Project 5: The Monte Carlo Casino (Probability Simulator)

Build a simulation suite that estimates probabilities through random trials.

Project Overview

Attribute Value
Difficulty Level 1: Beginner
Time Estimate Weekend
Main Language Python
Alternative Languages JavaScript, C++
Knowledge Area Probability
Tools Plotting tool
Main Book “Introduction to Probability” by Blitzstein & Hwang

What you’ll build: A simulator that estimates outcomes for dice, cards, and simple games.

Why it teaches math: Monte Carlo methods show that probability is about long-run frequency, not guesswork.

Core challenges you’ll face:

  • Running large numbers of trials efficiently
  • Tracking outcomes and probabilities
  • Visualizing convergence

Real World Outcome

You will simulate games like rolling two dice and estimate probabilities that match theoretical values.

Example Output:

$ python casino.py --game dice --trials 100000
Estimated P(sum=7): 0.1664
Theoretical P(sum=7): 0.1667

Verification steps:

  • Compare estimates to known probabilities
  • Increase trials to see convergence

The Core Question You’re Answering

“How does randomness become predictable when you repeat it enough times?”

This is the essence of statistical thinking.

Concepts You Must Understand First

Stop and research these before coding:

  1. Law of large numbers
    • Why do averages stabilize with more trials?
    • Book Reference: “Introduction to Probability” by Blitzstein & Hwang, Ch. 5
  2. Expected value
    • What does it mean to expect a value over randomness?
    • Book Reference: “Introduction to Probability” by Blitzstein & Hwang, Ch. 3
  3. Random variables
    • How do you model outcomes as random variables?
    • Book Reference: “Introduction to Probability” by Blitzstein & Hwang, Ch. 2

Questions to Guide Your Design

  1. Simulation accuracy
    • How many trials are needed for stable estimates?
    • How will you measure variance?
  2. Visualization
    • Will you plot convergence over trials?
    • How will you compare simulated vs theoretical values?

Thinking Exercise

Dice Outcomes

Compute the theoretical probability of rolling a sum of 7 with two dice.

Questions while working:

  • How many combinations produce 7?
  • How does symmetry help?

The Interview Questions They’ll Ask

Prepare to answer these:

  1. “What is the law of large numbers?”
  2. “How does Monte Carlo estimation work?”
  3. “What is expected value?”
  4. “How many trials are enough?”
  5. “Why do Monte Carlo methods work for complex problems?”

Hints in Layers

Hint 1: Starting Point Start with a simple dice simulation.

Hint 2: Next Level Track running averages to visualize convergence.

Hint 3: Technical Details Use a fixed random seed for reproducibility.

Hint 4: Tools/Debugging Compare output against known theoretical probabilities.

Books That Will Help

Topic Book Chapter
Large numbers “Introduction to Probability” by Blitzstein & Hwang Ch. 5
Expected value “Introduction to Probability” by Blitzstein & Hwang Ch. 3
Random variables “Introduction to Probability” by Blitzstein & Hwang Ch. 2

Implementation Hints

  • Keep random generation separate from outcome analysis.
  • Store intermediate results for convergence plots.
  • Validate with small cases where theory is easy.

Learning Milestones

  1. First milestone: You can simulate dice and match probabilities.
  2. Second milestone: You can explain convergence behavior.
  3. Final milestone: You can apply Monte Carlo to new games.