Section 1.1 Enumeration
Enumeration is a big fancy word for counting. If you've taken a course in probability and statistics, you've already covered some of the techniques and problems we'll be covering in this course. When a statistician (or other mathematician) is calculating the βprobabilityβ of a particular outcome in circumstances where all outcomes are equally likely, what they usually do is enumerate all possible outcomes, and then figure out how many of these include the outcome they are looking for.Example 1.1.1.
What is the probability of rolling a \(3\) on a \(6\)-sided die?
To figure this out, a mathematician would count the sides of the die (there are six) and count how many of those sides display a three (one of them). They would conclude that the probability of rolling a \(3\) on a \(6\)-sided die is \(1/6\) (one in six).
If you are playing Texas Hold'em poker against a single opponent, and the two cards in your hand are a pair, what is the probability that your opponent has a higher pair?
How many distinct Shidokus (4-by-4 Sudokus) are there?
How many different orders of five items can be made from a bakery that makes three kinds of cookies?
Male honeybees come from a queen bee's unfertilised eggs, so have only one parent (a female). Female honeybees have two parents (one male, one female). Assuming all ancestors were distinct, how many ancestors does a male honeybee have from 10 generations ago?