Random Walks
From drunkard's walk to stock market fluctuations
🚶 1D Random Walk
The Setup
Imagine a person starting at position \(x=0\). At each step, they flip a coin:
- Heads (prob \(p\)): Move Right (+1)
- Tails (prob \(1-p\)): Move Left (-1)
This simple process models Brownian motion, gambling fortunes, and more!
Left Biased
Unbiased
Right Biased
0
Position: 0
🧩 The Meeting Problem (2D)
Alice starts at top-left (A). Bob starts at bottom-right (B). Alice moves Right/Down. Bob moves Left/Up. What is the probability they meet?
Analysis
- They can only meet on the diagonal!
- They meet after exactly 5 steps each (on a 6x6 grid).
- Total paths for each = \(2^5 = 32\).
- Total combined paths = \(32 \times 32 = 1024\).
📝 Test Your Understanding
Question 1:
In a simple symmetric 1D random walk (p=0.5), what is the expected position after \(n\) steps?