1
+    dist_step = np.ones(n_side, dtype=float) / n_side
+
+    iter1 = 0
+    while iter1 < k_time:
+        prob_dist = np.convolve(prob_dist, dist_step)
+        iter1 = iter1 + 1
+
+    prob_index = np.concatenate((prob_dist, np.array([k_time, k_time * n_side])))
+
+    return prob_index
+
+
+"""
+# Extra code for the verification
+
+dist_index = outcome_of_rolling_n_sided_dice_k_time(6, 3)
+
+the_range = range(int(dist_index[-2]), int(dist_index[-1]+1))
+probabilities =  dist_index[:-2]
+print("Indexes:",the_range)
+
+print("Distribution:",probabilities, "Their summation:",np.sum(probabilities))
+
+import matplotlib.pyplot as plt
+plt.bar(the_range, probabilities)
+plt.xlabel("Summation of Outcomes")
+plt.ylabel("Probabilities")
+
+"""