An Introduction to Probability and Simulation
Why study probability?
- Probability is the study of uncertainty, and life is uncertain
- Probability is used in a wide variety of fields, including: statistics, physics, engineering, biology, medicine, finance, actuarial science, political science, law, sports , …
- Many topics and problems in probability are frequently misunderstood and sometimes counter intuitive, so it’s worthwhile to take a careful study
- “Probabilistic thinking” is an important component of statistical literacy (e.g. how to assess risk when making decisions)
- Probability provides the foundation for many important statistical concepts and methods such as p-values and confidence intervals
Why use simulation to study probability?
- Many concepts encountered in probability can seem esoteric; simulation helps make them more concrete.
- Simulation provides an effective tool for analyzing probability models and for exploring effects of changing assumptions
- Simulation can be used to check analytical solutions
- Simulation is often the best or only method for investigating many problems which are too complex to solve analytically
- Simulation-based reasoning is an important component of statistical literacy (e.g. understanding a p-value via simulation)
- Many statistical procedures employ simulation-based methods (e.g. bootstrapping)
The examples in this book are used to both motivate new topics and to help you practice your understanding of the material. You should attempt the examples on your own before reading the solutions. To encourage you to do so, the solutions have been hidden. You can reveal the solution by clicking on the Show/hide solution button.
Here is where a solution would be, but be sure to think about the problem on your own first!
(Careful: in your browser, the triangle for the Show/hide solution button might be close to the back button, so clicking on Show/hide might take you to the previous page. To avoid this, click on the words Show/hide.)
Some of the examples in this book feature a character named Donny Don’t. The moral of these examples is “Don’t do what Donny Don’t does”. This is a Simpson’s reference. Donny represents a student who makes many of the mistakes commonly made by students studying probability. The idea of these problems is for you to learn from the common mistakes that Donny makes, by identifying why he is wrong and by helping him understand and correct his mistakes. (But be careful: sometimes Donny is right!)
This book uses the Python package Symbulate which provides a user friendly framework for conducting simulations involving probability models. The syntax of Symbulate reflects the “language of probability” and makes it intuitive to specify, run, analyze, and visualize the results of a simulation. In Symbulate, probability spaces, events, random variables, and random processes are symbolic objects which can be manipulated, independently of their simulated realizations. Symbulate’s consistency with the mathematics of probability reinforces understanding of probabilistic concepts. The article Ross and Sun (2019) discusses Symbulate and its features in more detail.
The best way to interact with Symbulate is through Google Colab or Jupyter notebooks. A notebook is organized by cells which contain text or code that can be run interactively with output displayed after the cell.
Symbulate can be run online in a Colab notebook by including the following line in the first cell.
!pip install symbulate
To install Symbulate on your own computer, it is recommended that you first install the Anaconda distribution, which is a Python environment with many scientific packages installed (including all of the packages that Symbulate is built on). After installing Anaconda, you can install Symbulate using the
pip command above.
You should always include the following command once in each notebook to import Symbulate during a Python session.
from symbulate import *
You should also always include the following command once in each notebook to enable graphical displays.
The Symbulate command
plot() produces graphics. These graphics can be customized (by changing axis limits, adding titles, legends, etc) using Matplotlib, and in particular the
pyplot method, which can be imported by including the lines
import matplotlib import matplotlib.pyplot as plt
Colab or Jupyter notebooks provide a natural interface for Symbulate. The code in this book matches as closely as possible the commands that would be entered into cells in a notebook. However, certain commands that appear throughout the book are needed only to properly produce the output in this book, and not if working directly in notebooks. In particular, the command
plt.show() is not needed to display graphics in a notebook (as long as you have included the line
%matplotlib inline). In addition, notebooks automatically displays the result of the last line in a cell;
print() is generally not needed to display output (unless you wish to format it, or it is not the last line in the cell).
For example, a code snippet that appears in this book as
= RV(Binomial(4, 0.5)).sim(10000) x x.plot() plt.show() print(x)
would be entered in a Jupyter notebook as
= RV(Binomial(4, 0.5)).sim(10000) x x.plot() x