A "random" post

Before I get into the meat of this post, I suppose I should explain how exactly the game of Chutes and Ladders is played for anyone who is unfamiliar with it. Players begin by choosing one game piece each and placing it on the first square of the one hundred square game board. When it is a player's turn, he or she spins a spinner containing the numbers one through six and moves his or her piece forward the number of spaces that the spinner lands on. If a player lands on the bottom of a ladder or the top of a chute, he or she will then move his or her piece to the space corresponding with the top of the ladder or the bottom of the chute, respectively. A game is won when a player reaches the hundredth square. If a player spins a number that would require him or her to move beyond the hundredth square (for example, spinning a six while on square 98), the player does not move.

At the heart of making predictions about the outcome of Chutes and Ladders games is the assumption that the process is random. While in theory it may be possible for someone, with enough practice, to be able to control the outcome of a spin, I will be assuming that the outcome of the game does not depend on the experience of the players. Under this assumption, all players have an equal chance of winning when the game begins, and the winner is selected purely by chance.

Understanding how this chance works on a small scale is fairly simple. There are six possible outcomes on the spinner and each one is equally likely, so there is a one-sixth chance that a player will spin a one, a one-sixth chance that a player will spin a two, etc. on each turn, and the chance of a player landing on a certain square in that turn can be calculated accordingly based on the player's position on the board. However, computing chance becomes much more complicated when it is done across multiple turns. For instance, it's not nearly as easy to determine the probability of a player landing, say, exactly twenty squares away in six moves, especially when taking into account the influence of the chutes and ladders. Well, it turns out that the process of progressing along the games board is really a "Markov Chain", and this allows us to pull a few tricks that greatly simplify these sorts of calculations. However, as the margins of this blog are too small to contain such a discussion (ok, I'm really just near the word limit for this entry), I'm going to push that explanation off into the next post. Stay tuned.