1. 1 Monte Carlo methods Mike Sinclair

2. 2 Overview Monte Carlo –Based on roulette wheel probabilities –Used to describe large-scale interactions in biology and physics –Especially useful for solving differential and integral calculus problems –Uses a random number generator and random walks

3. 3 Example problem Attempt to find π Random walk; find ratio of points within the circle vs. entire unit square Time-step Expectation: Enough steps and ratio approaches π

4. 4 Another problem Integrating [sin ( 1 / x )] 2 is extremely difficult, but...... a Monte Carlo approach yields a value ~0.667.

5. 5 Approach Monte Carlo methods use lots and lots of random steps...... and usually compares values “within” a function to the total trials Although simple to design, may take lots and lots of time!

6. 6 Conclusions Monte Carlo methods are used extensively in biology and physics. It uses a combination of random walk and a ratio test. It can be visualized by a modeling large numbers of particles. Slow but very effective for studying complex systems.