Project 7: Probability Theory Engine

Build a probability engine that pairs analytic calculations with large-scale simulation.

Quick Reference

Attribute Value
Difficulty Level 2: Intermediate
Time Estimate 1 week
Main Programming Language Python
Alternative Programming Languages R, Julia
Coolness Level Level 4: Hardcore Tech Flex
Business Potential 1. The “Resume Gold”
Prerequisites Algebra, basic coding, Project 6
Key Topics Sample spaces, Bayes, random variables, LLN/CLT, covariance

1. Learning Objectives

  1. Compute conditional and posterior probabilities in realistic scenarios.
  2. Validate theoretical values via Monte Carlo.
  3. Analyze joint behavior using covariance/correlation.
  4. Explain where independence assumptions fail.

2. All Theory Needed (Per-Concept Breakdown)

2.1 Conditional Probability and Bayes

  • Fundamentals: Conditioning updates probabilities when evidence arrives.
  • Deep Dive into the concept: Posterior odds = prior odds × likelihood ratio. Low base rates can dominate seemingly strong tests.
  • Minimal concrete example: 
    Posterior = (Sensitivity * Prevalence) /
          ((Sensitivity * Prevalence) + (1-Specificity)*(1-Prevalence))

2.2 Random Variables and Distribution Behavior

  • Fundamentals: PMF/PDF/CDF describe uncertainty structure.
  • Deep Dive into the concept: Expectation summarizes long-run center; variance quantifies spread; LLN and CLT explain stabilization and approximation.

3. Project Specification

3.1 What You Will Build

A scenario-driven engine (screening tests, reliability, conversion funnels) with exact and simulated probability outputs.

3.2 Functional Requirements

  1. Exact calculators for core conditional/Bayes scenarios.
  2. Monte Carlo simulator with configurable trial counts.
  3. Joint distribution and covariance analysis module.
  4. Markdown report with analytic vs simulation deltas.

3.3 Non-Functional Requirements

  • Deterministic simulation seeds.
  • Runtime controls for trial budget.

3.4 Example Usage / Output

$ python probability_engine.py --scenario screening_test
Posterior P(disease | positive): 0.157
Monte Carlo estimate: 0.158
Absolute error: 0.001
Saved: outputs/probability_engine/screening_test_report.md

3.5 Real World Outcome

You gain a reusable tool for sanity-checking probability intuition in product and risk contexts.

4. Solution Architecture

Scenario config -> exact solver + simulator -> comparator -> report

5. Implementation Guide

5.1 Development Environment Setup

pip install numpy scipy

5.2 Project Structure

P07/
  probability_engine.py
  scenarios/
  outputs/

5.3 The Core Question You Are Answering

“Can I reason correctly about uncertainty when priors and evidence conflict?”

5.4 Concepts You Must Understand First

  1. Conditional probability
  2. Bayes theorem
  3. PMF/PDF/CDF
  4. LLN/CLT

5.5 Questions to Guide Your Design

  1. Which scenarios are best for exact computation?
  2. How many trials are enough for stable simulation estimates?
  3. How will you expose uncertainty around simulation outputs?

5.6 Thinking Exercise

Estimate a posterior manually before running the simulator; compare and explain difference.

5.7 The Interview Questions They’ll Ask

  1. Why can positive test results still imply low true probability?
  2. How does LLN differ from CLT?
  3. When does covariance miss dependence?
  4. Why use simulation at all?
  5. What is base-rate neglect?

5.8 Hints in Layers

  • Hint 1: Build one scenario end-to-end first.
  • Hint 2: Add simulation parity checks.
  • Hint 3: Expand to joint distributions.
  • Hint 4: Add report-level assumption logs.

5.9 Books That Will Help

Topic Book Chapter
Bayes intuition Think Bayes Ch. 1-3
Probability foundations Blitzstein & Hwang Ch. 1-5
Simulation literacy Think Stats simulation sections

6. Testing Strategy

  • Unit tests for known textbook examples.
  • Convergence checks as trial counts increase.
  • Distribution sanity checks (sum to 1, bounds).

7. Common Pitfalls & Debugging

Pitfall Symptom Solution
Misdefined events impossible probabilities explicit event diagrams
No seed control non-reproducible outputs fixed seeds
Too few simulations noisy conclusions adaptive replicate thresholds

8. Extensions & Challenges

  • Add rare-event variance-reduction strategies.
  • Add Bayesian updating dashboard.

9. Real-World Connections

  • Fraud detection triage.
  • Reliability and warranty modeling.

10. Resources

  • Blitzstein & Hwang, “Introduction to Probability”
  • Downey, “Think Bayes”

11. Self-Assessment Checklist

  • I can compute and explain posterior updates.
  • I can validate theory with simulation.
  • I can explain one independence failure case.

12. Submission / Completion Criteria

Minimum: three scenarios with exact/simulation agreement.

Full: includes dependency diagnostics and uncertainty commentary.