Lesson 7. Does God Play the Dice?

Objective:
Coin tossing and dice rolling are the familiar means to simulate events occurring randomly. They are however not a practical means to generate a long sequence of random events. Hence, a pseudo-random number generator is used to quickly simulate several thousands of coin tossing (Prog#7a), dice rolling (Prog#7b - 7d), and random tree branching (Prog#7e - 7g). For one die roll, the chances for facing up any side are the same and equal to one out of six, so that they have a uniform distribution

But, the sums of faced-up pips of two dice favor the median value 7, and obey a triangle distribution.

Moreover, the distribution of sums of three dice gets rounded in the middle and falls off at the edges.

As more and more dice are rolled, the distribution of their sums becomes a bell-shaped curve. This is the Central Limit Theorem. The branch scaling factor and spreading angle controls the growth of binary tree introduced in Lesson 1. When we assign only the scaling factor randomly the binary tree grows into all different sizes. But, when both the scaling factor and spreading angle are chosen randomly, there appear more middle-sized binary trees and less of the extreme small and large ones, as suggested by the Central Limit Theorem.

Scary words:
Randomizer, pseudo-random number generator, random walk, event, sample space, probability distribution, uniform distribution, bell-shaped (normal) distribution, random variable, Central Limit Theorem.

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