Coupon Collector and Random Graphs

Master coupon collector and random graphs with applications in probability and combinatorics.

25 min read
Advanced

Introduction

Learning Objectives:

  • Solve coupon collector problem
  • Understand random graph models
  • Apply Erdős-Rényi model

Coupon Collector

Collect nn distinct coupons (uniform random). Expected draws to complete set:

E[T]=ncdotHn=n(1+1/2+cdots+1/n)approxnlnnE[T] = n cdot H_n = n(1 + 1/2 + cdots + 1/n) approx n ln n

Erdős-Rényi Graphs

G(n,p)G(n,p): nn vertices, each edge exists independently with probability pp.

Phase transition: Graph becomes connected at p approx rac{ln n}{n}

Applications

Apply these concepts to solve real-world problems in probability and statistics.

python
import numpy as np
import matplotlib.pyplot as plt

# Example implementation
print("Apply concepts from Coupon Collector and Random Graphs")

Key Takeaways

Master these advanced concepts to complete your probability and combinatorics journey!

🎲

Combinatorics and Probability: From Counting to Inference

Master the mathematical foundations of counting, probability, and randomness. This comprehensive course bridges pure combinatorics with computational probability, covering everything from basic counting principles to advanced probabilistic methods, limit theorems, and real-world applications in algorithms, finance, and data science.