Infinite Dice Game

Imagine playing a dice game where the player who rolls the highest number wins. The game allows using any number of dice fixed at the start of the game. What do you think will be more interesting: using less number of dice or an infinitely large number of dice (assuming you can handle them).

A Single Die

Let’s start with a single six-sided die. Rolling it yields a natural number between 1 and 6 (inclusive). The probability of each number occurring is the same for a fair die: 1/6.

This is not very interesting due to two reasons:

  1. We are familiar with this concept.
  2. Rolling multiple dice is more interesting.

Two Dice

Rolling two dice results in two numbers (which may be the same). We will say the combined result is the sum of the results of both dice.

Now the range becomes 2 to 12. We get a 2 if and only if both the dice roll a 1. But notice that there are more ways to get a 7: 3 + 4, 1 + 6, 5 + 2, etc.

What is the probability of getting a particular number now? Below is a graph of the probability (a picture is worth a thousand numbers):

The graph above is a normal binomial curve maximum at 7.

An Infinite Number of Dice

Now imagine throwing an infinite number of dice at once. What is the range of possible outcomes? What is the probability of getting a number in the range?

The answers are neither surprising nor hard to guess especiallye you have been introduced to the concept of multiple dice. I will leave this up to you. (Hint: Think about throwing a large finite number of dice first).