Probability Part Deux, Monte Carlo Simulations
0Today we did an hour of lab time doing probability problems, and then we ran through the solutions to them.
We then moved on to discuss several important topics like the Central Limit theorem, where it says that if you have certain kinds of well-behaving random processes, like a die roll, then as you move towards rolling an infinite number of them it will turn into a normal/Gaussian/bell curve.
We then spent a bit more time on probability trees and problems that involve something more complicated than just adding a bunch of random numbers together. Drawing cards (with and without replacement) from a deck, rolling multiple dice and keeping the Y greatest ones, that sort of thing.
For simple problems I will tend to solve them directly or numerically, sometimes with computer assistance (like when I need to add up and average 100 numbers) but for complicated problems like, "What is the expected value if you roll 9 exploding d10s and add up the highest 3 of them?" I just sort of give up and write a program. This idea, called a Monte Carlo simulations, is used in physics when you want to simulate the random motion of atoms in a gas, or in finance to do stock price simulations, or in climatology to do climate modelling, and so forth. It's a very powerful notion - basically, if something is too hard to solve directly, simulate a random process and see what happens when you average it out over a large number of trials.