Computer Simulation of Probabilistic Experiments using Python

Introduction

• Lecture on computer simulation of probabilistic experiments
• Difference from regular computations and derivations
• Focus on hands-on Python programming
• No major Python structures; simple functions provided
• Aim: to simulate probabilistic experiments and verify theoretical probabilities using Monte Carlo simulations

Experiments and Simulations

Concept of an Experiment

• Defined as an action leading to some outcome
• Simulate experiments using a computer program
• Utilize Python routines to simulate and verify probabilities
• Example: Simulating coin toss to verify the probability of heads (0.5)

Monte Carlo Simulation

• Standard technique to estimate the probability of events
• Simulate the experiment multiple times to approximate the probability
• Formula: Probability ≈ (Number of successful events) / (Total number of experiments)
• Demonstrated with a coin toss and die throw example

Python Notebooks in Google Colab

• Introduction to Python Notebooks and Google Colab
• Creating and using cells (code or text)
• Importance of Python Notebooks in data science and other fields
• Example notebook provided and explained (probability in Python)

Using Python in Colab

• Importing necessary modules (e.g., numpy)
• Implementing functions to simulate experiments (e.g., uniform distribution for coin toss, die throw)
• Example code snippets for single and multiple tosses of a coin and die

Monte Carlo Techniques in Detail

• Estimating probability using repeated experiments
• Steps: Define repetitions, conduct the experiment, check the event, calculate the probability
• Examples:
  • Coin toss simulation to estimate the probability of heads
  • Die throw simulation to estimate the probability of getting a specific number or a range of numbers

Classic Probability Problems using Python

Birthday Problem

• Scenario: Probability of shared birthdays in a group
• Assumption: Birthdays uniformly distributed across 365 days, independent
• Calculation: Using combinatorial probability and Monte Carlo simulation
• Key Insight: Over 50% chance of shared birthdays with just 23 people
• Python simulation to verify theoretical probability

Monty Hall Problem

• Game show scenario with three doors, one car, and two goats
• Choice dynamics: Initially choose a door, host reveals a goat behind another door, choice to switch or stick
• Insight: Switching doors increases the probability of winning the car to 2/3
• Python simulation to verify theoretical result

Other Experiments

• Polya’s Urn Scheme (briefly mentioned)
• Simple Random Walk / Gambler's Ruin
  • Scenario: Gambler with k units of money, goal to reach n units or go bankrupt
  • Analysis: Using a fair/unfair coin, probability of reaching the goal or going bankrupt
  • Python simulation to verify theoretical probabilities

Activity for Students

• Understanding how to set parameters in Python code
• Recreating Monte Carlo simulations for chosen experiments
• Example: Modify code to simulate different probabilistic scenarios (e.g., binomial distribution)
• Goal: Compare theoretical and simulated probabilities
• Submission: Share the customized notebook on Google Sites

Conclusion

• Recap of Stats One concepts in Week One
• Preparation for new topics from Week Two
• Emphasis on practical skills and theoretical understanding in probabilistic simulations