1. Homework #5 — Monte Carlo Simulation
Professor Hugh C. Lauer CS-1004 — Introduction to Programming for Non-Majors
(Slides include materials from Python Programming: An Introduction to Computer Science, 2nd edition, by John Zelle and copyright notes by Prof. George Heineman of Worcester Polytechnic Institute)
CS-1004, A-Term 2014
Monte Carlo Simulation

2. Monte Carlo Simulation
A program that mimics the behavior of a real-world system
Or a system under design, …
… or a fantasy world!
Values of program variables model states of the system
Actions of program model behavior of the system
E.g., graphic ball of HW#3 models behavior of a bouncing ball in a 2-dimensional, frictionless world with no gravity
CS-1004, A-Term 2014
Monte Carlo Simulation

3. Monte Carlo Simulation
A simulation of a system whose behavior is characterized by probabilities
Examples
Stock market
Weather
Traffic patterns
Migratory bird patterns
Diseases and epidemics …
CS-1004, A-Term 2014
Monte Carlo Simulation

4. Monte Carlo Simulation
Homework #5
Model the game of Craps
Player rolls two (normal) dice
2, 3, 12  player loses
7, 11  player wins
Otherwise roll again
Initial roll value  player wins
7  player loses
Otherwise, repeat!
Questions to answer for this assignment:–
What is the probability that player wins eventually wins?
What percentage of games are decided on 1st roll? 2nd roll? 3rd roll? etc.?
What is average number of rolls per game? …
CS-1004, A-Term 2014
Monte Carlo Simulation

5. How to answer questions like these?
Roll the dice …
… many times …
(i.e., 1000s, 10,000s, or more games!)
… make experimental observations, and
… infer probabilities from them
… i.e., what percentage win, lose, etc.
CS-1004, A-Term 2014
Monte Carlo Simulation

6. How do we roll dice in a computer?
Answer:– Random number generator
Object of class random
Returns a different value each time its method is called
Seemingly random, unpredictable …
… but values exhibit a desired probability distribution
E.g.,
Uniform
Normal
Logarithmic
Exponential …
CS-1004, A-Term 2014
Monte Carlo Simulation

7. Random number generators
Extensive mathematical theory
Tests exist to measure the “quality” of the randomness
Detect bias
Etc.
CS-1004, A-Term 2014
Monte Carlo Simulation

8. Monte Carlo Simulation
Good news and bad news
Good news:–
Seemingly unpredictable sequence of numbers that do not repeat
Great for modeling probabilistic systems
Can run many, many experiments in a short amount of time
Without impacting physical environment!
Bad news:–
Terrible for debugging programs!
Non-reproducible results
CS-1004, A-Term 2014
Monte Carlo Simulation

9. Monte Carlo Simulation
Solution — the “seed”
Random number generators …
… generate sequences of numbers that only look random
Can be restarted to generate same sequence again …
… and again …
…etc.
… without compromising the appearance of randomness
Definition:– “seed”
Value for initializing internal state of random number generator
… so that it can be replayed exactly as before!
CS-1004, A-Term 2014
Monte Carlo Simulation

10. Python random number class
import random
random.random()
generates a new “random” number
to a specified distribution
random.seed(str)
Initializes random number generator
Examples
If no seed, random class initializes from internal variable in OS
Good quality random variable
Very, very difficult to reproduce
Very, very difficult to predict
CS-1004, A-Term 2014
Monte Carlo Simulation

11. Monte Carlo Simulation
Summary
Play many 1000’s or 10,000s of games of Craps
Use random number generator to provide data values that satisfy a probability distribution
Gather statistics
Use information to answer the questions
Justify your answer with words, graphs, and data!
CS-1004, A-Term 2014
Monte Carlo Simulation

12. Monte Carlo Simulation
Questions?
CS-1004, A-Term 2014
Monte Carlo Simulation