10 - What Are the Chances?

Summary

The world is infused with chance. Random events have underlying probabilities. A fair flip of a coin has the same chance of landing heads as tails. In this chapter, we use linear algebra to analyze situations that involve randomness.

Down the Chute

Ever play Monopoly for several days? How often does this happen? How short a game is possible? These types of questions can be analyzed using matrices to determine such things as average and shortest game length. Let's try this with a few different games.

First, we will look at Chutes and Ladders, also known as Snakes and Ladders. Chutes and Ladders originated in India as a game of knowledge and was known as Jñána Chaupár. Landing on a virtue resulted in climbing a ladder toward the god Vishnu. Rather than sliding down a chute, the game had a player swallowed by a snake, which resulted in death and a new start. This game entered middle-class Victorian parlors of England in the nineteenth century as Snakes and Ladders. Board game pioneer Milton Bradley introduced the game as Chutes and Ladders in the United States in 1943 promoting it as the “improved new version of snakes and ladders, England's famous indoor sport.”

The game is played with two or more players on a board with numbered squares on a grid as seen in Figure 10.1. Players begin on the starting square and the first to reach the finish square wins. This journey from beginning to end is helped and hindered by ladders and chutes (or snakes) that appear on the board.

Let's see how linear algebra can aid us in analyzing the game. We'll look at a smaller game board to simplify the computations. We'll analyze the game board in Figure 10.2. The game starts on square 1. Reaching square 9 results in winning. On each turn, roll a die. If you roll 1 or 2, do not move and stay on your current square.