snakes_probability, a MATLAB code which computes the game length probabilities for Snakes and Ladders, by Desmond Higham and Nicholas Higham.
Snakes and Ladders is a children's game played on a 10x10 numbered board. A player's turn consists of rolling a single die, and moving the indicated number of squares. If the final square is the foot of a ladder, the player moves up to a higher numbered square. If the final square is the mouth of a snake, the player moves downward.
For the one-player version of the game, it is interesting to pose the question of the probability that a particular game will take a certain number of moves.
By adding a square 0, where the player begins, the game board can be modeled as a vector of length 101, and the transitions from one square to another can be modeled by a transition matrix. Most commonly, the entries in row I will be zero except that columns I+1 through I+6 will have the value 1/6. However, rows which correspond to a snake or ladder, and rows for which I+6 is greater than 100, must be handled specially.
Given the transition matrix A, the one player game can be modeled as a Markov Chain Monte Carlo system. In particular, given an initial starting vector v, the probability distribution after one move is the vector A' * v, and repeated multiplication by A' will display the exact probability distribution at every step.
The computer code and data files made available on this web page are distributed under the GNU LGPL license.
snakes_probability is available in a MATLAB version.
dice_simulation, a MATLAB code which simulates n tosses of m dice, making a histogram of the results.
duel_simulation, a MATLAB code which simulates n repetitions of a duel between two players, each of whom has a known firing accuracy.
fair_dice_simulation, a MATLAB code which simulates n tosses of 2 dice, making a histogram of the results.
gamblers_ruin_simulation, a MATLAB code which simulates the game of gambler's ruin.
high_card_simulation, a MATLAB code which simulates a situation in which you see the cards in a deck one by one, and must select the one you think is the highest and stop.
ising_2d_simulation, a MATLAB code which carries out a monte carlo simulation of an ising model, a 2d array of positive and negative charges, each of which is likely to flip to be in agreement with neighbors.
poisson_simulation, a MATLAB code which simulates a poisson process in which events randomly occur with an average waiting time of lambda.
random_walk_1d_simulation, a MATLAB code which simulates a random walk in a 1-dimensional region.
random_walk_2d_avoid_simulation, a MATLAB code which simulates a self-avoiding random walk in a 2-dimensional region.
random_walk_2d_simulation, a MATLAB code which simulates a random walk in a 2-dimensional region.
random_walk_3d_simulation, a MATLAB code which simulates a random walk in a 3-dimensional region.
reactor_simulation, a MATLAB code which is a simple monte carlo simulation of the shielding effect of a slab of a certain thickness in front of a neutron source. this program was provided as an example with the book "numerical methods and software."
roulette_simulation, a MATLAB code which simulates the spinning of a roulette wheel and the evaluation of certain common roulette bets.
sir_simulation, a MATLAB code which simulates the spread of a disease through a hospital room of m by n beds, using the susceptible/infected/recovered (sir) model.
snakes_and_ladders, a MATLAB code which simulates the game of snakes and ladders.
snakes_bar, a MATLAB code which produces bar charts of the count, pdf and cdf estimates for the length of a one-player game of snakes and ladders, produced by simulating n games.
snakes_histogram, a MATLAB code which produces histograms of the count, pdf and cdf estimates for the length of a one-player game of snakes and ladders, produced by simulating n games.
snakes_matrix, a MATLAB code which computes the transition matrix for snakes and ladders
traffic_simulation, a MATLAB code which simulates the cars waiting to get through a traffic light.
truel_simulation, a MATLAB code which simulates n repetitions of a duel between three players, each of whom has a known firing accuracy.
Desmond Higham, Nicholas Higham