Generating Random Numbers with a Fixed Sum

The challenge posed is to generate a series of pseudo-random numbers whose sum equals a predefined value. Specifically, how to generate four numbers that, when added together, equal 40.

Instead of relying on a method that could bias the distribution of the first number, a more uniform approach is employed. The solution utilizes a strategy of dividing the predefined value into smaller segments, using randomly selected dividers.

Assume we have four random positive integers (e, f, g, and h) such that 0

a = e
b = f - e
c = g - f
d = 40 - g

This technique guarantees an equal probability for each set of numbers, ensuring a uniform distribution. The resulting random numbers meet the requirement of summing to the predefined value.

Extending this concept, the following Python function generates a random list of positive integers summing to a specified total:

import random

def constrained_sum_sample_pos(n, total):
    """Return a randomly chosen list of n positive integers summing to total.
    Each such list is equally likely to occur."""

    dividers = sorted(random.sample(range(1, total), n - 1))
    return [a - b for a, b in zip(dividers   [total], [0]   dividers)]

To generate non-negative integers, an additional transformation is employed:

def constrained_sum_sample_nonneg(n, total):
    """Return a randomly chosen list of n nonnegative integers summing to total.
    Each such list is equally likely to occur."""

    return [x - 1 for x in constrained_sum_sample_pos(n, total   n)]